Equations of motion

Classical mechanics
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Branches Statics Dynamics Kinetics Kinematics Applied mechanics Celestial mechanics Continuum mechanics Statistical mechanics
Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics (Lagrangian mechanics Hamiltonian mechanics)
Fundamental concepts Space Time Mass Inertia Velocity Speed Acceleration Force Momentum Impulse Torque / Moment / Couple Angular momentum Moment of inertia Frame of reference Energy (kinetic potential) Mechanical work Mechanical power Virtual work D'Alembert's principle
Core topics Rigid body (dynamics Euler's equations) Motion (linear) Euler's laws of motion Newton's laws of motion Newton's law of universal gravitation Equations of motion Inertial / Non-inertial reference frame Fictitious force Mechanics of planar particle motion Displacement (vector) Relative velocity Friction Simple harmonic motion Harmonic oscillator Vibration Damping (ratio)
Rotational motion Circular motion (uniform non-uniform) Rotating reference frame Centripetal force Centrifugal force (rotating reference frame reactive) Coriolis force Pendulum Tangential / Rotational speed Angular acceleration Angular velocity Angular frequency Angular displacement
Scientists Galileo Newton Kepler Horrocks Halley Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Daniel / Johann Bernoulli Cauchy
v t e

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 motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum 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 a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics.

There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term 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 circumstances of 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, the main types of motion are translations, rotations, oscillations, or any combinations of these.

Historically, equations of motion initiated in classical mechanics and the extension to celestial mechanics, to describe the motion of massive objects. Later they appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields. With the advent of general relativity, the classical equations of motion became modified. 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.^[3] However, the equations of quantum mechanics can also be considered equations of motion, since they are differential equations of the wavefunction, 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, notably waves. These equations are explained below.

Introduction [edit]

Qualitative [edit]

Equations of motion generally involve:

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 appropriately for 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 [edit]

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

$M\left[\mathbf{r}(t),\mathbf{\dot{r}}(t),\mathbf{\ddot{r}}(t),t\right]=0$

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

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

$\mathbf{r}(0), \quad \mathbf{\dot{r}}(0).$

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

$\tilde{M}\left[\mathbf{p}(t),\mathbf{\dot{p}}(t),\mathbf{\ddot{p}}(t),t\right]=0$

with initial conditions (again constant values)

$\mathbf{p}(0), \quad \mathbf{\dot{p}}(0).$

The solution r (or p) to the equation of motion, combined with the initial values, describes the system for all times after t = 0. For more than one particle, there are separate equations for each (this is contrary to a statistical ensemble of many particles in statistical mechanics, and a many-particle system in 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 is non-linear, and may lead to chaotic behaviour depending on how sensitive the system is to the initial conditions.

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 two first order equations in the generalized coordinates and momenta:

$M\left[\mathbf{q}(t),\mathbf{\dot{q}}(t),t\right]=0,\quad \tilde{M}\left[\mathbf{p}(t),\mathbf{\dot{p}}(t),t\right]=0$

where q is a tuple of generalized coordinates and similarly p is the tuple of generalized momenta. The initial conditions are similarly defined.

Kinematic equations for one particle [edit]

Kinematic quantities [edit]

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

From the instantaneous position r = r (t) (instantaneous meaning at an instant value of time t), the instantaneous velocity v = v (t) and acceleration a = a (t) have the general, coordinate-independent definitions;^[5]

$\mathbf{v} = \frac{{\rm d} \mathbf{r}}{{\rm d} t}, \quad \mathbf{a} = \frac{{\rm d} \mathbf{v}}{{\rm d} t} = \frac{{\rm d}^2 \mathbf{r}}{{\rm d} t^2} \,\!$

z 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):

$\boldsymbol{\omega} = \mathbf{\hat{n}}\frac{{\rm d} \theta}{{\rm d} t}, \quad \boldsymbol{\alpha} = \frac{{\rm d} \boldsymbol{\omega}}{{\rm d} t} = \mathbf{\hat{n}}\frac{{\rm d}^2 \theta}{{\rm d} t^2} \,\!$

where

$\mathbf{\hat{n}} = \mathbf{\hat{e}}_r\times\mathbf{\hat{e}}_\theta \,\!$

is a unit axial vector, pointing parallel to the axis of rotation, $\scriptstyle \mathbf{\hat{e}}_r \,\!$ = unit vector in direction of r, $\scriptstyle \mathbf{\hat{e}}_\theta \,\!$ = unit vector tangential to the angle.

NB: 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 polar angle used in polar coordinate systems.

The following relations hold for a rotating rigid body:^[6]

$\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} \,\!$

$\mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r} + \boldsymbol{\omega} \times \mathbf{v} \,\!$

where r is a radial position.

Uniform acceleration [edit]

Constant linear acceleration [edit]

These equations apply to a particle moving linearly, in three dimensions in a straight line, with constant acceleration.^[7] Since the vectors are collinear (parallel, and lie on the same line) - only the magnitudes of the vectors are necessary, hence non-bold letters are used for magnitudes, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.

Derivation

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

$\begin{align} \mathbf{v} & = \int \mathbf{a} {\rm d}t = \mathbf{a}t+\mathbf{v}_0 \quad [1] \\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) {\rm d}t = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \quad [2] \\ \end{align}$

in magnitudes:

$\begin{align} v & = at+v_0 \quad [1] \\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \quad [2] \\ \end{align}$

One is the average velocity - since the velocity increases linearly, the average velocity multiplied by time is the 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):

$\mathbf{r} = \left( \frac{\mathbf{v}+\mathbf{v}_0}{2} \right )t \quad [3] \,\!$

in magnitudes

$r = r_0 + \left( \frac{v+v_0}{2} \right )t \quad [3] \,\!$

From [3]

$t = \left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )\,\!$

substituting for t in [1]:

$\begin{align} v & = a\left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )+v_0 \\ v\left( v+v_0 \right ) & = 2a\left( r - r_0 \right)+v_0\left( v+v_0 \right ) \\ v^2+vv_0 & = 2a\left( r - r_0 \right)+v_0v+v_0^2 \\ v^2 & = v_0^2 + 2a\left( r - r_0 \right)\quad [4] \\ \end{align}$

From [3]:

$2\left(r - r_0\right) - vt = v_0 t \,\!$

substituting into [2]:

$\begin{align} r & = \frac{{a}t^2}{2}+2r - 2r_0 - vt +r_0 \\ 0 & = \frac{{a}t^2}{2}+r - r_0 - vt \\ r & = r_0 + vt - \frac{{a}t^2}{2} \quad [5] \end{align}\,\!$

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

$\begin{align} v & = at+v_0 \quad [1]\\ r & = r_0 + v_0 t + \frac{{a}t^2}{2} \quad [2]\\ r & = r_0 + \left( \frac{v+v_0}{2} \right )t \quad [3]\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) \quad [4]\\ r & = r_0 + vt - \frac{{a}t^2}{2} \quad [5]\\ \end{align}$

where r₀ and v₀ are the particle's initial position and velocity, r, v, a are the final position (displacement), velocity and acceleration of the particle after the time interval.

Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g 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.

SUVAT equations [edit]

In elementary physics the above formulae are frequently written as:

$\begin{align} v & = u + at \quad [1] \\ s & = ut + \frac{1}{2} at^2 \quad [2] \\ s & = \frac{1}{2}(u + v)t \quad [3] \\ v^2 & = u^2 + 2as \quad [4] \\ s & = vt - \frac{1}{2}at^2 \quad [5] \\ \end{align}$

where u has replaced v₀, s replaces r, and s₀ = 0. They are often referred to as the "SUVAT" equations, eponymous from to the variables: s = displacement (s₀ = initial displacement), u = initial velocity, v = final velocity, a = acceleration, t = time.^[8]^[9]

Applications [edit]

Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into 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 these quantities 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, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration 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, we have:

$s= \frac{v^2 - u^2}{-2g}.$

Substituting and cancelling minus signs gives:

$s = \frac{u^2}{2g}.$

Constant circular acceleration [edit]

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

$\begin{align} \omega & = \omega_0 + \alpha t \\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\ \end{align}\,\!$

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 [edit]

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 polar coordinates in the context of physical quantities (like angular velocity ω).

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

$\begin{align} \mathbf{r} & =\mathbf{r}\left ( r,\theta, t \right ) = r \mathbf{\hat{e}}_r \\ \mathbf{v} & = \mathbf{\hat{e}}_r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \mathbf{\hat{e}}_\theta \\ \mathbf{a} & =\left ( \frac{\mathrm{d}^2 r}{\mathrm{d}t^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{\mathrm{d}r}{{\rm d}t} \right )\mathbf{\hat{e}}_\theta \end{align} \,\!$

where $\scriptstyle \mathbf{\hat{e}}_r, \mathbf{\hat{e}}_\theta, \,\!$ are the polar unit vectors. Notice for a the components (–rω²) and 2ωdr/dt are the centripetal and Coriolis accelerations respectively.

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 angular components are zero, and the non-zero component of motion describes uniform acceleration.

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]

Principle of least action

All classical equations of motion can be derived from this variational principle:

$\delta S = 0$

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

Euler-Lagrange equations

The Euler-Lagrange equations are:^[2]^[17]

$\frac{{\rm d}}{{\rm d} t} \left ( \frac{\partial L}{\partial \mathbf{\dot{q}} } \right ) = \frac{\partial L}{\partial \mathbf{q}}$

After substituting for the Lagrangian, evaluating the partial derivatives, and simplifying, a second order ODE in each q_i is obtained.

Hamilton's equations

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

$\mathbf{\dot{p}} = -\frac{\partial H}{\partial \mathbf{q}} \quad \mathbf{\dot{q}} = + \frac{\partial H}{\partial \mathbf{p}}$

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

$\mathbf{p} \rightleftharpoons \mathbf{q}, \quad H \rightarrow -H .$

After substituting the Hamiltonian, evaluating the partial derivatives, and simplifying, two first order ODEs in q_i and p_i are obtained.

Hamilton–Jacobi equation

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

$H = - \frac{\partial S}{\partial t}$

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 for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether.

Electrodynamics [edit]

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.

In electrodynamics, the force on a charged particle of charge q is the Lorentz force:^[18]

$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) \,\!$

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

$m\frac{{\rm d}^2 \mathbf{r}}{{\rm d}t^2} = q\left(\mathbf{E} + \frac{{\rm d} \mathbf{r}}{{\rm d}t} \times \mathbf{B}\right) \,\!$

or its momentum:

$\frac{{\rm d}\mathbf{p}}{{\rm d}t} = q\left(\mathbf{E} + \frac{\mathbf{p} \times \mathbf{B}}{m}\right) \,\!$

The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q:^[19]

$L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{\dot{r}}-q\phi$

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

$\mathbf{P} = \frac{\partial L}{\partial \mathbf{\dot{r}}} = m \mathbf{\dot{r}} + q \mathbf{A}$

instead of just mv, 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):^[17]

$H = \frac{\left(\mathbf{P} - q \mathbf{A}\right)^2}{2m} - q\phi \,\!$

can derive the Lorentz force equation.

Geodesic equation of motion [edit]

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).

Main articles: General relativity and Geodesic equation

The above equations are valid in flat spacetime. In curved space spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a geodesic of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor g, the metric provides the notion of arc length (see line element for details), the differential arc length is given by:^[20]

$\mathrm{d}s = \sqrt{g_{\alpha\beta} \mathrm{d} x^\alpha \mathrm{d}x^\beta}$

and the geodesic equation is a second-order differential equation in the coordinates, the general solution is a family of geodesics:^[21]

$\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}s^2} = - \Gamma^\mu{}_{\alpha\beta}\frac{\mathrm{d} x^\alpha}{\mathrm{d}s}\frac{\mathrm{d} x^\beta}{\mathrm{d}s}$

where Γ^μ_αβ 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 T^αβ, the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of space time is equivalent to a gravitational field (see principle of equivalence). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation:

$\frac{\mathrm{D}^2\xi^\alpha}{\mathrm{d}s^2} = -R^\alpha{}_{\beta\gamma\delta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}s}\xi^\gamma\frac{\mathrm{d}x^\delta}{\mathrm{d}s}$

where ξ^α = (x₂)^α − (x₁)^α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and R^α_βγδ 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.^[22]

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 to Newton's law of gravity.

Analogues for waves and fields [edit]

Field equations

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

the Navier–Stokes equations for the velocity field of a fluid,
Maxwell's equations for the electromagnetic field,
the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles).

Wave equations

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

From classical equations of motion and field equations; mechanical and electromagnetic wave equations can be derived. The general linear wave equation in 3d is:

$\nabla^2 X = \frac{1}{v^2}\frac{\partial^2 X}{\partial^2 t}$

where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:^[23]

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 on amplitude, replacing v by v(X). There are other wave equations for very specific applications, non-linear equations arise in different mathematical forms (see for example the Korteweg–de Vries equation).

In quantum mechanics, the analogue of the equation of motion is the Schrödinger equation:

$\hat{H}\Psi = i\hbar\frac{\partial\Psi}{\partial t} \,\!$

where $\hat{H}\,\!$ is the Hamiltonian operator (rather than a function as above), Ψ is the wavefunction and ħ is the reduced Planck constant. Setting up the Hamiltonian and inserting it into the equation results in a differential equation, the solution is the wavefunction as a function of space and time. There are also relativistic wave equations used in quantum field theory.

External links [edit]

Equations of Motion Applet

References [edit]

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