Equations of Motion - Wikipedia, The Free Encyclopedia

Equations of motionFrom Wikipedia, the free encyclopedia

In mathematical physics, equations of motion are equations that describe the behaviour of a physical

system in terms of its motion as a function of time.[1] More specifically, the equations of motiondescribe the behaviour of a physical system as a set of mathematical functions in terms of dynamicvariables: normally spatial coordinates and time are used, but others are also possible, such asmomentum components and time. The most general choice are generalized coordinates which can be

any convenient variables characteristic of the physical system.[2] The functions are defined in aEuclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamicsof a system is known, the equations are the solutions to the differential equations describing themotion of the dynamics.

There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, sincemomenta, forces and energy of the particles are taken into account. In this instance, sometimes theterm refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.

However, kinematics is simpler as it concerns only spatial and time-related variables. In circumstancesof constant acceleration, these simpler equations of motion are usually referred to as the "SUVAT"equations, arising from the definitions of kinematic quantities: displacement (S), initial velocity (U),final velocity (V), acceleration (A), and time (T). (see below).

Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, themain types of motion are translations, rotations, oscillations, or any combinations of these.

Historically, equations of motion initiated in classical mechanics and the extension to celestialmechanics, to describe the motion of massive objects. Later they appeared in electrodynamics, whendescribing the motion of charged particles in electric and magnetic fields. With the advent of generalrelativity, the classical equations of motion became modified. In all these cases the differentialequations 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.[3] However, the equations of quantummechanics 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 timecoordinates of the particles. There are analogs of equations of motion in other areas of physics,notably waves. These equations are explained below.

Contents

1 Introduction

1.1 Qualitative

1.2 Quantitative

2 Kinematic equations for one particle

2.1 Kinematic quantities

Equations of motion - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Equations_of_motion

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2.2 Uniform acceleration

2.2.1 Constant linear acceleration: collinear vectors

2.2.2 Constant linear acceleration: non-collinear vectors

2.2.3 Applications

2.2.4 Constant circular acceleration

2.3 General planar motion

2.4 General 3d motion

2.5 Harmonic motion of one particle

2.5.1 Translation

2.5.2 Rotation

3 Dynamic equations of motion

3.1 Newtonian mechanics

3.1.1 Newton's second law for translation

3.1.2 Newton's (Euler's) second law for rotation

3.1.3 Applications

3.2 Eulerian mechanics

3.3 Newton–Euler equations

4 Analytical mechanics

4.1 Constraints and motion

4.2 Generalized classical equations of motion

5 Electrodynamics

6 General relativity

6.1 Geodesic equation of motion

6.2 Spinning objects

7 Analogues for waves and fields

7.1 Field equations

7.2 Wave equations

7.3 Quantum theory

8 See also

9 References

Introduction

Qualitative

Equations of motion typically involve:


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a differential equation of motion, usually identified as some physical law and applying definitions

of physical quantities, is used to set up an equation for the problem,

setting the boundary and initial value conditions,

a function of the position (or momentum) and time variables, describing the dynamics of the

system,

solving the resulting differential equation subject to the boundary and initial conditions.

The differential equation is a general description of the application and may be adjusted appropriatelyfor a specific situation, the solution describes exactly how the system will behave for all times after the

initial conditions, and according to the boundary conditions.[1][4]

Quantitative

In Newtonian mechanics, an equation of motion M takes the general form of a second order ordinarydifferential equation (ODE) in the position r (see below for details) of the object:

where t is time, and each overdot denotes a time derivative.

The initial conditions are given by the constant values at t = 0:

Another dynamical variable is the momentum p of the object, which can be used instead of r (thoughless commonly), i.e. a second order ODE in p:

with initial conditions (again constant values)

The solution r (or p) to the equation of motion, combined with the initial values, describes the systemfor all times after t = 0. For more than one particle, there are separate equations for each (this iscontrary to a statistical ensemble of many particles in statistical mechanics, and a many-particle systemin quantum mechanics - where all particles are described by a single probability distribution).Sometimes, the equation will be linear and can be solved exactly. However in general, the equation isnon-linear, and may lead to chaotic behaviour depending on how sensitive the system is to the initialconditions.

In the generalized Lagrangian mechanics, the generalized coordinates q (or generalized momenta p)replace the ordinary position (or momentum). Hamiltonian mechanics is slightly different, there are twofirst order equations in the generalized coordinates and momenta:

where q is a tuple of generalized coordinates and similarly p is the tuple of generalized momenta. The


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Kinematic quantities of a classical particle of

mass m: position r, velocity v, acceleration a.

initial conditions are similarly defined.

Kinematic equations for one particle

Kinematic quantities

From the instantaneous position r = r (t), instantaneousmeaning at an instant value of time t, the instantaneousvelocity v = v (t) and acceleration a = a (t) have the

general, coordinate-independent definitions;[5]

Notice that velocity always points in the direction ofmotion, in other words for a curved path it is thetangent vector. Loosely speaking, first order derivativesare related to tangents of curves. Still for curved paths,the acceleration is directed towards the center ofcurvature of the path. Again, loosely speaking, secondorder derivatives are related to curvature.

The rotational analogues are the angular position (angle the particle rotates about some axis) θ = θ(t),angular velocity ω = ω(t), and angular acceleration a = a(t):

where

is a unit axial vector, pointing parallel to the axis of rotation, is the unit vector in direction of r, and is the unit vector tangential to the angle. In these rotational definitions, the angle can be any angle

about the specified axis of rotation. It is customary to use θ, but this does not have to be the polarangle used in polar coordinate systems.

The following relations hold for a point-like particle, orbiting about some axis with angular velocity

ω:[6]

where r is a radial position, v the tangential velocity of the particle, and a the particle's acceleration.More generally, these relations hold for each point in a rotating continuum rigid body.

Uniform acceleration


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Constant linear acceleration: collinear vectors

These equations apply to a particle moving linearly, in three dimensions in a straight line, with constant

acceleration.[7] Since the position, velocity, and acceleration are collinear (parallel, and lie on the sameline) - only the magnitudes of these vectors are necessary, and because the motion is along a straightline, the problem effectively reduces from three dimensions to one.

where:

is the particle's initial position

is the particle's final position

is the particle's initial velocity

is the particle's final velocity

is the particle's acceleration

is the time interval

Derivation

Two arise from integrating the definitions of velocity and acceleration:[7]

in magnitudes:

One is the average velocity - since the velocity increases linearly, the average velocity multiplied by time isthe distance travelled while increasing the velocity from v0 to v (this can be illustrated graphically by

plotting velocity against time as a straight line graph):


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Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, thestandard gravity g is used. Note that each of the equations contains four of the five variables, so in thissituation it is sufficient to know three out of the five variables to calculate the remaining two.

In elementary physics the same formulae are frequently written in different notation as:

in magnitudes

From [3]

substituting for t in [1]:

From [3]:

substituting into [2]:

Usually only the first 4 are needed, the fifth is optional.


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where u has replaced v0, s replaces r, and s0 = 0. They are often referred to as the "SUVAT"

equations, where "SUVAT" is an acronym from the variables: s = displacement (s0 = initial

displacement), u = initial velocity, v = final velocity, a = acceleration, t = time.[8][9]

Constant linear acceleration: non-collinear vectors

In the case of the initial position, initial velocity, and acceration vectors not being collinear, theequations can be extended to account for the non-collinearity using the vector dot product. Thederivations are essentially the same as in the collinear case:

although the Torricelli equation [4] can be derived using the distributive property of the dot product asfollows:

Applications

Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwardsinto the air. Given initial speed u, one can calculate how high the ball will travel before it begins to fall.The acceleration is local acceleration of gravity g. At this point one must remember that while thesequantities appear to be scalars, the direction of displacement, speed and acceleration is important.They could in fact be considered as uni-directional vectors. Choosing s to measure up from the ground,


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the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also theacceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using equation [4] in the set above, wehave:

Substituting and cancelling minus signs gives:

Constant circular acceleration

The analogues of the above equations can be written for rotation. Again these axial vectors must all beparallel (to the axis of rotation), so only the magnitudes of the vectors are necessary:

where α is the constant angular acceleration, ω is the angular velocity, ω0 is the initial angular velocity,

θ is the angle turned through (angular displacement), θ0 is the initial angle, and t is the time taken to

rotate from the initial state to the final state.

General planar motion

These are the kinematic equations for a particle traversing a path in a plane, described by position r =

r(t).[10] They are actually no more than the time derivatives of the position vector in plane polarcoordinates using the definitions of physical quantities (like angular velocity ω).

The position, velocity and acceleration of the particle are respectively:

where are the polar unit vectors. Notice for a the components (–rω2) and 2ωdr/dt are thecentripetal and Coriolis accelerations respectively.


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Special cases of motion described be 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 angularcomponents are zero, and the non-zero component of motion describes uniform acceleration.

State of motion Constant r Linear r Quadratic r Non-linear r

Constant θ Stationary

Uniform

translation

(constant

translational

velocity)

Uniform

translational

acceleration

Non-uniform

translation

Linear θ

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

Quadratic θ

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

Non-linear θ

Non-uniform

angular

acceleration in a

circle

Non-uniform

angular

acceleration

in a spiral,constant radialvelocity

Non-uniform

angular

acceleration

in a spiral,constant radialacceleration

Non-uniform

angular

acceleration

in a spiral, varyingradial acceleration

General 3d motion

In 3d space, the equations become more complicated and unwieldy in spherical coordinates (r, θ, ϕ)with corresponding unit vectors , the position, velocity, and acceleration are respectively:


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In the case of a constant ϕ this reduces to the planar equations above.

Harmonic motion of one particle

Translation

The kinematic equation of motion for a simple harmonic oscillator (SHO), oscillating in one dimension(the ±x direction) in a straight line is:

where ω is the angular frequency of the oscillatory motion, related to the general frequency f and thetime period T (time taken for one cycle of oscillation):

Many systems approximately execute simple harmonic motion (SHM). The complex harmonic oscillator

is a superposition of simple harmonic oscillators:[7]

It is possible for simple harmonic motions to occur in any direction:[11]

known as a multidimensional harmonic oscillator. In cartesian coordinates, each component of theposition will be a superposition of sinusiodal SHM.

Rotation


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The rotational analogue of SHM in a straight line is angular oscillation about an axle or fulcrum:

where ω is still the angular frequency of the oscillatory motion - though not the angular velocity whichis the rate of change of θ.

This form can be identified (at least approximately) as libration. The complex analogue is again asuperposition of simple harmonic oscillators:

Dynamic equations of motion

Newtonian mechanics

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 themis not easy. Often there is an excess of variables to solve for the problem completely, so Newton's lawsare not the most efficient method for generally finding and solving for the motion of a particle. Insimple cases of rectangular geometry, the use of Cartesian coordinates works fine, but othercoordinate systems can become dramatically complex.

Newton's second law for translation

The first developed and most famous is Newton's second law of motion, there are several ways to write

and use it, the most general is:[12]

where p = p(t) is the momentum of the particle and F = F(t) is the resultant external force acting onthe particle (not any force the particle exerts) - in each case at time t. The law is also written morefamously as:

since m is a constant in Newtonian mechanics. However the momentum form is preferable since this isreadily generalized to more complex systems, generalizes to special and general relativity (seefour-momentum), and since momentum is a conserved quantity; with deeper fundamental significance

than the position vector or its time derivatives.[12]

For a number of particles (see many body problem), the equation of motion for one particle i

influenced by other particles is:[5][13]


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where pi = momentum of particle i, Fij = force on particle i by particle j, and FE = resultant external

force (due to any agent not part of system). Particle i does not exert a force on itself.

Newton's (Euler's) second law for rotation

For rigid bodies, Newton's second law for rotation takes the same form as for translation:[14]

where L is the angular momentum. Analogous to force and acceleration:

where I is the moment of inertia tensor. Likewise, for a number of particles, the equation of motion for

one particle i is:[15]

where Li = angular momentum of particle i, τij = torque on particle i by particle j, and τE = resultant

external torque (due to any agent not part of system). Particle i does not exert a torque on itself.

Applications

Some examples[11] of Newton's law include describing the motion of a pendulum:

a damped, driven harmonic oscillator:

or a ball thrown in the air, in air currents (such as wind) described by a vector field of resistive forces R= R(r, t):

where G = gravitational constant, M = mass of the Earth and A = R/m is the acceleration of theprojectile due to the air currents at position r and time t. Newton's law of gravity has been used. The


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mass m of the ball cancels.

Eulerian mechanics

Euler developed Euler's laws of motion, analogous to Newton's laws, for the motion of rigid bodies.

Newton–Euler equations

The Newton–Euler equations combine Euler's equations into one.

Analytical mechanics

Constraints and motion

Using all three coordinates of 3d space is unnecessary if there are constraints on the system.Generalized coordinates q(t) = [q1(t), q2(t) ... qN(t)], where N is the total number of degrees of

freedom the system has, are any set of coordinates used to define the configuration of the system, inthe form of arc lengths or angles. They are a considerable simplification to describe motion since theytake advantage of the intrinsic constraints that limit the system's motion - i.e. the number ofcoordinates is reduced to a minimum, rather than demanding rote algebra to describe the constraintsand the motion using all three coordinates.

Corresponding to generalized coordinates are:

their time derivatives, the generalized velocities: ,

conjugate "generalized" momenta: ,

(see matrix calculus for the denominator notation) where

the Lagrangian is a function of the configuration q, the rate of change of configuration dq/dt, and

time t: ,

the Hamiltonian is a function of the configuration q, motion p, and time t:

, and

Hamilton's principal function, also called the classical action is a functional of L:

.

The Lagrangian or Hamiltonian function is set up for the system using the q and p variables, thenthese are inserted into the Euler–Lagrange or Hamilton's equations to obtain differential equations ofthe system. These are solved for the coordinates and momenta.

Generalized classical equations of motion

Principle of least action


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As the system evolves, q traces a path

through configuration 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).[16]

All classical equations of motion can be derived from thisvariational principle:

stating the path the system takes through the configurationspace is the one with the least action.

Euler–Lagrange equations

The Euler–Lagrange equations are:[2][17]

After substituting for the Lagrangian, evaluating the partialderivatives, and simplifying, a second order ODE in each qi is

obtained.

Hamilton's equations

Hamilton's equations are:[2][17]

Notice the equations are symmetric (remain in the same form) by making these interchangessimultaneously:

After substituting the Hamiltonian, evaluating the partial derivatives, and simplifying, two first orderODEs in qi and pi are obtained.

Hamilton–Jacobi equation

Hamilton's formalism can be rewritten as:[2]

Although the equation has a simple form, it's actually a non-linear PDE, first order in N + 1 variables,rather than 2N such equations. Due to the action S, it can be used to identify conserved quantities formechanical systems, even when the mechanical problem itself cannot be solved fully, because anydifferentiable symmetry of the action of a physical system has a corresponding conservation law, atheorem due to Emmy Noether.


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Lorentz force f on a charged

particle (of charge q) in motion

(instantaneous velocity v). The

E field and B field vary in

space and time.

Electrodynamics

In electrodynamics, the force on a charged particle of charge q is the

Lorentz force:[18]

Combining with Newton's second law gives a first order differentialequation of motion, in terms of position of the particle:

or its momentum:

The same equation can be obtained using the Lagrangian (andapplying Lagrange's equations above) for a charged particle of mass

m and charge q:[19]

where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates anadditional detail: the canonical momentum in Lagrangian mechanics is given by:

instead of just mv, implying the motion of a charged particle is fundamentally determined by the massand charge of the particle. The Lagrangian expression was first used to derive the force equation.

Alternatively the Hamiltonian (and substituting into the equations):[17]

can derive the Lorentz force equation.

General relativity

Geodesic equation of motion

The above equations are valid in flat spacetime. In curved space spacetime, things becomemathematically more complicated since there is no straight line; this is generalized and replaced by a


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Geodesics on a sphere are arcs

of great circles (yellow curve).

On a 2d–manifold (such as the

sphere shown), the direction of

the accelerating geodesic is

uniquely fixed if the separation

vector ξ is orthogonal to the

"fiducial geodesic" (green

curve). As the separation

vector ξ0 changes to ξ after a

distance s, the geodesics are

not parallel (geodesic

deviation).[20]

geodesic of the curved spacetime (the shortest length of curvebetween two points). For curved manifolds with a metric tensor g, themetric provides the notion of arc length (see line element for details),

the differential arc length is given by:[21]

and the geodesic equation is a second-order differential equation in

the coordinates, the general solution is a family of geodesics:[22]

where Γμαβ is a Christoffel symbol of the second kind, which contains

the metric (with respect to the coordinate system).

Given the mass-energy distribution provided by the stress–energy

tensor Tαβ, the Einstein field equations are a set of non-linearsecond-order partial differential equations in the metric, and imply thecurvature of space time is equivalent to a gravitational field (seeprinciple of equivalence). Mass falling in curved spacetime isequivalent to a mass falling in a gravitational field - because gravity isa fictitious force. The relative acceleration of one geodesic to anotherin curved spacetime is given by the geodesic deviation equation:

where ξα = (x2)α − (x1)

α is the separation vector between two geodesics, D/ds (not just d/ds) is the

covariant derivative, and Rαβγδ is the Riemann 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.[23]

For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesicequation has the solutions of straight lines. This is also the limiting case when masses move accordingto Newton's law of gravity.

Spinning objects

In general relativity, rotational motion is described by the relativistic angular momentum tensor,including the spin tensor, which enter the equations of motion under covariant derivatives with respectto proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objectsmoving in a gravitational field.

Analogues for waves and fields


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Unlike the equations of motion for describing particle mechanics, which are systems of coupledordinary differential equations, the analogous equations governing the dynamics of waves and fieldsare always partial differential equations, since the waves or fields are functions of space and time.Sometimes in the following contexts, the wave or field equations are also called "equations of motion".

Field equations

Equations that describe the spatial dependence and time evolution of fields are called field equations.These include

Maxwell's equations for the electromagnetic field,

Poisson's equation for Newtonian gravitational or electrostatic field potentials,

the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak

gravitational fields and low velocities of particles).

This terminology is not universal: for example although the Navier–Stokes equations govern thevelocity field of a fluid, they are not usually called "field equations", since in this context they representthe momentum of the fluid and are called the "momentum equations" instead.

Wave equations

Equations of wave motion are called wave equations. The solutions to a wave equation give thetime-evolution and spatial dependence of the amplitude. Boundary conditions determine if thesolutions describe traveling waves or standing 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:

where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:[24]

the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc.,

the fluctuating pressure of a medium, sound pressure,

the electric fields E or D, or the magnetic fields B or H,

the voltage V or current I in an alternating current circuit,

and v is the phase velocity. Non-linear equations model the dependence of phase velocity onamplitude, replacing v by v(X). There are other linear and non-linear wave equations for very specificapplications, see for example the Korteweg–de Vries equation.

Quantum theory

In quantum theory, the wave and field concepts both appear.

In quantum mechanics, in which particles also have wave-like properties according to wave–particle


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duality, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation,Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form:

where Ψ is the wavefunction of the system, is the quantum Hamiltonian operator, rather than a

function as in classical mechanics, and ħ is the Planck constant divided by 2π. Setting up theHamiltonian and inserting it into the equation results in a wave equation, the solution is thewavefunction as a function of space and time. The Schrödinger equation itself reduces to theHamilton–Jacobi equation in when one considers the correspondence principle, in the limit that ħbecomes zero.

Applying special relativity to quantum mechanics results in their unification as relativistic quantummechanics; this is achieved by inserting relativistic Hamiltonians into the Schrödinger equation, leadingto relativistic wave equations.

In the context of relativistic and non-relativistic quantum field theory, in which particles are interpretedand treated as fields rather than waves, the Schrödinger equation above has solutions Ψ which areinterpreted as fields.

Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulationsalternative to the Schrödinger equation that govern the time evolution and behavior of a quantumsystem, for instance:

the Heisenberg equation of motion resembles the time evolution of classical observables as

functions of position, momentum, and time, if one replaces dynamical observables by their

quantum operators and the classical Poisson bracket by the commutator,

the phase space formulation closely follows classical Hamiltonian mechanics, placing position and

momentum on equal footing,

the Feynman path integral formulation extends the principle of least action to quantum

mechanics and field theory, placing emphasis on the use of a Lagrangians rather than

Hamiltonians.

See also

Scalar (physics)

Vector

Distance

Displacement

Speed

Velocity

Acceleration

Angular displacement

Equations for a falling body

Parabolic trajectory

Curvilinear coordinates

Orthogonal coordinates

Newton's laws of motion

Torricelli's equation

Euler–Lagrange equation

Generalized forces


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Angular speed

Angular velocity

Angular acceleration

Defining equation (physics)

Newton–Euler laws of motion for a rigid

body

References

Encyclopaedia of Physics (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN

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Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-02.

Halliday, David; Resnick, Robert; Walker, Jearl (2004-06-16). Fundamentals of Physics (7 Sub ed.). Wiley.

ISBN 0-471-23231-9.

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Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-85.

M.R. Spiegel, S. Lipcshutz, D. Spellman (2009). Vector Analysis. Schaum's Outlines (2nd ed.). McGraw Hill.

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"Relativity, J.R. Forshaw 2009"15.

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Electromagnetism (second edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN

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