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, 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. 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).
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. 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.
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.
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 (though less 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 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:
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 
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;
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):
is a unit axial vector, pointing parallel to the axis of rotation, = unit vector in direction of r, = 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.
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.
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 as:
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.
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  in the set above, we have:
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 be parallel (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). 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:
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.
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