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Inphysics,equations of motion areequations that describe the behavior of aphysical system in terms of itsmotion as afunction of time.[1] More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may includemomentum components. The most general choice aregeneralized coordinates which can be any convenient variables characteristic of the physical system.[2] The functions are defined in aEuclidean space inclassical mechanics, but are replaced bycurved spaces inrelativity. If thedynamics of a system is known, the equations are the solutions for thedifferential equations describing the motion of the dynamics.
There are two main descriptions of motion: dynamics andkinematics. Dynamics is general, since the momenta,forces andenergy of theparticles are taken into account. In this instance, sometimes the termdynamics refers to the differential equations that the system satisfies (e.g.,Newton's second law orEuler–Lagrange equations), and sometimes to the solutions to those equations.
However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as theSUVAT equations, arising from the definitions ofkinematic quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
A differential equation of motion, usually identified as somephysical law (for example, F = ma), and applying definitions ofphysical quantities, is used to set up an equation to solve a kinematics problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a set of solutions. A particular solution can be obtained by setting theinitial values, which fixes the values of the constants.
Stated formally, in general, an equation of motionM is afunction of thepositionr of the object, itsvelocity (the first timederivative ofr,v =dr/dt), and its acceleration (the secondderivative ofr,a =d2r/dt2), and timet.Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion inr is a second-orderordinary differential equation (ODE) inr,
wheret is time, and each overdot denotes onetime derivative. Theinitial conditions are given by theconstant values att = 0,
The solutionr(t) to the equation of motion, with specified initial values, describes the system for all timest aftert = 0. Other dynamical variables like themomentump of the object, or quantities derived fromr andp likeangular momentum, can be used in place ofr as the quantity to solve for from some equation of motion, although the position of the object at timet is by far the most sought-after quantity.
Sometimes, the equation will belinear and is more likely to be exactly solvable. In general, the equation will benon-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may showchaotic behavior depending on howsensitive the system is to the initial conditions.
Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity,priests,astrologers andastronomers predicted solar and lunareclipses, the solstices and the equinoxes of theSun and the period of theMoon. But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years.
Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop a new body of knowledge, now called physics.
At Oxford,Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris.Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time.Nicholas Oresme further extended Bradwardine's arguments. TheMerton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion.
For writers on kinematics beforeGalileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as a result of greater elevation. OnlyDomingo de Soto, a Spanish theologian, in his commentary onAristotle'sPhysics published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. De Soto's comments are remarkably correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that acceleration would be negative during ascent.
Discourses such as these spread throughout Europe, shaping the work ofGalileo Galilei and others, and helped in laying the foundation of kinematics.[3] Galileo deduced the equations =1/2gt2 in his work geometrically,[4] using theMerton rule, now known as a special case of one of the equations of kinematics.
Galileo was the first to show that the path of a projectile is aparabola. Galileo had an understanding ofcentrifugal force and gave a correct definition ofmomentum. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says inDiscourses[5] that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution.
The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.)
Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. WithStevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope.
Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum.
More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum.
Thus we arrive atRené Descartes,Isaac Newton,Gottfried Leibniz, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones.
Later the equations of motion also appeared inelectrodynamics, when describing the motion of charged particles in electric and magnetic fields, theLorentz force is the general equation which serves as the definition of what is meant by anelectric field andmagnetic field. With the advent ofspecial relativity andgeneral relativity, the theoretical modifications tospacetime meant the classical equations of motion were also modified to account for the finitespeed of light, andcurvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations.[6]
However, the equations ofquantum mechanics can also be considered "equations of motion", since they are differential equations of thewavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields.
From theinstantaneous positionr =r(t), instantaneous meaning at an instant value of timet, the instantaneous velocityv =v(t) and accelerationa =a(t) have the general, coordinate-independent definitions;[7]
Notice that velocity always points in the direction of motion, in other words for a curved path it is thetangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards thecenter of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature.
The rotational analogues are the "angular vector" (angle the particle rotates about some axis)θ =θ(t), angular velocityω =ω(t), and angular accelerationα =α(t):
wheren̂ is aunit vector in the direction of the axis of rotation, andθ is the angle the object turns through about the axis.
The following relation holds for a point-like particle, orbiting about some axis with angular velocityω:[8]
wherer is the position vector of the particle (radial from the rotation axis) andv the tangential velocity of the particle. For a rotating continuumrigid body, these relations hold for each point in the rigid body.
The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.
These equations apply to a particle moving linearly, in three dimensions in a straight line with constantacceleration.[9] Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.
where:
Equations [1] and [2] are from integrating the definitions of velocity and acceleration,[9] subject to the initial conditionsr(t0) =r0 andv(t0) =v0;
in magnitudes,
Equation [3] involves the average velocityv +v0/2. Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity fromv0 tov, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for
and substituting into [2]
then simplifying to get
or in magnitudes
From [3],
substituting fort in [1]:
From [3],
substituting into [2]:
Usually only the first 4 are needed, the fifth is optional.
Herea isconstant acceleration, or in the case of bodies moving under the influence ofgravity, thestandard gravityg is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two.
In some programs, such as theIGCSE Physics andIB DPPhysics programs (international programs but especially popular in the UK and Europe), the same formulae would be written with a different set of preferred variables. Thereu replacesv0 ands replacesr -r0. They are often referred to as theSUVAT equations, where "SUVAT" is anacronym from the variables:s = displacement,u = initial velocity,v = final velocity,a = acceleration,t = time.[10][11] In these variables, the equations of motion would be written
The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical form. The only difference is that the square magnitudes of the velocities require thedot product. The derivations are essentially the same as in the collinear case,
although theTorricelli equation [4] can be derived using thedistributive property of the dot product as follows:
Elementary and frequent examples in kinematics involveprojectiles, for example a ball thrown upwards into the air. Given initial velocityu, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravityg. While these quantities appear to bescalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosings to measure up from the ground, the accelerationa must be in fact−g, since the force ofgravity acts downwards and therefore also the acceleration on the ball due to it.
At the highest point, the ball will be at rest: thereforev = 0. Using equation [4] in the set above, we have:
Substituting and cancelling minus signs gives:
The analogues of the above equations can be written forrotation. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary,
whereα is the constantangular acceleration,ω is theangular velocity,ω0 is the initial angular velocity,θ is the angle turned through (angular displacement),θ0 is the initial angle, andt is the time taken to rotate from the initial state to the final state.
These are the kinematic equations for a particle traversing a path in a plane, described by positionr =r(t).[12] They are simply the time derivatives of the position vector in planepolar coordinates using the definitions of physical quantities above for angular velocityω and angular accelerationα. These are instantaneous quantities which change with time.
The position of the particle is
whereêr andêθ are thepolar unit vectors. Differentiating with respect to time gives the velocity
with radial componentdr/dt and an additional componentrω due to the rotation. Differentiating with respect to time again obtains the acceleration
which breaks into the radial accelerationd2r/dt2,centripetal acceleration–rω2,Coriolis acceleration2ωdr/dt, and angular accelerationrα.
Special cases of motion described by these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.
| State of motion | Constantr | r linear int | r quadratic int | r non-linear int |
|---|---|---|---|---|
| Constantθ | Stationary | Uniform translation (constant translational velocity) | Uniform translational acceleration | Non-uniform translation |
| θ linear int | Uniform angular motion in a circle (constant angular velocity) | Uniform angular motion in a spiral, constant radial velocity | Angular motion in a spiral, constant radial acceleration | Angular motion in a spiral, varying radial acceleration |
| θ quadratic int | Uniform angular acceleration in a circle | Uniform angular acceleration in a spiral, constant radial velocity | Uniform angular acceleration in a spiral, constant radial acceleration | Uniform angular acceleration in a spiral, varying radial acceleration |
| θ non-linear int | Non-uniform angular acceleration in a circle | Non-uniform angular acceleration in a spiral, constant radial velocity | Non-uniform angular acceleration in a spiral, constant radial acceleration | Non-uniform angular acceleration in a spiral, varying radial acceleration |
In 3D space, the equations in spherical coordinates(r,θ,φ) with corresponding unit vectorsêr,êθ andêφ, the position, velocity, and acceleration generalize respectively to
In the case of a constantφ this reduces to the planar equations above.
The first general equation of motion developed wasNewton's second law of motion. In its most general form it states the rate of change of momentump =p(t) =mv(t) of an object equals the forceF =F(x(t),v(t),t) acting on it,[13]: 1112
The force in the equation isnot the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as
sincem is a constant inNewtonian mechanics.
Newton's second law applies to point-like particles, and to all points in arigid body. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; seematerial derivative. In the case the mass is not constant, it is not sufficient to use theproduct rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent withconservation of momentum; seevariable-mass system.
It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex.
The momentum form is preferable since this is readily generalized to more complex systems, such asspecial andgeneral relativity (seefour-momentum).[13]: 112 It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces.
For a number of particles (seemany body problem), the equation of motion for one particlei influenced by other particles is[7][1]
wherepi is the momentum of particlei,Fij is the force on particlei by particlej, andFE is the resultant external force due to any agent not part of system. Particlei does not exert a force on itself.
Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion ofrigid bodies. TheNewton–Euler equations combine the forces and torques acting on a rigid body into a single equation.
Newton's second law for rotation takes a similar form to the translational case,[13]
by equating thetorque acting on the body to the rate of change of itsangular momentumL. Analogous to mass times acceleration, themoment of inertiatensorI depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity,
Again, these equations apply to point like particles, or at each point of a rigid body.
Likewise, for a number of particles, the equation of motion for one particlei is[7]
whereLi is the angular momentum of particlei,τij the torque on particlei by particlej, andτE is resultant external torque (due to any agent not part of system). Particlei does not exert a torque on itself.
Some examples[14] of Newton's law include describing the motion of asimple pendulum,
and adamped, sinusoidally driven harmonic oscillator,
For describing the motion of masses due to gravity,Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of massm thrown in the air, in air currents (such as wind) described by a vector field of resistive forcesR =R(r,t),
whereG is thegravitational constant,M the mass of the Earth, andA =R/m is the acceleration of the projectile due to the air currents at positionr and timet.
The classicalN-body problem forN particles each interacting with each other due to gravity is a set ofN nonlinear coupled second order ODEs,
wherei = 1, 2, ...,N labels the quantities (mass, position, etc.) associated with each particle.
Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system hasNdegrees of freedom, then one can use a set ofNgeneralized coordinatesq(t) = [q1(t),q2(t) ...qN(t)], to define the configuration of the system. They can be in the form ofarc lengths orangles. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. Thetime derivatives of the generalized coordinates are thegeneralized velocities
TheEuler–Lagrange equations are[2][16]
where theLagrangian is a function of the configurationq and its time rate of changedq/dt (and possibly timet)
Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupledN second orderODEs in the coordinates are obtained.
Hamilton's equations are[2][16]
where the Hamiltonian
is a function of the configurationq and conjugate"generalized" momenta
in which∂/∂q =(∂/∂q1,∂/∂q2, …,∂/∂qN) is a shorthand notation for a vector ofpartial derivatives with respect to the indicated variables (see for examplematrix calculus for this denominator notation), and possibly timet,
Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled2N first order ODEs in the coordinatesqi and momentapi are obtained.
TheHamilton–Jacobi equation is[2]
where
isHamilton's principal function, also called theclassical action is afunctional ofL. In this case, the momenta are given by
Although the equation has a simple general form, for a given Hamiltonian it is actually a single first ordernon-linearPDE, inN + 1 variables. The actionS allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because anydifferentiablesymmetry of theaction of a physical system has a correspondingconservation law, a theorem due toEmmy Noether.
All classical equations of motion can be derived from thevariational principle known asHamilton's principle of least action
stating the path the system takes through theconfiguration space is the one with the least actionS.
In electrodynamics, the force on a charged particle of chargeq is theLorentz force:[17]
Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle:
or its momentum:
The same equation can be obtained using theLagrangian (and applying Lagrange's equations above) for a charged particle of massm and chargeq:[16]
whereA andϕ are the electromagneticscalar andvector potential fields. The Lagrangian indicates an additional detail: thecanonical momentum in Lagrangian mechanics is given by:instead of justmv, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.
Alternatively the Hamiltonian (and substituting into the equations):[16]can derive the Lorentz force equation.
The above equations are valid in flat spacetime. Incurvedspacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by ageodesic of the curved spacetime (the shortest length of curve between two points). For curvedmanifolds with ametric tensorg, the metric provides the notion of arc length (seeline element for details). Thedifferential arc length is given by:[19]: 1199
and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics:[19]: 1200
whereΓ μαβ is aChristoffel symbol of the second kind, which contains the metric (with respect to the coordinate system).
Given themass-energy distribution provided by thestress–energy tensorT αβ, theEinstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (seeequivalence principle). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - becausegravity is a fictitious force. Therelative acceleration of one geodesic to another in curved spacetime is given by thegeodesic deviation equation:
whereξα =x2α −x1α is the separation vector between two geodesics,D/ds (not justd/ds) is thecovariant derivative, andRαβγδ is theRiemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.[18]: 34–35
For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according toNewton's law of gravity.
In general relativity, rotational motion is described by therelativistic angular momentum tensor, including thespin tensor, which enter the equations of motion undercovariant derivatives with respect toproper time. TheMathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in agravitational field.
Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics ofwaves andfields are alwayspartial differential equations, since the waves or fields are functions of space and time. For a particular solution,boundary conditions along with initial conditions need to be specified.
Sometimes in the following contexts, the wave or field equations are also called "equations of motion".
Equations that describe the spatial dependence andtime evolution of fields are calledfield equations. These include
This terminology is not universal: for example although theNavier–Stokes equations govern thevelocity field of afluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.
Equations of wave motion are calledwave equations. The solutions to a wave equation give the time-evolution and spatial dependence of theamplitude. Boundary conditions determine if the solutions describetraveling waves orstanding waves.
From classical equations of motion and field equations; mechanical,gravitational wave, andelectromagnetic wave equations can be derived. The general linear wave equation in 3D is:
whereX =X(r,t) is any mechanical or electromagnetic field amplitude, say:[20]
andv is thephase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacingv byv(X). There are other linear and nonlinear wave equations for very specific applications, see for example theKorteweg–de Vries equation.
In quantum theory, the wave and field concepts both appear.
Inquantum mechanics the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is theSchrödinger equation in its most general form:
whereΨ is thewavefunction of the system,Ĥ is the quantumHamiltonian operator, rather than a function as in classical mechanics, andħ is thePlanck constant divided by 2π. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers thecorrespondence principle, in the limit thatħ becomes zero. To compare to measurements, operators for observables must be applied the quantum wavefunction according to the experiment performed, leading to either wave-like or particle-like results.
Throughout all aspects of quantum theory, relativistic or non-relativistic, there arevarious formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance:
The 5 symbols are remembered by "suvat". Given any three, the other two can be found.
