Equations of motion

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. More specifically, they are usually differential equations in terms of some function 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.

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 as they do from the definitions of kinematic quantities: displacement (S), velocity (U and 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 become 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. 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 analogous of equations of motion in other areas of physics, notably waves. These equations are explained below.

Introduction

Equations of motion generally involve the following scheme. A general differential equation of motion, identified as some physical law, is used to set up a specific equation to the problem, in doing so the boundary and initial value conditions are set. Some function describing the system as a function of the position and time coordinates. The resulting differential equation is then solved for the function. The differential equation is a general description of the situation, the solution describes exactly how the system will behave for all times after the initial conditions.

Kinematic equations for one particle

Position vector

In any are differential equations of motion, the position vector is the most sought-after quantity becuase this function describes the motion of the particle - its location relative to a given coordinate system at some time t. In three dimensions, it is a function of any set of spatial coordinates, such as Cartesian, spherical polar and Cylindrical polar coordinates, and time, which can also be a parameter (common variable) of each coordinate;

${\begin{aligned}{\mathbf {r}}&\equiv {\mathbf {r}}\left(x,y,z,t\right)\equiv x(t){\mathbf {\hat {e}}}_{x}+y(t){\mathbf {\hat {e}}}_{y}+z(t){\mathbf {\hat {e}}}_{z}\\&\equiv {\mathbf {r}}\left(r,\theta ,\phi ,t\right)\equiv r(t){\mathbf {\hat {e}}}_{r}+\theta (t){\mathbf {\hat {e}}}_{\theta }+\phi (t){\mathbf {\hat {e}}}_{\phi }\\&\equiv {\mathbf {r}}\left(r,\theta ,z,t\right)\equiv r(t){\mathbf {\hat {e}}}_{r}+\theta (t){\mathbf {\hat {e}}}_{\theta }+z(t){\mathbf {\hat {e}}}_{z}\\&\,\!\cdots \\\end{aligned}}$

These are different representations for the position vector. Any set of three dimensional coordinates and their corresponding unit vectors can be used to define the motion - whichever is the simplist may be used. Each coordinate can be parameterized using time since each succesive value of time corresponds to a sequence of succesive spatial locations given by the coordinates, so the continuum limit of many succesive locations is the path the particle traces.

NB: The coordinates are the magnitudes of the vector components, which are scalar multiplied by corresponding unit vectors, then these are vector-added to obatian the full vector. Coordinates and vectors are not exactly the same thing, but are closley related (see linear algebra, coordinate vector, and basis vector). In the above representations, x, y, z, r, θ, ϕ are coordinates, and :$\scriptstyle {\mathbf {\hat {e}}}_{x},{\mathbf {\hat {e}}}_{y},{\mathbf {\hat {e}}}_{z},{\mathbf {\hat {e}}}_{r},{\mathbf {\hat {e}}}_{\theta }{\mathbf {\hat {e}}}_{\phi }\,\!$ are unit vectors in the directions of the matching coordinate axes respectivley.

In the case of one dimension, the position has only one component, so it effectively degenerates to a scalar coordinate. It could be, say, a vector in the x-direction, or the radial direction. Frequently s is used for an arbitary one-dimensional displacement vector. Explicitley;

${\mathbf {x}}\equiv x\equiv x(t),\quad r\equiv r(t),\quad s\equiv s(t)\cdots \,\!$

Kinematic quantities

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;

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

The rotational analogues are the angular position (angle the particle rotates about some axis) θ = θ(t), angualar 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 costomary to use θ, but this does not have to be the polar angle used in polar coordinate systems.

For a rotating rigid body, the following relations useful for describing the motion hold:

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

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

where r is a radial position.

Uniform acceleration

Constant linear acceleration

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

${\begin{aligned}{\mathbf {v} }&=\int {\mathbf {a} }{\rm {d}}t={\mathbf {a} }t+{\mathbf {v} }_{0}\quad [1]\\{\mathbf {r} }&=\int ({\mathbf {a} }t+{\mathbf {v} }_{0}){\rm {d}}t={\frac {{\mathbf {a} }t^{2}}{2}}+{\mathbf {v} }_{0}t+{\mathbf {r} }_{0}\quad [2]\\\end{aligned}}$

in magnitudes:

${\begin{aligned}v&=at+v_{0}\quad [1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\quad [2]\\\end{aligned}}$

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]:

${\begin{aligned}v&=a\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)+v_{0}\\v\left(v+v_{0}\right)&=2a\left(r-r_{0}\right)+v_{0}\left(v+v_{0}\right)\\v^{2}+vv_{0}&=2a\left(r-r_{0}\right)+v_{0}v+v_{0}^{2}\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\\end{aligned}}$

From [3]:

$2\left(r-r_{0}\right)-vt=v_{0}t\,\!$

substituting into [2]:

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

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

${\begin{aligned}v&=at+v_{0}\quad [1]\\r&={\frac {at^{2}}{2}}+v_{0}t+r_{0}\quad [2]\\r&=\left({\frac {v+v_{0}}{2}}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\r&=r_{0}+vt-{\frac {{a}t^{2}}{2}}\quad [5]\\\end{aligned}}$

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

In elementary physics the above formulae are frequently written:

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

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.

Applications

These formulae have limited application - they can only be used in situations where velocities are much less that the speed of light and accelerations can be neglected.

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

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:

${\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}\,\!$

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

Main article: General planar motion

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t). Although useful in physics, these are no more than the time derivatives of plane polar coordinates.

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

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

where $\scriptstyle {\mathbf {\hat {e}}}_{r},{\mathbf {\hat {e}}}_{\theta },\,\!$ are the polar unit vectors. They can be readily derived by vector geometry and using kinematic definitions above, and prove to be very useful. Special cases of the forms of motion which result are summarized 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	$r=r\left(t\right)\,\!$ Constant	$r=r\left(t\right)\,\!$ Varies linearly	$r=r\left(t\right)\,\!$ Varies quadratically	$r=r\left(t\right)\,\!$ Varies non-linearly
$\theta =\theta \left(t\right)\,\!$ Constant	Stationary	Uniform translation (constant translational velocity)	Uniform translational acceleration	Non-uniform translation
$\theta =\theta \left(t\right)\,\!$ Varies linearly	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
$\theta =\theta \left(t\right)\,\!$ Varies quadratically	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
$\theta =\theta \left(t\right)\,\!$ Varies non-linearly	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

General 3d motion

Main article: Spherical coordinate system

It is certainly possible to derive analogue equations for motion in 3d space, but the equations become more complicated and unwieldy. Using the spherical coordinates (r, θ, ϕ) with corresponding unit vectors $\scriptstyle {\mathbf {\hat {e}}}_{r},{\mathbf {\hat {e}}}_{\theta },{\mathbf {\hat {e}}}_{\phi }\,\!$.

The position, velocity, and acceleration are respectively:

${\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r{\mathbf {\hat {e}}}_{r}\\\mathbf {v} &=v{\mathbf {\hat {e}}}_{r}+r\,{\frac {{\rm {d}}\theta }{{\rm {d}}t}}{\mathbf {\hat {e}}}_{\theta }+r\,{\frac {{\rm {d}}\phi }{{\rm {d}}t}}\,\sin \theta {\mathbf {\hat {e}}}_{\phi }\\\mathbf {a} &=\left(a-r\left({\frac {{\rm {d}}\theta }{{\rm {d}}t}}\right)^{2}-r\left({\frac {{\rm {d}}\phi }{{\rm {d}}t}}\right)^{2}\sin ^{2}\theta \right){\mathbf {\hat {e}}}_{r}\\&+\left(r{\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}+2v{\frac {{\rm {d}}\theta }{{\rm {d}}t}}-r\left({\frac {{\rm {d}}\phi }{{\rm {d}}t}}\right)^{2}\sin \theta \cos \theta \right){\mathbf {\hat {e}}}_{\theta }\\&+\left(r{\frac {{\rm {d}}^{2}\phi }{{\rm {d}}t^{2}}}\,\sin \theta +2v\,{\frac {{\rm {d}}\phi }{{\rm {d}}t}}\,\sin \theta +2r\,{\frac {{\rm {d}}\theta }{{\rm {d}}t}}\,{\frac {{\rm {d}}\phi }{{\rm {d}}t}}\,\cos \theta \right){\mathbf {\hat {e}}}_{\phi }\end{aligned}}\,\!$

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

Harmonic motion of one particle

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

${\frac {{\rm {d}}^{2}x}{{\rm {d}}t}}=\omega ^{2}x$

where ω is the angular frequency of the oscillator. Many systems approximately execute simple harmonic motion (SHM). The complex harmonic oscillator is a superposition of simple harmonic oscillators:

${\frac {{\rm {d}}^{2}x}{{\rm {d}}t}}=\sum _{n}\omega _{n}^{2}x$

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

${\frac {{\rm {d}}^{2}{\mathbf {r}}}{{\rm {d}}t}}=\sum _{n}\omega _{n}^{2}{\mathbf {r}}_{n}$

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

Dynamic equations of motion

Dynamic quantities

Some important dynamic quantities needed to describe forces are momentum p, angular momentum L and moment of inertia I are as follows:

${\mathbf {p}}=m{\mathbf {v}},\quad {\mathbf {L}}={\mathbf {r}}\times {\mathbf {p}},\quad I=\sum _{i}m_{i}r_{i}^{2}\,\!$

Another quantity, which is useful for simplification of forces but not essential, is the instantaneous mass moment:

$\mathbf {m} =mR{\mathbf {\hat {e}}}_{R}\,\!$

where R = R (t) = instantaneous radius of curvature at r on the curve, and $\scriptstyle {\mathbf {\hat {e}}}_{r}\,\!$ = unit vector directed to centre of circle of curvature.

Some useful relations for describing the motion of rotating rigid bodies, analogous to the above, are:

$\mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} \,\!$

$\mathbf {L} =\mathbf {I} \cdot {\boldsymbol {\omega }}\,\!$

Newtonian mechanics

Main article: 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 them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not the most efficient method for generally finding and solving for the motion of a particle. In simple cases of rectangular geometry, the use of Cartesian coordinates works fine, but other coordinate systems can become dramatically complex.

Newton's 2nd law for translation

The first developed and most famous is Newton's 2nd law of motion:

$\mathbf {F} ={\frac {{\rm {d}}\mathbf {p} }{{\rm {d}}t}}\,\!$

where p = p(t) is the momentum of the particle and F = F(t) is the resultant external force acting on the particle (not any force the particle exerts) - in each case at time t. This is a differential equation of momentum, so solving this equation obtains the momentum vector as a function of time. This version of the formula is actually not much use, since the momentum is simply

$\mathbf {p} =m{\mathbf {v}}=m{\frac {{\rm {d}}\mathbf {r} }{{\rm {d}}t}}\,\!$

for a particle at position r = r(t), velocity v = v(t), and of mass m = m(t), again each at time t. In general mass can vary with time, if the system gains or loses mass in some way. Velocity v is usually known as a function of time. Hence in usual application, Newton's 2nd law is a differential equation of position and time, rather than momentum and time, using the product rule for differentiation:

${\begin{aligned}\mathbf {F} &=m{\frac {{\rm {d}}\mathbf {v} }{{\rm {d}}t}}+\mathbf {v} {\frac {{\rm {d}}m}{{\rm {d}}t}}\\&=m{\frac {{\rm {d}}^{2}\mathbf {r} }{{\rm {d}}t^{2}}}+{\frac {{\rm {d}}\mathbf {r} }{{\rm {d}}t}}{\frac {{\rm {d}}m}{{\rm {d}}t}}\\\end{aligned}}\,\!$

where a = acceleration of particle. This is a differential equation in terms of position r. The solution is r as a function of the spatial coordinates and time (in general), for example in cartesian coordinates r = (x, y, z, t), or in spherical polar coordinates r = (r, θ, ϕ, t). Time is usually a parameter of the spatial coordinates, so explicitly r = (x(t), y(t), z(t), t).

For a number of particles, the equation of motion for one particle i is:

${\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!$

where p_i = momentum of particle i, F_ij = force ON particle i BY particle j, F_ij = force ON body j BY particle i, and F_E = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself.

Newton's 2nd law for rotation

The analogue for rotation is:

${\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}\,\!$

where $\scriptstyle {\boldsymbol {\tau }}\,\!$ denotes the torque acting on the body. Again this form isn't much use, so re-writing it:

${\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {r} }{{\rm {d}}t}}\times {\mathbf {p}}+\mathbf {r} \times {\frac {{\rm {d}}\mathbf {p} }{{\rm {d}}t}}\,\!$

that is, the resultant torque acting on the system is the moment of the resultant force:

${\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \,\!$

since momentum and velocity are always parallel by definition:

$\mathbf {v} \times {\mathbf {p}}={\boldsymbol {0}}\,\!$

For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation:

${\mathbf {L}}=I{\boldsymbol {\omega }}\,\!$

again by the product rule:

${\begin{aligned}{\boldsymbol {\tau }}&=I{\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}+{\boldsymbol {\omega }}{\frac {{\rm {d}}I}{{\rm {d}}t}}\\&={\mathbf {\hat {n}}}\left(I{\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}+{\frac {{\rm {d}}\theta }{{\rm {d}}t}}{\frac {{\rm {d}}I}{{\rm {d}}t}}\right)\\\end{aligned}}\,\!$

Likewise, for a number of particles, the equation of motion for one particle i is:

${\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,\!$

where L_i = angular momentum of particle i, $\scriptstyle {\boldsymbol {\tau }}_{ij}\,\!$ = torque ON particle i BY particle j, $\scriptstyle {\boldsymbol {\tau }}_{ji}\,\!$ = torque ON body j BY particle i, and $\scriptstyle {\boldsymbol {\tau }}_{E}\,\!$ = resultant external torque (due to any agent not part of system). Particle i does not exert a torque on itself.

Applications

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

$mg\sin \theta =m{\frac {{\rm {d}}^{2}(\ell \theta )}{{\rm {d}}t^{2}}}+0\Rightarrow g\sin \theta =\ell {\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}\,\!$

a damped, driven harmonic oscillator:

$F_{0}\sin {\omega _{0}t}=m{\frac {{\rm {d}}^{2}x}{{\rm {d}}^{2}t}}+bm{\frac {{\rm {d}}x}{{\rm {d}}t}}+\omega ^{2}x\,\!$

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

$-{\frac {GmM}{|{\mathbf {r} }|^{2}}}{\mathbf {\hat {e}} }_{r}+{\mathbf {R} }=m{\frac {{\rm {d}}^{2}{\mathbf {r} }}{{\rm {d}}t^{2}}}+0\Rightarrow -{\frac {GM}{|{\mathbf {r} }|^{2}}}{\mathbf {\hat {e}} }_{r}+{\mathbf {A} }={\frac {{\rm {d}}^{2}{\mathbf {r} }}{{\rm {d}}t^{2}}}\,\!$

where G = gravitational constant, M = mass of the earth and A is acceleration. Newton's law of gravity has been used. The mass m of the ball cancels.

Euler's equations for rigid body dynamics

Main article: Euler's equations (rigid body dynamics)

Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. An new equation Euler formulated is:

$\mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}\,\!$

General planar motion

The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

$\mathbf {r} ={\mathbf {r}}(t)=r{\mathbf {\hat {e}}}_{r}\,\!$

the following are general dynamic results.

The momentum and angular momenta are:

${\begin{aligned}\mathbf {p} &=m\left({\mathbf {\hat {e}}}_{r}{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}+r\omega {\mathbf {\hat {e}}}_{\theta }\right)\\\mathbf {L} &=m{\mathbf {r}}\times \left({\mathbf {\hat {e}}}_{r}{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}+r\omega {\mathbf {\hat {e}}}_{\theta }\right)\\\end{aligned}}\,\!$

and the centripetal force is

$\mathbf {F} _{\bot }=-m\omega ^{2}R{\mathbf {\hat {e}}}_{r}=-\omega ^{2}\mathbf {m} \,\!$

where again m is the mass moment.

Lagrangian and Hamiltonian mechanics

Main articles: Lagrangian mechanics and Hamiltonian mechanics

More powerful equations of motion are the Euler-Lagrange equations and Hamilton's equations.

Generalized coordinates

Main article: Generalized coordinates

In Newtonian mechanics, it is customary to use fully all three Cartesian coordinates (or other 3d coordinate systems) to define the position of a particle. In a mechanical situation - there are normally constraints of motion, so using a full set of Cartesian coordinates is often unnecessary: they will be related to each other by equations corresponding to the constraint. For example a particle may be confined to move on a curved surface - the equation of the curve relates the coordinates into a constraint equation: the particle must have a position on the curve and not any other (no position off the curve).

The Eulerian-Lagrangian and Hamiltonian formalisms the incorporate the constraints of the situation into the geometry of the motion, and in doing so the number of coordinates is reduced to only the minimum number needed to define the motion. There are almost always multiple choices of this - it makes no difference which is chosen. These are known as generalized coordinates, denoted q_i (for i = 1, 2, 3...) and are governed only by convenience.

For a one-particle system, the generalized coordinates define the position of the particle. For many particle systems, each particle has its own subset of the full set of generalized coordinates. Each generalized coordinate is not for each particle in a many-particle system, the set of all generalized coordinates unique defines the configuration of the system: two may be needed for one particle, three may be needed for another, and so on. The number of generalized coordinates N equals the number of dimensions, minus the number of constraint equations.

Corresponding to generalized coordinates are their time derivatives: generalized velocities. There are also generalized momenta, denoted p_i, and their time derivatives: generalized forces, often denoted by Q_i.

There are two functions of energy: the Lagrangian and Hamiltonian functions, in terms of generalized coordinates, and velocities or momenta, and time. The Lagrangian or Hamiltonian function is set up from the system using these coordinates, then these are inserted into the Euler-Lagrange or Hamilton's equations to obtain differential equations of the system. These are solved for the function of the coordinates (inserted to begin with).

Euler-Lagrange equations

Main article: Euler-Lagrange equations

The Euler-Lagrange equations are:

${\frac {\rm {d}}{{\rm {d}}t}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)=-{\frac {\partial L}{\partial q_{i}}}$

where L is the Lagrangian function (function of generalized coordinates and momenta) - which generally has the form;

$L=\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]$

in which;

$\mathbf {q} (t)=[q_{1}(t),q_{2}(t)\cdots q_{N}(t)],\quad \mathbf {\dot {q}} (t)=[{\dot {q}}_{1}(t),{\dot {q}}_{2}(t)\cdots {\dot {q}}_{N}(t)]$

are vectors whose components are generalized coordinates and velocities respectively; wherein the generalized velocities are;

${\dot {q}}_{i}={{\rm {d}}q_{i} \over {\rm {d}}t}$

Hamilton's equations

Main article: Hamilton's equations

Alternatively Hamilton's equations can be used:

${\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}\quad {\dot {q}}_{i}=+{\frac {\partial H}{\partial p_{i}}}$

where p and q are as above, and H is the Hamiltonian function (function of generalized coordinates and momenta) - which generally has the form

$H=\left[\mathbf {q} (t),\mathbf {p} (t),t\right]$

where

${\mathbf {p}}(t)=[p_{1}(t),p_{2}(t)\cdots p_{N}(t)],\quad \mathbf {\dot {p}} (t)=[{\dot {p}}_{1}(t),{\dot {p}}_{2}(t)\cdots {\dot {p}}_{N}(t)]$

are vectors whose components are generalized momenta and forces respectively, and

$p_{i}(t)={\partial L \over {\dot {q}}_{i}}$

are the generalized momenta. Notice the anti-symmetry in p_i and p_i, that is; interchanging them reverses the sign in the following way:

$p_{i}\rightleftharpoons q_{i},\quad H\rightleftharpoons -H.$

Electrodynamics

In electrodynamics, the force on a charged particle is the Lorentz force:

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

combining with Newton's 2nd law gives a differential equation of motion, in terms of the momentum of the particle:

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

or in terms of position:

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

The same equation can be obtained using the Hamiltonian:

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

and applying Hamilton's equations.

Non-relativistic physics - wave equations

Mechanical waves

Main article: wave equation

The analogue of an equation of motion for waves is a wave equation. From Newton's law the classical mechanical wave equation can be derived, the solutions to the wave equation describe wave motion for some prescribed wave, for both travelling and standing waves. There are other wave equations for very specific applications (aside from quantum mechanics).

Quantum mechanics

Main article: Schrödinger 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.

See also

• Scalar (physics)
• Vector
• Distance
• Displacement
• Speed
• Velocity
• Acceleration
• Angular displacement
• Angular speed
• Angular velocity
• Angular acceleration

• Equations for a falling body
• Parabolic trajectory
• Curvilinear coordinates
• Orthogonal coordinates
• Newton's laws of motion
• Torricelli's Equation
• Euler–Lagrange equation
• Generalized forces
• Defining equation (physics)

External links

• Equations of Motion Applet