Editing Equations of motion (section)

==Types==
There are two main descriptions of motion: dynamics and [[kinematics]]. Dynamics is general, since the momenta, [[force]]s and [[energy]] of the [[particles]] are taken into account. In this instance, sometimes the term ''dynamics'' 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. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the [[#Constant translational acceleration in a straight line|SUVAT<!---THERE ARE NUMEROUS NAMES - PLEASE LEAVE THESE EQUATIONS AS "SUVAT" ---> equations]], arising from the definitions of [[kinematic quantity|kinematic quantities]]: displacement ({{math|''s''}}), initial velocity ({{math|''u''}}), final velocity ({{math|''v''}}), acceleration ({{math|''a''}}), and time ({{math|''t''}}).

A differential equation of motion, usually identified as some [[physical law]] (for example, F = ma), and applying definitions of [[physical quantities]], is used to set up an equation to solve a kinematics problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a set of solutions. A particular solution can be obtained by setting the [[Initial value problem|initial value]]s, which fixes the values of the constants.

Stated formally, in general, an equation of motion {{math|''M''}} is a [[function (mathematics)|function]] of the [[Position (vector)|position]] {{math|'''r'''}} of the object, its [[velocity]] (the first time [[derivative]] of {{math|'''r'''}}, {{math|'''v''' {{=}} {{sfrac|''d'''''r'''|''dt''}}}}), and its acceleration (the second [[derivative]] of {{math|'''r'''}}, {{math|'''a''' {{=}} {{sfrac|''d''<sup>2</sup>'''r'''|''dt''<sup>2</sup>}}}}), and time {{math|''t''}}. [[Euclidean vector]]s in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in {{math|'''r'''}} is a second-order [[ordinary differential equation]] (ODE) in {{math|'''r'''}},

<math display="block">M\left[\mathbf{r}(t),\mathbf{\dot{r}}(t),\mathbf{\ddot{r}}(t),t\right]=0\,,</math>

where {{math|''t''}} is time, and each overdot denotes one [[time derivative]]. The [[Initial value problem|initial conditions]] are given by the ''constant'' values at {{math|''t'' {{=}} 0}},

<math display="block"> \mathbf{r}(0) \,, \quad \mathbf{\dot{r}}(0) \,. </math>

The solution {{math|'''r'''(''t'')}} to the equation of motion, with specified initial values, describes the system for all times {{math|''t''}} after {{math|''t'' {{=}} 0}}. Other dynamical variables like the [[momentum]] {{math|'''p'''}} of the object, or quantities derived from {{math|'''r'''}} and {{math|'''p'''}} like [[angular momentum]], can be used in place of {{math|'''r'''}} as the quantity to solve for from some equation of motion, although the position of the object at time {{math|''t''}} is by far the most sought-after quantity.

Sometimes, the equation will be [[linear differential equation|linear]] and is more likely to be exactly solvable. In general, the equation will be [[Non-linear differential equation|non-linear]], and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show [[Chaos theory|chaotic]] behavior depending on how ''sensitive'' the system is to the initial conditions.