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Lagrangian mechanics

From Wikipedia, the free encyclopedia
Formulation of classical mechanics

Joseph-Louis Lagrange (1736–1813)
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inphysics,Lagrangian mechanics is an alternate formulation ofclassical mechanics founded on thed'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomerJoseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760[1] culminating in his 1788 grand opus,Mécanique analytique.[2] Lagrange's approach greatly simplifies the analysis of many problems in mechanics, and it had crucial influence on other branches of physics, including relativity and quantum field theory.

Lagrangian mechanics describes a mechanical system as a pair(M,L) consisting of aconfiguration spaceM and a smooth functionL{\textstyle L} within that space called aLagrangian. For many systems,L =T −V, whereT andV are thekinetic andpotential energy of the system, respectively.[3]

The stationary action principle requires that theaction functional of the system derived fromL must remain at astationary point (specifically, amaximum,minimum, orsaddle point) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.[4]

Introduction

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Bead constrained to move on a frictionless wire. The wire exerts a reaction forceC on the bead to keep it on the wire. The non-constraint forceN in this case is gravity. Notice the initial position of the bead on the wire can lead to different motions.
Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equationf(x,y) = 0, the constraint forceC is the tension in the rod. Again the non-constraint forceN in this case is gravity.

Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems.[5] This method works well for many problems, but for others the approach isnightmarishly complicated.[6] For example, in calculation of the motion of atorus rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like theangular velocity of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations.[7] Lagrangian mechanics adopts energy rather than force as its basic ingredient,[5] leading to more abstract equations capable of tackling more complex problems.[6]

Particularly, Lagrange's approach was to set up independentgeneralized coordinates for the position and speed of every object, which allows the writing down of a general form of Lagrangian (total kinetic energy minus potential energy of the system) and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a generalized set of equations. This summed quantity is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalizedsystem of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.[7]

For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as apoint particle. For a system ofN point particles withmassesm1,m2, ...,mN, each particle has aposition vector, denotedr1,r2, ...,rN.Cartesian coordinates are often sufficient, sor1 = (x1,y1,z1),r2 = (x2,y2,z2) and so on. Inthree-dimensional space, each position vector requires threecoordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is writtenr = (x,y,z). Thevelocity of each particle is how fast the particle moves along its path of motion, and is thetime derivative of its position, thusv1=dr1dt,v2=dr2dt,…,vN=drNdt.{\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.}In Newtonian mechanics, theequations of motion are given byNewton's laws. The second law "netforce equals mass timesacceleration",∑F=md2rdt2,{\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},}applies to each particle. For anN-particle system in 3 dimensions, there are 3N second-orderordinary differential equations in the positions of the particles to solve for.

Lagrangian

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Instead of forces, Lagrangian mechanics uses theenergies in the system. The central quantity of Lagrangian mechanics is theLagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. Thenon-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is given by[8]L=T−V,{\displaystyle L=T-V,}whereT=12∑k=1Nmkvk2{\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}is the totalkinetic energy of the system, equaling thesum Σ of the kinetic energies of theN{\displaystyle N} particles. Each particle labeledk{\displaystyle k} has massmk,{\displaystyle m_{k},} andvk2 =vk ·vk is the magnitude squared of its velocity, equivalent to thedot product of the velocity with itself.[9]

Kinetic energyT is the energy of the system's motion and is a function only of the velocitiesvk, not the positionsrk, nor timet, soT =T(v1,v2, ...).

V, thepotential energy of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any external influences. Forconservative forces (e.g.Newtonian gravity), it is a function of the position vectors of the particles only, soV =V(r1,r2, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g.electromagnetic potential), the velocities will appear also,V =V(r1,r2, ...,v1,v2, ...). If there is some external field or external driving force changing with time, the potential changes with time, so most generallyV =V(r1,r2, ...,v1,v2, ...,t).

Mathematical formulation (for finite particle systems)

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An equivalent but more mathematically formal definition of the Lagrangian is as follows.[10] For a system ofN particles in three-dimensional space, theconfiguration space of the system is a smooth manifoldQ⊆R3N{\displaystyle Q\subseteq \mathbb {R} ^{3N}}, where each configurationq∈Q{\displaystyle q\in Q} specifies the spatial positions of each of the particles at a given instant of time, and the manifold is composed of all the configurations that are allowed by the constraints on the system.

The Lagrangian is a smooth function:L:TQ×R→R,{\displaystyle L:TQ\times \mathbb {R} \to \mathbb {R} ,}whereTQ⊆R6N{\displaystyle TQ\subseteq \mathbb {R} ^{6N}} is thetangent bundle of the configuration space. That is, each element inTQ{\displaystyle TQ} represents both the positions and the velocities of the particles, and can be written as a tuple(q1,…,qN,q˙1,…q˙N){\displaystyle (q_{1},\ldots ,q_{N},{\dot {q}}_{1},\ldots {\dot {q}}_{N})} withqi{\displaystyle q_{i}} andq˙i{\displaystyle {\dot {q}}_{i}} specifying a position and a velocity of the i'th particle respectively. The time dependencet∈R{\displaystyle t\in \mathbb {R} } allows for the Lagrangian to describe time-dependent forces or potentials.

Atrajectory of the system is a smooth functionq:[t0,t1]→Q,{\displaystyle q:[t_{0},t_{1}]\to Q,}describing the evolution of the configuration over time. Its velocity is the time derivativeq˙(t)∈Tq(t)Q{\displaystyle {\dot {q}}(t)\in T_{q(t)}Q} ofq(t){\displaystyle q(t)}, and the pair(q(t),q˙(t))∈TQ{\displaystyle (q(t),{\dot {q}}(t))\in TQ} is thus an element of the bundle for anyt{\displaystyle t}. Theaction functional of the trajectory can therefore be defined as the integral of the Lagrangian along the path:S[q]=∫t0t1L(q(t),q˙(t),t)dt.{\displaystyle S[q]=\int _{t_{0}}^{t_{1}}L(q(t),{\dot {q}}(t),t)\,dt.}

The laws of motion in Lagrangian mechanics are derived from the postulate that among all trajectories between two given configurations, the actual one that will be taken by the system must be a critical point (often but not necessarily a local minimum) of the action functional. This leads to theEuler–Lagrange equations (see also below).

Equations of motion

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For a system of particles with massesm1,…,mN{\displaystyle m_{1},\ldots ,m_{N}}, thekinetic energy is:T(q,q˙)=12∑i=1Nmi‖q˙i‖2,{\displaystyle T(q,{\dot {q}})={\frac {1}{2}}\sum _{i=1}^{N}m_{i}\|{\dot {q}}_{i}\|^{2},}whereq˙i∈R3{\displaystyle {\dot {q}}_{i}\in \mathbb {R} ^{3}} is the velocity of particlei.

Thepotential energyV(q,t){\displaystyle V(q,t)} depends only on the configuration (and possibly on time), and typically arises from conservative forces.

The standardLagrangian is given by the difference:L(q,q˙,t)=T(q,q˙)−V(q,t).{\displaystyle L(q,{\dot {q}},t)=T(q,{\dot {q}})-V(q,t).}

This formulation covers both conservative and time-dependent systems and forms the basis for generalizations to continuous systems (fields), constrained systems, and systems with curved configuration spaces.

IfT orV or both depend explicitly on time due to time-varying constraints or external influences, the LagrangianL(r1,r2, ...v1,v2, ...t) isexplicitly time-dependent. If neither the potential nor the kinetic energy depend on time, then the LagrangianL(r1,r2, ...v1,v2, ...) isexplicitly independent of time. In either case, the Lagrangian always has implicit time dependence through the generalized coordinates.

With these definitions,Lagrange's equations are[11]

Lagrange's equations

∂L∂rk−ddt∂L∂r˙k+∑i=1Cλi∂fi∂rk=0,{\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}

wherek = 1, 2, ...,N labels the particles, there is aLagrange multiplierλi for each constraint equationfi, and∂∂rk≡(∂∂xk,∂∂yk,∂∂zk),∂∂r˙k≡(∂∂x˙k,∂∂y˙k,∂∂z˙k){\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)}are each shorthands for a vector ofpartial derivatives∂/∂ with respect to the indicated variables (not a derivative with respect to the entire vector).[nb 1] Each overdot is a shorthand for atime derivative. This procedure does increase the number of equations to solve compared to Newton's laws, from 3N to3N +C, because there are 3N coupled second-order differential equations in the position coordinates and multipliers, plusC constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.

In the Lagrangian, the position coordinates and velocity components are allindependent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usualdifferentiation rules (e.g. the partial derivative ofL with respect to thez velocity component of particle 2, defined byvz,2 =dz2/dt, is just∂L/∂vz,2; no awkwardchain rules or total derivatives need to be used to relate the velocity component to the corresponding coordinatez2).

In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number ofindependent coordinates is thereforen = 3N −C. We can transform each position vector to a common set ofngeneralized coordinates, conveniently written as ann-tupleq = (q1,q2, ...qn), by expressing each position vector, and hence the position coordinates, asfunctions of the generalized coordinates and time:rk=rk(q,t)=(xk(q,t),yk(q,t),zk(q,t),t).{\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.}

The vectorq is a point in theconfiguration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, thetotal derivative of its position with respect to time, isq˙j=dqjdt,vk=∑j=1n∂rk∂qjq˙j+∂rk∂t.{\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.}

Given thisvk, the kinetic energyin generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, soT=T(q,q˙,t).{\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).}

With these definitions, theEuler–Lagrange equations,[12][13]

Lagrange's equations

ddt(∂L∂q˙j)=∂L∂qj{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}}

are mathematical results from thecalculus of variations, which can also be used in mechanics. Substituting in the LagrangianL(q, dq/dt,t) gives theequations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3N ton = 3N −C coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.

Although the equations of motion includepartial derivatives, the results of the partial derivatives are stillordinary differential equations in the position coordinates of the particles. Thetotal time derivative denoted d/dt often involvesimplicit differentiation. Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates.

Extensions

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As already noted, this form ofL is applicable to many important classes of system, but not everywhere. Forrelativistic Lagrangian mechanics it must be replaced as a whole by a function consistent with special relativity (scalar under Lorentz transformations) or general relativity (4-scalar).[14] Where a magnetic field is present, the expression for the potential energy needs restating.[citation needed] And for dissipative forces (e.g.,friction), another function must be introduced alongside Lagrangian often referred to as a "Rayleigh dissipation function" to account for the loss of energy.[15]

One or more of the particles may each be subject to one or moreholonomic constraints; such a constraint is described by an equation of the formf(r,t) = 0. If the number of constraints in the system isC, then each constraint has an equationf1(r,t) = 0,f2(r,t) = 0, ...,fC(r,t) = 0, each of which could apply to any of the particles. If particlek is subject to constrainti, thenfi(rk,t) = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time.Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanicscan only be applied to systems whose constraints, if any, are all holonomic. Three examples of nonholonomic constraints are:[16] when the constraint equations are non-integrable, when the constraints have inequalities, or when the constraints involve complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert toNewtonian mechanics or use other methods.[17]

From Newtonian to Lagrangian mechanics

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Newton's laws

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Isaac Newton (1642–1727)

For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system ofN particles, all of these equations apply to each particle in the system). Theequation of motion for a particle of constant massm isNewton's second law of 1687, in modern vector notationF=ma,{\displaystyle \mathbf {F} =m\mathbf {a} ,}wherea is its acceleration andF the resultant force actingon it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-orderordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vectorr of the particle at timet, subject to theinitial conditions ofr andv whent = 0.

Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set ofcurvilinear coordinatesξ = (ξ1,ξ2,ξ3), the law intensor index notation is the"Lagrangian form"[18][19]Fa=m(d2ξadt2+Γabcdξbdtdξcdt)=gak(ddt∂T∂ξ˙k−∂T∂ξk),ξ˙a≡dξadt,{\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},}whereFa is thea-thcontravariant component of the resultant force acting on the particle, Γabc are theChristoffel symbols of the second kind,T=12mgbcdξbdtdξcdt{\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}is the kinetic energy of the particle, andgbc thecovariant components of themetric tensor of the curvilinear coordinate system. All the indicesa,b,c, each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates.

It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle,F =0, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation aregeodesics, the curves of extremal length between two points in space (these may end up being minimal, that is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forcesF ≠0, the particle accelerates due to forces acting on it and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4Dcurved spacetime, the above form of Newton's law also carries over toEinstein'sgeneral relativity, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense.[20]

However, we still need to know the total resultant forceF acting on the particle, which in turn requires the resultant non-constraint forceN plus the resultant constraint forceC,F=C+N.{\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .}

The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations.

The constraint forces can either be eliminated from the equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.

D'Alembert's principle

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Jean d'Alembert (1717–1783)
One degree of freedom.
Two degrees of freedom.
Constraint forceC and virtual displacementδr for a particle of massm confined to a curve. The resultant non-constraint force isN.

A fundamental result inanalytical mechanics isD'Alembert's principle, introduced in 1708 byJacques Bernoulli to understandstatic equilibrium, and developed byD'Alembert in 1743 to solve dynamical problems.[21] The principle asserts forN particles the virtual work, i.e. the work along a virtual displacement,δrk, is zero:[9]∑k=1N(Nk+Ck−mkak)⋅δrk=0.{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}

Thevirtual displacements,δrk, are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the systemat an instant of time,[22] i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.[nb 2]Virtual work is the work done along a virtual displacement for any force (constraint or non-constraint).

Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:[23][nb 3]∑k=1NCk⋅δrk=0,{\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,}so that∑k=1N(Nk−mkak)⋅δrk=0.{\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.}

Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion.[23][24] The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacementsδrk might be connected by a constraint equation, which prevents us from setting theN individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.

Equations of motion from D'Alembert's principle

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If there are constraints on particlek, then since the coordinates of the positionrk = (xk,yk,zk) are linked together by a constraint equation, so are those of thevirtual displacementsδrk = (δxk,δyk,δzk). Since the generalized coordinates are independent, we can avoid the complications with theδrk by converting to virtual displacements in the generalized coordinates. These are related in the same form as atotal differential,[9]δrk=∑j=1n∂rk∂qjδqj.{\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.}

There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in aninstant of time.

The first term in D'Alembert's principle above is the virtual work done by the non-constraint forcesNk along the virtual displacementsδrk, and can without loss of generality be converted into the generalized analogues by the definition ofgeneralized forcesQj=∑k=1NNk⋅∂rk∂qj,{\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},}so that∑k=1NNk⋅δrk=∑k=1NNk⋅∑j=1n∂rk∂qjδqj=∑j=1nQjδqj.{\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.}

This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result:[9]∑k=1Nmkak⋅∂rk∂qj=ddt∂T∂q˙j−∂T∂qj.{\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.}

Now D'Alembert's principle is in the generalized coordinates as required,∑j=1n[Qj−(ddt∂T∂q˙j−∂T∂qj)]δqj=0,{\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0,}and since these virtual displacementsδqj are independent and nonzero, the coefficients can be equated to zero, resulting inLagrange's equations[25][26] or thegeneralized equations of motion,[27]Qj=ddt∂T∂q˙j−∂T∂qj{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}}

These equations are equivalent to Newton's lawsfor the non-constraint forces. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.[28]

Euler–Lagrange equations and Hamilton's principle

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As the system evolves,q traces a path throughconfiguration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).[29]

For a non-conservative force which depends on velocity, itmay be possible to find a potential energy functionV that depends on positions and velocities. If the generalized forcesQi can be derived from a potentialV such that[30][31]Qj=ddt∂V∂q˙j−∂V∂qj,{\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},}equating to Lagrange's equations and defining the Lagrangian asL =T −V obtainsLagrange's equations of the second kind or theEuler–Lagrange equations of motion∂L∂qj−ddt∂L∂q˙j=0.{\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.}

However, the Euler–Lagrange equations can only account for non-conservative forcesif a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.

The Euler–Lagrange equations also follow from thecalculus of variations. Thevariation of the Lagrangian isδL=∑j=1n(∂L∂qjδqj+∂L∂q˙jδq˙j),δq˙j≡δdqjdt≡d(δqj)dt,{\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},}which has a form similar to thetotal differential ofL, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. Anintegration by parts with respect to time can transfer the time derivative ofδqj to the ∂L/∂(dqj/dt), in the process exchanging d(δqj)/dt forδqj, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,∫t1t2δLdt=∫t1t2∑j=1n(∂L∂qjδqj+ddt(∂L∂q˙jδqj)−ddt∂L∂q˙jδqj)dt=∑j=1n[∂L∂q˙jδqj]t1t2+∫t1t2∑j=1n(∂L∂qj−ddt∂L∂q˙j)δqjdt.{\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\end{aligned}}}

Now, if the conditionδqj(t1) =δqj(t2) = 0 holds for allj, the terms not integrated are zero. If in addition the entire time integral ofδL is zero, then because theδqj are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients ofδqj must also be zero. Then we obtain the equations of motion. This can be summarized byHamilton's principle:∫t1t2δLdt=0.{\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.}

The time integral of the Lagrangian is another quantity called theaction, defined as[32]S=∫t1t2Ldt,{\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,}which is afunctional; it takes in the Lagrangian function for all times betweent1 andt2 and returns a scalar value. Its dimensions are the same as [angular momentum], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle isδS=0.{\displaystyle \delta S=0.}

Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of severalaction principles.[33]

Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of thecalculus of variations to mechanical problems, such as theBrachistochrone problem solved byJean Bernoulli in 1696, as well asLeibniz,Daniel Bernoulli,L'Hôpital around the same time, andNewton the following year.[34] Newton himself was thinking along the lines of the variational calculus, but did not publish.[34] These ideas in turn lead to thevariational principles of mechanics, ofFermat,Maupertuis,Euler,Hamilton, and others.

Hamilton's principle can be applied tononholonomic constraints if the constraint equations can be put into a certain form, alinear combination of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation.[35] This will not be given here.

Lagrange multipliers and constraints

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The LagrangianL can be varied in the Cartesianrk coordinates, forN particles,∫t1t2∑k=1N(∂L∂rk−ddt∂L∂r˙k)⋅δrkdt=0.{\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}

Hamilton's principle is still valid even if the coordinatesL is expressed in are not independent, hererk, but the constraints are still assumed to be holonomic.[36] As always the end points are fixedδrk(t1) =δrk(t2) =0 for allk. What cannot be done is to simply equate the coefficients ofδrk to zero because theδrk are not independent. Instead, the method ofLagrange multipliers can be used to include the constraints. Multiplying each constraint equationfi(rk,t) = 0 by a Lagrange multiplierλi fori = 1, 2, ...,C, and adding the results to the original Lagrangian, gives the new LagrangianL′=L(r1,r2,…,r˙1,r˙2,…,t)+∑i=1Cλi(t)fi(rk,t).{\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).}

The Lagrange multipliers are arbitrary functions of timet, but not functions of the coordinatesrk, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives∫t1t2δL′dt=∫t1t2∑k=1N(∂L∂rk−ddt∂L∂r˙k+∑i=1Cλi∂fi∂rk)⋅δrkdt=0.{\displaystyle \int _{t_{1}}^{t_{2}}\delta L'\mathrm {d} t=\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.}

The introduced multipliers can be found so that the coefficients ofδrk are zero, even though therk are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement∂L′∂rk−ddt∂L′∂r˙k=0⇒∂L∂rk−ddt∂L∂r˙k+∑i=1Cλi∂fi∂rk=0,{\displaystyle {\frac {\partial L'}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\mathbf {r} }}_{k}}}=0\quad \Rightarrow \quad {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}which areLagrange's equations of the first kind. Also, theλi Euler-Lagrange equations for the new Lagrangian return the constraint equations∂L′∂λi−ddt∂L′∂λ˙i=0⇒fi(rk,t)=0.{\displaystyle {\frac {\partial L'}{\partial \lambda _{i}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {\lambda }}_{i}}}=0\quad \Rightarrow \quad f_{i}(\mathbf {r} _{k},t)=0.}

For the case of a conservative force given by the gradient of some potential energyV, a function of therk coordinates only, substituting the LagrangianL =T −V gives∂T∂rk−ddt∂T∂r˙k⏟−Fk+−∂V∂rk⏟Nk+∑i=1Cλi∂fi∂rk=0,{\displaystyle \underbrace {{\frac {\partial T}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\mathbf {r} }}_{k}}}} _{-\mathbf {F} _{k}}+\underbrace {-{\frac {\partial V}{\partial \mathbf {r} _{k}}}} _{\mathbf {N} _{k}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,}and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces areCk=∑i=1Cλi∂fi∂rk,{\displaystyle \mathbf {C} _{k}=\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}},}thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.

Properties of the Lagrangian

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Non-uniqueness

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The Lagrangian of a given system is not unique. A LagrangianL can be multiplied by a nonzero constanta and shifted by an arbitrary constantb, and the new LagrangianL′ =aL +b will describe the same motion asL. If one restricts as above to trajectoriesq over a given time interval[tst,tfin] and fixed end pointsPst =q(tst) andPfin =q(tfin), then two Lagrangians describing the same system can differ by the "total time derivative" of a functionf(q,t):[37]L′(q,q˙,t)=L(q,q˙,t)+df(q,t)dt,{\displaystyle L'(\mathbf {q} ,{\dot {\mathbf {q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t)+{\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}},}wheredf(q,t)dt{\textstyle {\frac {\mathrm {d} f(\mathbf {q} ,t)}{\mathrm {d} t}}} means∂f(q,t)∂t+∑i∂f(q,t)∂qiq˙i.{\textstyle {\frac {\partial f(\mathbf {q} ,t)}{\partial t}}+\sum _{i}{\frac {\partial f(\mathbf {q} ,t)}{\partial q_{i}}}{\dot {q}}_{i}.}

Both LagrangiansL andL′ produce the same equations of motion[38][39] since the corresponding actionsS andS′ are related viaS′[q]=∫tsttfinL′(q(t),q˙(t),t)dt=∫tsttfinL(q(t),q˙(t),t)dt+∫tsttfindf(q(t),t)dtdt=S[q]+f(Pfin,tfin)−f(Pst,tst),{\displaystyle {\begin{aligned}S'[\mathbf {q} ]&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L'(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt\\&=\int _{t_{\text{st}}}^{t_{\text{fin}}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt+\int _{t_{\text{st}}}^{t_{\text{fin}}}{\frac {\mathrm {d} f(\mathbf {q} (t),t)}{\mathrm {d} t}}\,dt\\&=S[\mathbf {q} ]+f(P_{\text{fin}},t_{\text{fin}})-f(P_{\text{st}},t_{\text{st}}),\end{aligned}}}with the last two componentsf(Pfin,tfin) andf(Pst,tst) independent ofq.

Invariance under point transformations

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Given a set of generalized coordinatesq, if we change these variables to a new set of generalized coordinatesQ according to apoint transformationQ =Q(q,t) which is invertible asq =q(Q,t), the new LagrangianL′ is a function of the new coordinates and similarly for the constraintsL′(Q,Q˙,t)=L(q(Q,t),q˙(Q,Q˙,t),t),ϕj′(Q,t)=ϕj(q(Q,t),t){\displaystyle {\begin{aligned}L'(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&=L(\mathbf {q} (\mathbf {Q} ,t),{\dot {\mathbf {q} }}(\mathbf {Q} ,{\dot {\mathbf {Q} }},t),t),\\\phi _{j}'(\mathbf {Q} ,t)&=\phi _{j}(\mathbf {q} (\mathbf {Q} ,t),t)\end{aligned}}}and by thechain rule for partial differentiation, Lagrange's equations are invariant under this transformation;[40][citation needed]

ddt∂L′∂Q˙i=∂L′∂Qi+∑jλj∂ϕj′∂Qi.{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L'}{\partial {\dot {Q}}_{i}}}={\frac {\partial L'}{\partial Q_{i}}}+\sum _{j}\lambda _{j}{\frac {\partial \phi '_{j}}{\partial Q_{i}}}.}

Proof

For a coordinate transformationQi=Qi(q,t){\displaystyle Q_{i}=Q_{i}(\mathbf {q} ,t)}, we havedQi=∑k∂Qi∂qkdqk+∂Qk∂tdt,{\displaystyle d{Q_{i}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}d{q_{k}}+{\frac {\partial Q_{k}}{\partial t}}dt,}which implies thatQi˙=∑k∂Qi∂qk(q,t)q˙k+∂Qk∂t(q,t){\displaystyle {\dot {Q_{i}}}=\sum _{k}{\frac {\partial Q_{i}}{\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial Q_{k}}{\partial t}}(\mathbf {q} ,t)} which implies that∂Qi˙∂q˙k=∂Qi∂qk{\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial {\dot {q}}_{k}}}={\frac {\partial Q_{i}}{\partial q_{k}}}}.

It also follows that:∂Qi˙∂qj=∑k∂2Qi∂qj∂qk(q,t)q˙k+∂2Qk∂qj∂t(q,t){\displaystyle {\frac {\partial {\dot {Q_{i}}}}{\partial q_{j}}}=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{j}\partial q_{k}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial q_{j}\partial t}}(\mathbf {q} ,t)}and similarly:ddt(∂Qi∂qj)=∑k∂2Qi∂qk∂qj(q,t)q˙k+∂2Qk∂t∂qj(q,t){\displaystyle {\frac {d}{dt}}\left({\frac {\partial {Q_{i}}}{\partial q_{j}}}\right)=\sum _{k}{\frac {\partial ^{2}Q_{i}}{\partial q_{k}\partial q_{j}}}(\mathbf {q} ,t)\,{\dot {q}}_{k}+{\frac {\partial ^{2}Q_{k}}{\partial t\partial q_{j}}}(\mathbf {q} ,t)}which imply thatddt(∂Qi∂qk)=∂Q˙i∂qk{\displaystyle {\frac {d}{dt}}\left({\frac {\partial Q_{i}}{\partial q_{k}}}\right)={\frac {\partial {\dot {Q}}_{i}}{\partial q_{k}}}}. The two derived relations can be employed in the proof.

Starting from Euler Lagrange equations in initial set of generalized coordinates, we have:ddt∂L∂q˙i−∂L∂qi−∑jλj∂ϕj∂qi=0∑k(ddt(∂L∂Q˙k∂Q˙k∂q˙i)−∂L∂Qk∂Qk∂qi−∂L∂Q˙k∂Q˙k∂qi−∑jλj∂ϕj∂Qk∂Qk∂qi)=0∑k(ddt(∂L∂Q˙k)∂Q˙k∂q˙i+∂L∂Q˙kddt(∂Q˙k∂q˙i)−∂L∂Qk∂Q˙k∂q˙i−∂L∂Q˙kddt(∂Qk∂qi)−∑jλj∂ϕj∂Qk∂Qk∂qi)=0∑k(ddt(∂L∂Q˙k)−∂L∂Qk−∑jλj∂ϕj∂Qk)∂Qk∂qi=0{\displaystyle {\begin{aligned}{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial q_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial q_{i}}}=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial {q}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {q}_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial q_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right){\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}+{\frac {\partial L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}\right)-{\frac {{\partial }L}{\partial Q_{k}}}{\frac {\partial {\dot {Q}}_{k}}{\partial {\dot {q}}_{i}}}-{\frac {{\partial }L}{\partial {\dot {Q}}_{k}}}{\frac {d}{dt}}\left({\frac {\partial Q_{k}}{\partial q_{i}}}\right)-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}{\frac {\partial Q_{k}}{\partial q_{i}}}\right)=0\\\sum _{k}\left({\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {Q}}_{k}}}\right)-{\frac {\partial L}{\partial Q_{k}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{k}}}\right){\frac {\partial Q_{k}}{\partial q_{i}}}=0\\\end{aligned}}}

Since the transformation fromq→Q{\displaystyle q\rightarrow Q} is invertible, it follows that the form of the Euler-Lagrange equation is invariant i.e.,ddt∂L∂Q˙i−∂L∂Qi−∑jλj∂ϕj∂Qi=0.{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {Q}}_{i}}}-{\frac {{\partial }L}{\partial Q_{i}}}-\sum _{j}\lambda _{j}{\frac {\partial \phi _{j}}{\partial Q_{i}}}=0.}

Cyclic coordinates and conserved momenta

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An important property of the Lagrangian is thatconserved quantities can easily be read off from it. Thegeneralized momentum "canonically conjugate to" the coordinateqi is defined bypi=∂L∂q˙i.{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}

If the LagrangianL doesnot depend on some coordinateqi, it follows immediately from the Euler–Lagrange equations thatp˙i=ddt∂L∂q˙i=∂L∂qi=0{\displaystyle {\dot {p}}_{i}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}=0}and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case ofNoether's theorem. Such coordinates are called "cyclic" or "ignorable".

For example, a system may have a LagrangianL(r,θ,s˙,z˙,r˙,θ˙,ϕ˙,t),{\displaystyle L(r,\theta ,{\dot {s}},{\dot {z}},{\dot {r}},{\dot {\theta }},{\dot {\phi }},t),}wherer andz are lengths along straight lines,s is an arc length along some curve, andθ andφ are angles. Noticez,s, andφ are all absent in the Lagrangian even though their velocities are not. Then the momentapz=∂L∂z˙,ps=∂L∂s˙,pϕ=∂L∂ϕ˙,{\displaystyle p_{z}={\frac {\partial L}{\partial {\dot {z}}}},\quad p_{s}={\frac {\partial L}{\partial {\dot {s}}}},\quad p_{\phi }={\frac {\partial L}{\partial {\dot {\phi }}}},}are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this casepz is a translational momentum in thez direction,ps is also a translational momentum along the curves is measured, andpφ is an angular momentum in the plane the angleφ is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.

Energy

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Given a LagrangianL,{\displaystyle L,} theHamiltonian of the corresponding mechanical system is, by definition,H=(∑i=1nq˙i∂L∂q˙i)−L.{\displaystyle H={\biggl (}\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\biggr )}-L.}This quantity will be equivalent to energy if the generalized coordinates are natural coordinates, i.e., they have no explicit time dependence when expressing position vector:r=r(q1,⋯,qn){\displaystyle \mathbf {r} =\mathbf {r} (q_{1},\cdots ,q_{n})}. From:T=m2v2=m2∑i,j(∂r→∂qiq˙i)⋅(∂r→∂qjq˙j)=m2∑i,jaijq˙iq˙j{\displaystyle T={\frac {m}{2}}v^{2}={\frac {m}{2}}\sum _{i,j}\left({\frac {\partial {\vec {r}}}{\partial q_{i}}}{\dot {q}}_{i}\right)\cdot \left({\frac {\partial {\vec {r}}}{\partial q_{j}}}{\dot {q}}_{j}\right)={\frac {m}{2}}\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}∑k=1nq˙k∂L∂q˙k=∑k=1nq˙k∂T∂q˙k=m2(2∑i,jaijq˙iq˙j)=2T{\displaystyle \sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=\sum _{k=1}^{n}{\dot {q}}_{k}{\frac {\partial T}{\partial {\dot {q}}_{k}}}={\frac {m}{2}}\left(2\sum _{i,j}a_{ij}{\dot {q}}_{i}{\dot {q}}_{j}\right)=2T}H=(∑i=1nq˙i∂L∂q˙i)−L=2T−(T−V)=T+V=E{\displaystyle H=\left(\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-L=2T-(T-V)=T+V=E}whereaij=∂r∂qi⋅∂r∂qj{\displaystyle a_{ij}={\frac {\partial \mathbf {r} }{\partial q_{i}}}\cdot {\frac {\partial \mathbf {r} }{\partial q_{j}}}} is a symmetric matrix that is defined for the derivation.

Invariance under coordinate transformations

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At every time instantt, the energy is invariant underconfiguration space coordinate changesq →Q, i.e. (using natural coordinates)E(q,q˙,t)=E(Q,Q˙,t).{\displaystyle E(\mathbf {q} ,{\dot {\mathbf {q} }},t)=E(\mathbf {Q} ,{\dot {\mathbf {Q} }},t).}Besides this result, the proof below shows that, under such change of coordinates, the derivatives∂L/∂q˙i{\displaystyle \partial L/\partial {\dot {q}}_{i}} change as coefficients of a linear form.

Proof

For a coordinate transformationQ =F(q), we havedQ=F∗(q)dq,{\displaystyle d\mathbf {Q} =F_{*}(\mathbf {q} )d\mathbf {q} ,}whereF∗(q){\displaystyle F_{*}(\mathbf {q} )} is thetangent map of the vector space{∑i=1nq˙i⋅(∂∂qi|q) | q˙i∈R}{\displaystyle \left\{\sum _{i=1}^{n}{\dot {q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial q_{i}}}\right|_{\mathbf {q} }\right)\ {\biggl |}\ {\dot {q}}_{i}\in \mathbb {R} \right\}}to the vector space{∑i=1nQ˙i⋅(∂∂Qi|F(q)) | Q˙i∈R},{\displaystyle \left\{\sum _{i=1}^{n}{\dot {Q}}_{i}\cdot \left(\left.{\frac {\partial }{\partial Q_{i}}}\right|_{F(\mathbf {q} )}\right)\ {\biggl |}\ {\dot {Q}}_{i}\in \mathbb {R} \right\},}andF∗(q)=(∂Fi∂qj|q)i,j=1n{\displaystyle \textstyle F_{*}(\mathbf {q} )=\left(\left.{\frac {\partial F_{i}}{\partial q_{j}}}\right|_{\mathbf {q} }\right)_{i,j=1}^{n}}is the Jacobian. In the coordinatesq˙i{\displaystyle {\dot {q}}_{i}} andQ˙i,{\displaystyle {\dot {Q}}_{i},} the previous formula fordQ{\displaystyle d\mathbf {Q} } has the formQ˙=F∗(q)q˙.{\displaystyle {\dot {\mathbf {Q} }}=F_{*}(\mathbf {q} ){\dot {\mathbf {q} }}.} After differentiation involving the product rule,dQ˙=G(q,q˙)dq+F∗(q)dq˙,{\displaystyle d{\dot {\mathbf {Q} }}=G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} +F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }},}whereG(q,q˙)dq=defd(F∗(q))q˙=(∑k=1n∂2Fi∂qj∂qk|qdqk)i,j=1nq˙=(∑j=1nq˙j∑k=1n∂2Fi∂qj∂qk|qdqk)i=1,…,nT=(∑k=1ndqk∑j=1n∂2Fi∂qj∂qk|qq˙j)i=1,…,nT=(∑j=1n∂2Fi∂qj∂qk|qq˙j)i,k=1ndq.{\displaystyle {\begin{aligned}G(\mathbf {q} ,{\dot {\mathbf {q} }})d\mathbf {q} &\,{\stackrel {\text{def}}{=}}\,d(F_{*}(\mathbf {q} )){\dot {\mathbf {q} }}=\left(\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i,j=1}^{n}{\dot {\mathbf {q} }}=\left(\sum _{j=1}^{n}{\dot {q}}_{j}\sum _{k=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }dq_{k}\right)_{i=1,\ldots ,n}^{T}\\&=\left(\sum _{k=1}^{n}dq_{k}\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i=1,\ldots ,n}^{T}=\left(\sum _{j=1}^{n}{\frac {\partial ^{2}F_{i}}{\partial q_{j}\partial q_{k}}}{\biggl |}_{\mathbf {q} }{\dot {q}}_{j}\right)_{i,k=1}^{n}d\mathbf {q} .\end{aligned}}}

In vector notation,dL(Q,Q˙,t)=∂L∂QdQ+∂L∂Q˙dQ˙+∂L∂tdt=(∂L∂QF∗(q)+∂L∂Q˙G(q,q˙))dq+∂L∂Q˙F∗(q)dq˙+∂L∂t.{\displaystyle {\begin{aligned}dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)&={\frac {\partial L}{\partial \mathbf {Q} }}d\mathbf {Q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}d{\dot {\mathbf {Q} }}+{\frac {\partial L}{\partial t}}dt\\&=\left({\frac {\partial L}{\partial \mathbf {Q} }}F_{*}(\mathbf {q} )+{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}G(\mathbf {q} ,{\dot {\mathbf {q} }})\right)d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}.\end{aligned}}}

On the other hand,dL(q,q˙,t)=∂L∂qdq+∂L∂q˙dq˙+∂L∂tdt.{\displaystyle dL(\mathbf {q} ,{\dot {\mathbf {q} }},t)={\frac {\partial L}{\partial \mathbf {q} }}d\mathbf {q} +{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}d{\dot {\mathbf {q} }}+{\frac {\partial L}{\partial t}}dt.}

It was mentioned earlier that Lagrangians do not depend on the choice of configuration space coordinates, i.e.L(Q,Q˙,t)=L(q,q˙,t).{\displaystyle L(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=L(\mathbf {q} ,{\dot {\mathbf {q} }},t).} One implication of this is thatdL(Q,Q˙,t)=dL(q,q˙,t),{\displaystyle dL(\mathbf {Q} ,{\dot {\mathbf {Q} }},t)=dL(\mathbf {q} ,{\dot {\mathbf {q} }},t),} and∂L∂Q˙F∗(q)=∂L∂q˙.{\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}F_{*}(\mathbf {q} )={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}This demonstrates that, for eachq,{\displaystyle \mathbf {q} ,}q˙,{\displaystyle {\dot {\mathbf {q} }},} andt,{\displaystyle t,}∑i=1n∂L∂q˙idq˙i{\displaystyle \textstyle \sum \limits _{i=1}^{n}{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}} is a well-defined linear form whose coefficients∂L∂q˙i{\displaystyle \textstyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}} are contravariant 1-tensors. Applying both sides of the equation toq˙{\displaystyle {\dot {\mathbf {q} }}} and using the above formula forQ˙{\displaystyle {\dot {\mathbf {Q} }}} yieldsQ˙∂L∂Q˙=q˙∂L∂q˙.{\displaystyle {\dot {\mathbf {Q} }}{\frac {\partial L}{\partial {\dot {\mathbf {Q} }}}}={\dot {\mathbf {q} }}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}.}The invariance of the energyE{\displaystyle E} follows.

Conservation

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In Lagrangian mechanics, the system isclosed if and only if its LagrangianL{\displaystyle L} does not explicitly depend on time. Theenergy conservation law states that the energyE{\displaystyle E} of a closed system is anintegral of motion.

More precisely, letq =q(t) be anextremal. (In other words,q satisfies the Euler–Lagrange equations). Taking the total time-derivative ofL along this extremal and using the EL equations leads todLdt=q˙∂L∂q+q¨∂L∂q˙+∂L∂t−∂L∂t=ddt(∂L∂q˙)q˙+q¨∂L∂q˙−L˙−∂L∂t=ddt(∂L∂q˙q˙−L)=dHdt{\displaystyle {\begin{aligned}{\frac {dL}{dt}}&={\dot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}+{\frac {\partial L}{\partial t}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right){\dot {\mathbf {q} }}+{\ddot {\mathbf {q} }}{\frac {\partial L}{\partial \mathbf {\dot {q}} }}-{\dot {L}}\\-{\frac {\partial L}{\partial t}}&={\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\mathbf {\dot {q}} -L\right)={\frac {dH}{dt}}\end{aligned}}}

If the LagrangianL does not explicitly depend on time, then∂L/∂t = 0, thenH does not vary with time evolution of particle, indeed, an integral of motion, meaning thatH(q(t),q˙(t),t)=constant of time.{\displaystyle H(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)={\text{constant of time}}.}Hence, if the chosen coordinates were natural coordinates, the energy is conserved.

Kinetic and potential energies

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Under all these circumstances,[41] the constantE=T+V{\displaystyle E=T+V}is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energyE is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates.

Mechanical similarity

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Main article:Mechanical similarity

If the potential energy is ahomogeneous function of the coordinates and independent of time,[42] and all position vectors are scaled by the same nonzero constantα,rk′ =αrk, so thatV(αr1,αr2,…,αrN)=αNV(r1,r2,…,rN){\displaystyle V(\alpha \mathbf {r} _{1},\alpha \mathbf {r} _{2},\ldots ,\alpha \mathbf {r} _{N})=\alpha ^{N}V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})}and time is scaled by a factorβ,t′ =βt, then the velocitiesvk are scaled by a factor ofα/β and the kinetic energyT by (α/β)2. The entire Lagrangian has been scaled by the same factor ifα2β2=αN⇒β=α1−N2.{\displaystyle {\frac {\alpha ^{2}}{\beta ^{2}}}=\alpha ^{N}\quad \Rightarrow \quad \beta =\alpha ^{1-{\frac {N}{2}}}.}

Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The lengthl traversed in timet in the original trajectory corresponds to a new lengthl′ traversed in timet′ in the new trajectory, given by the ratiost′t=(l′l)1−N2.{\displaystyle {\frac {t'}{t}}=\left({\frac {l'}{l}}\right)^{1-{\frac {N}{2}}}.}

Interacting particles

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For a given system, if two subsystemsA andB are non-interacting, the LagrangianL of the overall system is the sum of the LagrangiansLA andLB for the subsystems:[37]L=LA+LB.{\displaystyle L=L_{A}+L_{B}.}

If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the systemL into the sum of non-interacting Lagrangians, plus another LagrangianLAB containing information about the interaction,L=LA+LB+LAB.{\displaystyle L=L_{A}+L_{B}+L_{AB}.}

This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction,LAB tends to zero reducing to the non-interacting case above.

The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.

Consequences of singular Lagrangians

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From the Euler-Lagrange equations, it follows that:ddt∂L∂q˙i−∂L∂qi=0∂2L∂qj∂q˙idqjdt+∂2L∂q˙j∂q˙idq˙jdt+∂L∂t−∂L∂qi=0∑jWij(q,q˙,t)q¨j=∂L∂qi−∂L∂t−∑j∂2L∂q˙i∂qjq˙j,{\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-{\frac {\partial L}{\partial q_{i}}}=0\\&{\frac {\partial ^{2}L}{\partial q_{j}\partial {\dot {q}}_{i}}}{\frac {dq_{j}}{dt}}+{\frac {\partial ^{2}L}{\partial {\dot {q}}_{j}\partial {\dot {q}}_{i}}}{\frac {d{\dot {q}}_{j}}{dt}}+{\frac {\partial L}{\partial t}}-{\frac {\partial L}{\partial q_{i}}}=0\\&\sum _{j}W_{ij}(q,{\dot {q}},t){\ddot {q}}_{j}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial L}{\partial t}}-\sum _{j}{\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial q_{j}}}{\dot {q}}_{j},\\\end{aligned}}}

where the matrix is defined asWij=∂2L∂q˙i∂q˙j{\displaystyle W_{ij}={\frac {\partial ^{2}L}{\partial {\dot {q}}_{i}\partial {\dot {q}}_{j}}}}. If the matrixW{\displaystyle W} is non-singular, the above equations can be solved to representq¨{\displaystyle {\ddot {q}}} as a function of(q˙,q,t){\displaystyle ({\dot {q}},q,t)}. If the matrix is non-invertible, it would not be possible to represent allq¨{\displaystyle {\ddot {q}}}'s as a function of(q˙,q,t){\displaystyle ({\dot {q}},q,t)} but also, the Hamiltonian equations of motions will not take the standard form.[43]

Examples

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The following examples apply Lagrange's equations of the second kind to mechanical problems.

Conservative force

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A particle of massm moves under the influence of aconservative force derived from thegradient ∇ of ascalar potential,F=−∇V(r).{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}V(\mathbf {r} ).}

If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.

Cartesian coordinates

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The Lagrangian of the particle can be writtenL(x,y,z,x˙,y˙,z˙)=12m(x˙2+y˙2+z˙2)−V(x,y,z).{\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})={\frac {1}{2}}m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-V(x,y,z).}

The equations of motion for the particle are found by applying theEuler–Lagrange equation, for thex coordinateddt(∂L∂x˙)=∂L∂x,{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\partial L}{\partial x}},}with derivatives∂L∂x=−∂V∂x,∂L∂x˙=mx˙,ddt(∂L∂x˙)=mx¨,{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial V}{\partial x}},\quad {\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}},\quad {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)=m{\ddot {x}},}hencemx¨=−∂V∂x,{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}},}and similarly for they andz coordinates. Collecting the equations in vector form we findmr¨=−∇V{\displaystyle m{\ddot {\mathbf {r} }}=-{\boldsymbol {\nabla }}V}which isNewton's second law of motion for a particle subject to a conservative force.

Polar coordinates in 2D and 3D

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Using the spherical coordinates(r,θ,φ) as commonly used in physics (ISO 80000-2:2019 convention), wherer is the radial distance to origin,θ is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), andφ is the azimuthal angle, the Lagrangian for a central potential isL=m2(r˙2+r2θ˙2+r2sin2⁡θφ˙2)−V(r).{\displaystyle L={\frac {m}{2}}({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}+r^{2}\sin ^{2}\theta \,{\dot {\varphi }}^{2})-V(r).}So, in spherical coordinates, the Euler–Lagrange equations aremr¨−mr(θ˙2+sin2⁡θφ˙2)+∂V∂r=0,{\displaystyle m{\ddot {r}}-mr({\dot {\theta }}^{2}+\sin ^{2}\theta \,{\dot {\varphi }}^{2})+{\frac {\partial V}{\partial r}}=0,}ddt(mr2θ˙)−mr2sin⁡θcos⁡θφ˙2=0,{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}{\dot {\theta }})-mr^{2}\sin \theta \cos \theta \,{\dot {\varphi }}^{2}=0,}ddt(mr2sin2⁡θφ˙)=0.{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(mr^{2}\sin ^{2}\theta \,{\dot {\varphi }})=0.}Theφ coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentumpφ=∂L∂φ˙=mr2sin2⁡θφ˙,{\displaystyle p_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=mr^{2}\sin ^{2}\theta {\dot {\varphi }},}in whichr,θ anddφ/dt can all vary with time, but only in such a way thatpφ is constant.

The Lagrangian in two-dimensional polar coordinates is recovered by fixingθ to the constant valueπ/2.

Pendulum on a movable support

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Sketch of the situation with definition of the coordinates (click to enlarge)

Consider a pendulum of massm and lengthℓ, which is attached to a support with massM, which can move along a line in thex{\displaystyle x}-direction. Letx{\displaystyle x} be the coordinate along the line of the support, and let us denote the position of the pendulum by the angleθ{\displaystyle \theta } from the vertical. The coordinates and velocity components of the pendulum bob arexpend=x+ℓsin⁡θ⇒x˙pend=x˙+ℓθ˙cos⁡θypend=−ℓcos⁡θ⇒y˙pend=ℓθ˙sin⁡θ.{\displaystyle {\begin{array}{rll}&x_{\mathrm {pend} }=x+\ell \sin \theta &\quad \Rightarrow \quad {\dot {x}}_{\mathrm {pend} }={\dot {x}}+\ell {\dot {\theta }}\cos \theta \\&y_{\mathrm {pend} }=-\ell \cos \theta &\quad \Rightarrow \quad {\dot {y}}_{\mathrm {pend} }=\ell {\dot {\theta }}\sin \theta .\end{array}}}

The generalized coordinates can be taken to bex{\displaystyle x} andθ{\displaystyle \theta }. The kinetic energy of the system is thenT=12Mx˙2+12m(x˙pend2+y˙pend2){\displaystyle T={\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left({\dot {x}}_{\mathrm {pend} }^{2}+{\dot {y}}_{\mathrm {pend} }^{2}\right)}and the potential energy isV=mgypend{\displaystyle V=mgy_{\mathrm {pend} }}giving the LagrangianL=T−V=12Mx˙2+12m[(x˙+ℓθ˙cos⁡θ)2+(ℓθ˙sin⁡θ)2]+mgℓcos⁡θ=12(M+m)x˙2+mx˙ℓθ˙cos⁡θ+12mℓ2θ˙2+mgℓcos⁡θ.{\displaystyle {\begin{array}{rcl}L&=&T-V\\&=&{\frac {1}{2}}M{\dot {x}}^{2}+{\frac {1}{2}}m\left[\left({\dot {x}}+\ell {\dot {\theta }}\cos \theta \right)^{2}+\left(\ell {\dot {\theta }}\sin \theta \right)^{2}\right]+mg\ell \cos \theta \\&=&{\frac {1}{2}}\left(M+m\right){\dot {x}}^{2}+m{\dot {x}}\ell {\dot {\theta }}\cos \theta +{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+mg\ell \cos \theta .\end{array}}}

Sincex is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum ispx=∂L∂x˙=(M+m)x˙+mℓθ˙cos⁡θ,{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=(M+m){\dot {x}}+m\ell {\dot {\theta }}\cos \theta ,}and the Lagrange equation for the support coordinatex{\displaystyle x} is(M+m)x¨+mℓθ¨cos⁡θ−mℓθ˙2sin⁡θ=0.{\displaystyle (M+m){\ddot {x}}+m\ell {\ddot {\theta }}\cos \theta -m\ell {\dot {\theta }}^{2}\sin \theta =0.}

The Lagrange equation for the angleθ isddt[m(x˙ℓcos⁡θ+ℓ2θ˙)]+mℓ(x˙θ˙+g)sin⁡θ=0;{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[m({\dot {x}}\ell \cos \theta +\ell ^{2}{\dot {\theta }})\right]+m\ell ({\dot {x}}{\dot {\theta }}+g)\sin \theta =0;}and simplifyingθ¨+x¨ℓcos⁡θ+gℓsin⁡θ=0.{\displaystyle {\ddot {\theta }}+{\frac {\ddot {x}}{\ell }}\cos \theta +{\frac {g}{\ell }}\sin \theta =0.}

These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example,x¨→0{\displaystyle {\ddot {x}}\to 0} should give the equations of motion for asimple pendulum that is at rest in someinertial frame, whileθ¨→0{\displaystyle {\ddot {\theta }}\to 0} should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, bystepping through the results iteratively.

Two-body central force problem

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Main articles:Two-body problem andCentral force

Two bodies of massesm1 andm2 with position vectorsr1 andr2 are in orbit about each other due to an attractivecentral potentialV. We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce theJacobi coordinates; the separation of the bodiesr =r2 −r1 and the location of thecenter of massR = (m1r1 +m2r2)/(m1 +m2). The Lagrangian is then[44][45][nb 4]L=12MR˙2⏟Lcm+12μr˙2−V(|r|)⏟Lrel{\displaystyle L=\underbrace {{\frac {1}{2}}M{\dot {\mathbf {R} }}^{2}} _{L_{\text{cm}}}+\underbrace {{\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}-V(|\mathbf {r} |)} _{L_{\text{rel}}}}whereM =m1 +m2 is the total mass,μ =m1m2/(m1 +m2) is thereduced mass, andV the potential of the radial force, which depends only on themagnitude of the separation|r| = |r2 −r1|. The Lagrangian splits into acenter-of-mass termLcm and arelative motion termLrel.

The Euler–Lagrange equation forR is simplyMR¨=0,{\displaystyle M{\ddot {\mathbf {R} }}=0,}which states the center of mass moves in a straight line at constant velocity.

Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates(r,θ) and taker = |r|,Lrel=12μ(r˙2+r2θ˙2)−V(r),{\displaystyle L_{\text{rel}}={\frac {1}{2}}\mu \left({\dot {r}}^{2}+r^{2}{\dot {\theta }}^{2}\right)-V(r),}soθ is a cyclic coordinate with the corresponding conserved (angular) momentumpθ=∂Lrel∂θ˙=μr2θ˙=ℓ.{\displaystyle p_{\theta }={\frac {\partial L_{\text{rel}}}{\partial {\dot {\theta }}}}=\mu r^{2}{\dot {\theta }}=\ell .}

The radial coordinater and angular velocitydθ/dt can vary with time, but only in such a way thatℓ is constant. The Lagrange equation forr isμrθ˙2−dVdr=μr¨.{\displaystyle \mu r{\dot {\theta }}^{2}-{\frac {dV}{dr}}=\mu {\ddot {r}}.}

This equation is identical to the radial equation obtained using Newton's laws in aco-rotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocitydθ/dt from this radial equation,[46]μr¨=−dVdr+ℓ2μr3.{\displaystyle \mu {\ddot {r}}=-{\frac {\mathrm {d} V}{\mathrm {d} r}}+{\frac {\ell ^{2}}{\mu r^{3}}}.}which is the equation of motion for a one-dimensional problem in which a particle of massμ is subjected to the inward central force−dV/dr and a second outward force, called in this context the(Lagrangian) centrifugal force (seecentrifugal force#Other uses of the term):Fcf=μrθ˙2=ℓ2μr3.{\displaystyle F_{\mathrm {cf} }=\mu r{\dot {\theta }}^{2}={\frac {\ell ^{2}}{\mu r^{3}}}.}

Of course, if one remains entirely within the one-dimensional formulation,ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.

If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates(r,θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth ofusing polar coordinates. As Hildebrand says:[47]

"Since such quantities are not true physical forces, they are often calledinertia forces. Their presence or absence depends, not upon the particular problem at hand, butupon the coordinate system chosen." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides thecentripetal force for a curved motion.

This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see[48] for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in.[49] Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not aninertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces asgeneralized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we dealalways withgeneralized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."

It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.[50]

Extensions to include non-conservative forces

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Dissipative forces

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Dissipation (i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom.[51][52][53][54]

In a more general formulation, the forces could be both conservative andviscous. If an appropriate transformation can be found from theFi,Rayleigh suggests using adissipation function,D, of the following form:[55]D=12∑j=1m∑k=1mCjkq˙jq˙k,{\displaystyle D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k},}whereCjk are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. IfD is defined this way, then[55]Qj=−∂V∂qj−∂D∂q˙j{\displaystyle Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}}andddt(∂L∂q˙j)−∂L∂qj+∂D∂q˙j=0.{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}}+{\frac {\partial D}{\partial {\dot {q}}_{j}}}=0.}

Electromagnetism

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A test particle is a particle whosemass andcharge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles likeelectrons andup quarks are more complex and have additional terms in their Lagrangians. Not only can the fields form non conservative potentials, these potentials can also be velocity dependent.

The Lagrangian for acharged particle withelectrical chargeq, interacting with anelectromagnetic field, is the prototypical example of a velocity-dependent potential. The electricscalar potentialϕ =ϕ(r,t) andmagnetic vector potentialA =A(r,t) are defined from theelectric fieldE =E(r,t) andmagnetic fieldB =B(r,t) as follows:E=−∇ϕ−∂A∂t,B=∇×A.{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\phi -{\frac {\partial \mathbf {A} }{\partial t}},\quad \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}

The Lagrangian of a massive charged test particle in an electromagnetic fieldL=12mr˙2+qr˙⋅A−qϕ,{\displaystyle L={\tfrac {1}{2}}m{\dot {\mathbf {r} }}^{2}+q\,{\dot {\mathbf {r} }}\cdot \mathbf {A} -q\phi ,}is calledminimal coupling. This is a good example of when the commonrule of thumb that the Lagrangian is the kinetic energy minus the potential energy is incorrect. Combined withEuler–Lagrange equation, it produces theLorentz force lawmr¨=qE+qr˙×B{\displaystyle m{\ddot {\mathbf {r} }}=q\mathbf {E} +q{\dot {\mathbf {r} }}\times \mathbf {B} }

Undergauge transformation:A→A+∇f,ϕ→ϕ−f˙,{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +{\boldsymbol {\nabla }}f,\quad \phi \rightarrow \phi -{\dot {f}},}wheref(r,t) is any scalar function of space and time, the aforementioned Lagrangian transforms like:L→L+q(r˙⋅∇+∂∂t)f=L+qdfdt,{\displaystyle L\rightarrow L+q\left({\dot {\mathbf {r} }}\cdot {\boldsymbol {\nabla }}+{\frac {\partial }{\partial t}}\right)f=L+q{\frac {df}{dt}},}which still produces the same Lorentz force law.

Note that thecanonical momentum (conjugate to positionr) is thekinetic momentum plus a contribution from theA field (known as the potential momentum):p=∂L∂r˙=mr˙+qA.{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} .}

This relation is also used in theminimal coupling prescription inquantum mechanics andquantum field theory. From this expression, we can see that thecanonical momentump is not gauge invariant, and therefore not a measurable physical quantity; However, ifr is cyclic (i.e. Lagrangian is independent of positionr), which happens if theϕ andA fields are uniform, then this canonical momentump given here is the conserved momentum, while the measurable physical kinetic momentummv is not.

Other contexts and formulations

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The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.

Alternative formulations of classical mechanics

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A closely related formulation of classical mechanics isHamiltonian mechanics. The Hamiltonian is defined byH=∑i=1nq˙i∂L∂q˙i−L{\displaystyle H=\sum _{i=1}^{n}{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}-L}and can be obtained by performing aLegendre transformation on the Lagrangian, which introduces new variablescanonically conjugate to the original variables. For example, given a set of generalized coordinates, the variablescanonically conjugate are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity inquantum mechanics (seeHamiltonian (quantum mechanics)).

Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.

Momentum space formulation

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Main article:Position and momentum space § Lagrangian mechanics

The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate LagrangianL(q, dq/dt,t) obtains the generalized momenta LagrangianL′(p, dp/dt,t) in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta.

Higher derivatives of generalized coordinates

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There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, seeEuler–Lagrange equation for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, seeOstrogradsky instability

Optics

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Main article:Hamiltonian optics

Lagrangian mechanics can be applied togeometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.

Relativistic formulation

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Main article:Relativistic Lagrangian mechanics

Lagrangian mechanics can be formulated inspecial relativity andgeneral relativity. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in amanifestly covariant way, it may be possible if a particular frame of reference is singled out.

Quantum mechanics

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Inquantum mechanics,action and quantum-mechanicalphase are related via thePlanck constant, and theprinciple of stationary action can be understood in terms ofconstructive interference ofwave functions.

In 1948,Feynman discovered thepath integral formulation extending theprinciple of least action toquantum mechanics forelectrons andphotons. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, andFermat's principle inoptics.

Classical field theory

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In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. Inclassical field theory, the physical system is not a set of discrete particles, but rather a continuous fieldϕ(r,t) defined over a region of 3D space. Associated with the field is aLagrangian densityL(ϕ,∇ϕ,ϕ˙,r,t){\displaystyle {\mathcal {L}}(\phi ,\nabla \phi ,{\dot {\phi }},\mathbf {r} ,t)}defined in terms of the field and its space and time derivatives at a locationr and timet. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then thevolume integral of the Lagrangian density over 3D spaceL(t)=∫Ld3r{\displaystyle L(t)=\int {\mathcal {L}}\,\mathrm {d} ^{3}\mathbf {r} }where d3r is a 3Ddifferentialvolume element. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.

Noether's theorem

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The action principle, and the Lagrangian formalism, are tied closely toNoether's theorem, which connects physicalconserved quantities to continuoussymmetries of a physical system.

If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with eitherspecial relativity orgeneral relativity.

See also

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Footnotes

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  1. ^Sometimes in this context thevariational derivative denoted and defined asδδrk≡∂∂rk−ddt∂∂r˙k{\displaystyle {\frac {\delta }{\delta \mathbf {r} _{k}}}\equiv {\frac {\partial }{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}}is used. Throughout this article only partial and total derivatives are used.
  2. ^Here the virtual displacements are assumed reversible, it is possible for some systems to have non-reversible virtual displacements that violate this principle, seeUdwadia–Kalaba equation.
  3. ^In other wordsCk⋅δrk=0{\displaystyle \mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0}for particlek subject to a constraint force, howeverCkxδxk≠0,Ckyδyk≠0,Ckzδzk≠0{\displaystyle C_{k\,x}\delta x_{k}\neq 0,\quad C_{k\,y}\delta y_{k}\neq 0,\quad C_{k\,z}\delta z_{k}\neq 0}because of the constraint equations on therk coordinates.
  4. ^The Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000.

Notes

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  1. ^Fraser, Craig. "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics". Archive for History of Exact Sciences, vol. 28, no. 3, 1983, pp. 197–241. JSTOR,http://www.jstor.org/stable/41133689. Accessed 3 Nov. 2023.
  2. ^Hand & Finch 1998, p. 23
  3. ^Hand & Finch 1998, pp. 18–20
  4. ^Hand & Finch 1998, pp. 46, 51
  5. ^abBall, Philip (2019-09-13)."Teaching Energy Before Forces".Physics.12 (2): 100.Bibcode:2019PRPER..15b0126L.doi:10.1103/PhysRevPhysEducRes.15.020126.hdl:1808/29610. Retrieved2024-09-27.
  6. ^abTatum, J. B."Lagrangian mechanics"(PDF).www.astro.uvic.ca. Retrieved2024-09-27.
  7. ^abParsons, Paul; Dixon, Gail (2016).50 ideas you really need to know: science. London:Quercus. pp. 4–7.ISBN 978-1-78429-614-8.
  8. ^Torby 1984, p. 270
  9. ^abcdTorby 1984, p. 269
  10. ^Cortés, Vicente; Haupt, Alexander S. (2023). "2".Mathematical Methods of Classical Physics. Springer.arXiv:1612.03100. Retrieved2025-06-24.
  11. ^Hand & Finch 1998, p. 60–61
  12. ^Penrose 2007
  13. ^Hand & Finch 1998, p. 19
  14. ^Cremaschini, Claudio; Tessarotto, Massimo (2015-06-30)."Synchronous Lagrangian variational principles in General Relativity".The European Physical Journal Plus.130 (6): 123.arXiv:1609.04418.Bibcode:2015EPJP..130..123C.doi:10.1140/epjp/i2015-15123-4.ISSN 2190-5444.
  15. ^Bersani AM, Caressa P. Lagrangian descriptions of dissipative systems: a review.Mathematics and Mechanics of Solids. 2021;26(6):785-803. Doi: 10.1177/1081286520971834
  16. ^Hand & Finch 1998, p. 36–40
  17. ^Pfeiffer, Friedrich (2008), Pfeiffer, Friedrich (ed.),"Constraint Systems",Mechanical System Dynamics, Lecture Notes in Applied and Computational Mechanics, vol. 40, Berlin, Heidelberg: Springer, pp. 85–186,doi:10.1007/978-3-540-79436-3_3,ISBN 978-3-540-79436-3, retrieved2024-09-23
  18. ^Kay 1988, p. 156
  19. ^Synge & Schild 1949, p. 150–152
  20. ^Foster & Nightingale 1995, p. 89
  21. ^Hand & Finch 1998, p. 4
  22. ^Goldstein 1980, pp. 16–18
  23. ^abHand & Finch 1998, p. 15
  24. ^Fetter & Walecka 1980, p. 53
  25. ^Kibble & Berkshire 2004, p. 234
  26. ^Fetter & Walecka 1980, p. 56
  27. ^Hand & Finch 1998, p. 17
  28. ^Hand & Finch 1998, p. 15–17
  29. ^R. Penrose (2007).The Road to Reality. Vintage books. p. 474.ISBN 978-0-679-77631-4.
  30. ^Goldstein 1980, p. 23
  31. ^Kibble & Berkshire 2004, p. 234–235
  32. ^Hand & Finch 1998, p. 51
  33. ^Hanc, Jozef; Taylor, Edwin F.; Tuleja, Slavomir (2005-07-01)."Variational mechanics in one and two dimensions".American Journal of Physics.73 (7):603–610.Bibcode:2005AmJPh..73..603H.doi:10.1119/1.1848516.ISSN 0002-9505.
  34. ^abHand & Finch 1998, p. 44–45
  35. ^Goldstein 1980
  36. ^Fetter & Walecka 1980, pp. 68–70
  37. ^abLandau & Lifshitz 1976, p. 4
  38. ^Goldstein, Poole & Safko 2002, p. 21
  39. ^Landau & Lifshitz 1976, p. 4
  40. ^Goldstein 1980, p. 21
  41. ^Landau & Lifshitz 1976, p. 14
  42. ^Landau & Lifshitz 1976, p. 22
  43. ^Rothe, Heinz J; Rothe, Klaus D (2010).Classical and Quantum Dynamics of Constrained Hamiltonian Systems. World Scientific Lecture Notes in Physics. Vol. 81. WORLD SCIENTIFIC. p. 7.Bibcode:2010cqdc.book.....R.doi:10.1142/7689.ISBN 978-981-4299-64-0.
  44. ^Taylor 2005, p. 297
  45. ^Padmanabhan 2000, p. 48
  46. ^Hand & Finch 1998, pp. 140–141
  47. ^Hildebrand 1992, p. 156
  48. ^Zak, Zbilut & Meyers 1997, pp. 202
  49. ^Shabana 2008, pp. 118–119
  50. ^Gannon 2006, p. 267
  51. ^Kosyakov 2007
  52. ^Galley 2013
  53. ^Birnholtz, Hadar & Kol 2014
  54. ^Birnholtz, Hadar & Kol 2013
  55. ^abTorby 1984, p. 271

References

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Further reading

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  • Gupta, Kiran Chandra,Classical mechanics of particles and rigid bodies (Wiley, 1988).
  • Cassel, Kevin (2013).Variational methods with applications in science and engineering. Cambridge: Cambridge University Press.ISBN 978-1-107-02258-4.
  • Goldstein, Herbert, et al.Classical Mechanics. 3rd ed., Pearson, 2002.

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