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Intheoretical physics andmathematical physics,analytical mechanics, ortheoretical mechanics is a collection of closely related formulations ofclassical mechanics. Analytical mechanics usesscalar properties of motion representing the system as a whole—usually itskinetic energy andpotential energy. Theequations of motion are derived from the scalar quantity by some underlying principle about the scalar'svariation.
Analytical mechanics was developed by many scientists and mathematicians during the 18th century and onward, afterNewtonian mechanics. Newtonian mechanics considersvector quantities of motion, particularlyaccelerations,momenta,forces, of the constituents of the system; it can also be calledvectorial mechanics.[1] A scalar is a quantity, whereas a vector is represented by quantity and direction. The results of these two different approaches are equivalent, but the analytical mechanics approach has many advantages for complex problems.
Analytical mechanics takes advantage of a system'sconstraints to solve problems. The constraints limit thedegrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context asgeneralized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-conservative forces or dissipative forces likefriction, in which case one may revert to Newtonian mechanics.
Two dominant branches of analytical mechanics areLagrangian mechanics (using generalized coordinates and corresponding generalized velocities inconfiguration space) andHamiltonian mechanics (using coordinates and corresponding momenta inphase space). Both formulations are equivalent by aLegendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such asHamilton–Jacobi theory,Routhian mechanics, andAppell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called theprinciple of least action. One result isNoether's theorem, a statement which connectsconservation laws to their associatedsymmetries.
Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used inrelativistic mechanics andgeneral relativity, and with some modifications,quantum mechanics andquantum field theory.
Analytical mechanics is used widely, from fundamental physics toapplied mathematics, particularlychaos theory.
The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.
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The goal of mechanical theory is to solve mechanical problems, such as arise in physics and engineering. Starting from a physical system—such as a mechanism or a star system—amathematical model is developed in the form of a differential equation. The model can be solved numerically or analytically to determine the motion of the system.
Newton's vectorial approach to mechanics describes motion with the help ofvector quantities such asforce,velocity,acceleration. These quantities characterise themotion of a body idealised as a"mass point" or a "particle" understood as a single point to which a mass is attached. Newton's method has been successfully applied to a wide range of physical problems, including the motion of a particle inEarth'sgravitational field and the motion of planets around the Sun. In this approach, Newton's laws describe the motion by a differential equation and then the problem is reduced to the solving of that equation.
When a mechanical system contains many particles, however (such as a complex mechanism or afluid), Newton's approach is difficult to apply. Using a Newtonian approach is possible, under proper precautions, namely isolating each single particle from the others, and determining all the forces acting on it. Such analysis is cumbersome even in relatively simple systems. Newton thought thathis third law "action equals reaction" would take care of all complications.[citation needed] This is false even for such simple system asrotations of asolid body.[clarification needed] In more complicated systems, the vectorial approach cannot give an adequate description.
The analytical approach simplifies problems by treatingmechanical systems as ensembles of particles that interact with each other, rather considering each particle as an isolated unit. In the vectorial approach, forces must be determined individually for each particle, whereas in the analytical approach it is enough to know one single function which contains implicitly all the forces acting on and in the system. Such simplification is often done using certain kinematic conditions which are stateda priori. However, the analytical treatment does not require the knowledge of these forces and takes these kinematic conditions for granted.[citation needed]
Still, deriving the equations of motion of a complicated mechanical system requires a unifying basis from which they follow.[clarification needed] This is provided by variousvariational principles: behind each set of equations there is a principle that expresses the meaning of the entire set. Given a fundamental and universal quantity calledaction, the principle that this action be stationary under small variation of some other mechanical quantity generates the required set of differential equations. The statement of the principle does not require any specialcoordinate system, and all results are expressed ingeneralized coordinates. This means that the analytical equations of motion do not change upon acoordinate transformation, aninvariance property that is lacking in the vectorial equations of motion.[2]
It is not altogether clear what is meant by 'solving' a set of differential equations. A problem is regarded as solved when the particles coordinates at timet are expressed as simple functions oft and of parameters defining the initial positions and velocities. However, 'simple function' is not awell-defined concept: nowadays, afunctionf(t) is not regarded as a formal expression int (elementary function) as in the time of Newton but most generally as a quantity determined byt, and it is not possible to draw a sharp line between 'simple' and 'not simple' functions. If one speaks merely of 'functions', then every mechanical problem is solved as soon as it has been well stated in differential equations, because given the initial conditions andt determine the coordinates att. This is a fact especially at present with the modern methods ofcomputer modelling which provide arithmetical solutions to mechanical problems to any desired degree of accuracy, thedifferential equations being replaced bydifference equations.
Still, though lacking precise definitions, it is obvious that thetwo-body problem has a simple solution, whereas thethree-body problem has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, themathematical structure of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.
Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which Lagrangian or Hamiltonian equations of motion are applicable and that include a very wide range of problems indeed.[3]
Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration on specific problems for which methods have already been designed.
InNewtonian mechanics, one customarily uses all threeCartesian coordinates, or other 3Dcoordinate system, to refer to a body'sposition during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known asgeneralized coordinates, denotedqi (i = 1, 2, 3...).[4]: 231
Generalized coordinates incorporate constraints on the system. There is one generalized coordinateqi for eachdegree of freedom (for convenience labelled by an indexi = 1, 2...N), i.e. each way the system can change itsconfiguration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number ofcurvilinear coordinates equals thedimension of the position space in question (usually 3 for 3d space), while the number ofgeneralized coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:[5][dubious –discuss]
For a system withN degrees of freedom, the generalized coordinates can be collected into anN-tuple:and thetime derivative (here denoted by an overdot) of this tuple give thegeneralized velocities:
D'Alembert's principle states that infinitesimalvirtual work done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:[6]: 265 whereare thegeneralized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) andq are the generalized coordinates. This leads to the generalized form ofNewton's laws in the language of analytical mechanics:
whereT is the totalkinetic energy of the system, and the notationis a useful shorthand (seematrix calculus for this notation).
If the curvilinear coordinate system is defined by the standardposition vectorr, and if the position vector can be written in terms of the generalized coordinatesq and timet in the form: and this relation holds for all timest, thenq are calledholonomic constraints.[7] Vectorr is explicitly dependent ont in cases when the constraints vary with time, not just because ofq(t). For time-independent situations, the constraints are also calledscleronomic, for time-dependent cases they are calledrheonomic.[5]
The introduction of generalized coordinates and the fundamental Lagrangian function:
whereT is the totalkinetic energy andV is the totalpotential energy of the entire system, then either following thecalculus of variations or using the above formula – lead to theEuler–Lagrange equations;
which are a set ofN second-orderordinary differential equations, one for eachqi(t).
This formulation identifies the actual path followed by the motion as a selection of the path over which thetime integral ofkinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.
The Lagrangian formulation uses theconfiguration space of the system, theset of all possible generalized coordinates:
where isN-dimensionalreal space (see alsoset-builder notation). The particular solution to the Euler–Lagrange equations is called a(configuration) path or trajectory, i.e. one particularq(t) subject to the requiredinitial conditions. The general solutions form a set of possible configurations as functions of time:
The configuration space can be defined more generally, and indeed more deeply, in terms oftopologicalmanifolds and thetangent bundle.
TheLegendre transformation of the Lagrangian replaces the generalized coordinates and velocities (q,q̇) with (q,p); the generalized coordinates and thegeneralized momentums conjugate to the generalized coordinates:
and introduces the Hamiltonian (which is in terms of generalized coordinates and momentums):
where denotes thedot product, also leading toHamilton's equations:
which are now a set of 2N first-order ordinary differential equations, one for eachqi(t) andpi(t). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:
which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momentums can be written in terms of the generalized forces in the same way as Newton's second law:
Analogous to the configuration space, the set of all momentums is thegeneralizedmomentum space:
("Momentum space" also refers to "k-space"; the set of allwave vectors (given byDe Broglie relations) as used in quantum mechanics and theory ofwaves)
The set of all positions and momentums form thephase space:
that is, theCartesian product of the configuration space and generalized momentum space.
A particular solution to Hamilton's equations is called aphase path, a particular curve (q(t),p(t)) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is thephase portrait:
All dynamical variables can be derived from positionq, momentump, and timet, and written as a function of these:A =A(q,p,t). IfA(q,p,t) andB(q,p,t) are two scalar valued dynamical variables, thePoisson bracket is defined by the generalized coordinates and momentums:
Calculating thetotal derivative of one of these, sayA, and substituting Hamilton's equations into the result leads to the time evolution ofA:
This equation inA is closely related to the equation of motion in theHeisenberg picture ofquantum mechanics, in which classical dynamical variables becomequantum operators (indicated by hats (^)), and the Poisson bracket is replaced by thecommutator of operators via Dirac'scanonical quantization:
Following are overlapping properties between the Lagrangian and Hamiltonian functions.[5][8]
Action is another quantity in analytical mechanics defined as afunctional of the Lagrangian:
A general way to find the equations of motion from the action is theprinciple of least action:[10]
where the departuret1 and arrivalt2 times are fixed.[1] The term "path" or "trajectory" refers to thetime evolution of the system as a path through configuration space, in other wordsq(t) tracing out a path in. The path for which action is least is the path taken by the system.
From this principle,allequations of motion in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies thepath integral formulation ofquantum mechanics,[11][12] and is used for calculatinggeodesic motion ingeneral relativity.[13]
The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function ofp,q, andt) allows the Hamiltonian in one set of coordinatesq and momentap to be transformed into a new setQ =Q(q,p,t) andP =P(q,p,t), in four possible ways:
With the restriction onP andQ such that the transformed Hamiltonian system is:
the above transformations are calledcanonical transformations, each functionGn is called agenerating function of the "nth kind" or "type-n". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.
The choice ofQ andP is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformationq →Q andp →P to be canonical is the Poisson bracket be unity,
for alli = 1, 2,...N. If this does not hold then the transformation is not canonical.[5]
By setting the canonically transformed HamiltonianK = 0, and the type-2 generating function equal toHamilton's principal function (also the action) plus an arbitrary constantC:
the generalized momenta become:
andP is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:
whereH is the Hamiltonian as before:
Another related function isHamilton's characteristic function
used to solve the HJE byadditive separation of variables for a time-independent HamiltonianH.
The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study ofsymplectic manifolds andsymplectic topology.[14][15] In this formulation, the solutions of the Hamilton–Jacobi equations are theintegral curves ofHamiltonian vector fields.
Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates.[citation needed] If the Lagrangian of a system hass cyclic coordinatesq =q1,q2, ...qs with conjugate momentap =p1,p2, ...ps, with the rest of the coordinates non-cyclic and denotedζ =ζ1,ζ1, ...,ζN − s, they can be removed by introducing theRouthian:
which leads to a set of 2s Hamiltonian equations for the cyclic coordinatesq,
andN −s Lagrangian equations in the non cyclic coordinatesζ.
Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian withN −s degrees of freedom.
The coordinatesq do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion.
Appell's equation of motion involve generalized accelerations, the second time derivatives of the generalized coordinates:
as well as generalized forces mentioned above in D'Alembert's principle. The equations are
where
is the acceleration of thek particle, the second time derivative of its position vector. Each accelerationak is expressed in terms of the generalized accelerationsαr, likewise eachrk are expressed in terms the generalized coordinatesqr.
Generalized coordinates apply to discrete particles. ForNscalar fieldsφi(r,t) wherei = 1, 2, ...N, theLagrangian density is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:and the Euler–Lagrange equations have an analogue for fields:where∂μ denotes the4-gradient and thesummation convention has been used. ForN scalar fields, these Lagrangian field equations are a set ofN second order partial differential equations in the fields, which in general will be coupled and nonlinear.
This scalar field formulation can be extended tovector fields,tensor fields, andspinor fields.
The Lagrangian is thevolume integral of the Lagrangian density:[12][16]
Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such asNewtonian gravity,classical electromagnetism,general relativity, andquantum field theory. It is a question of determining the correct Lagrangian density to generate the correct field equation.
The corresponding "momentum" field densities conjugate to theN scalar fieldsφi(r,t) are:[12]where in this context the overdot denotes a partial time derivative, not a total time derivative. TheHamiltonian density is defined by analogy with mechanics:
The equations of motion are:where thevariational derivativemust be used instead of merely partial derivatives. ForN fields, these Hamiltonian field equations are a set of 2N first order partial differential equations, which in general will be coupled and nonlinear.
Again, the volume integral of the Hamiltonian density is the Hamiltonian
Each transformation can be described by an operator (i.e. function acting on the positionr or momentump variables to change them). The following are the cases when the operator does not changer orp, i.e. symmetries.[11]
| Transformation | Operator | Position | Momentum |
|---|---|---|---|
| Translational symmetry | |||
| Time translation | |||
| Rotational invariance | |||
| Galilean transformations | |||
| Parity | |||
| T-symmetry |
whereR(n̂, θ) is therotation matrix about an axis defined by theunit vectorn̂ and angle θ.
Noether's theorem states that acontinuous symmetry transformation of the action corresponds to aconservation law, i.e. the action (and hence the Lagrangian) does not change under a transformation parameterized by aparameters:the Lagrangian describes the same motion independent ofs, which can be length, angle of rotation, or time. The corresponding momenta toq will be conserved.[5]
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