Editing Equations of motion (section)

==Dynamic equations of motion==

===Newtonian mechanics===

{{main|Newtonian mechanics}}

The first general equation of motion developed was [[Newton's second law]] of motion.  In its most general form it states the rate of change of momentum {{math|1='''p''' = '''p'''(''t'') = ''m'''''v'''(''t'')}} of an object equals the force {{math|1='''F''' = '''F'''('''x'''(''t''), '''v'''(''t''), ''t'')}} acting on it,<ref name="Mechanics, D. Kleppner 2010">{{cite book | last = Kleppner | first = Daniel | url = https://www.worldcat.org/oclc/573196466 | title = An Introduction to Mechanics | date = 2010 | publisher = Cambridge University Press | author2 = Robert J. Kolenkow | isbn = 978-0-521-19821-9 | location = Cambridge | oclc = 573196466}}</ref>{{rp|p=1112}}

<math display="block" qid=Q2397319> \mathbf{F} = \frac{d\mathbf{p}}{dt} </math>

The force in the equation is ''not'' the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as

<math display="block"> \mathbf{F} = m\mathbf{a} </math>

since {{math|''m''}} is a constant in [[Newtonian mechanics]].

Newton's second law applies to point-like particles, and to all points in a [[rigid body]]. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; see [[material derivative]]. In the case the mass is not constant, it is not sufficient to use the [[product rule]] for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with [[conservation of momentum]]; see [[variable-mass system]].

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 always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex.

The momentum form is preferable since this is readily generalized to more complex systems, such as [[special relativity|special]] and [[general relativity]] (see [[four-momentum]]).<ref name="Mechanics, D. Kleppner 2010"/>{{rp|p=112}} It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces.

For a number of particles (see [[many body problem]]), the equation of motion for one particle {{math|''i''}} influenced by other particles is<ref name="Relativity"/><ref name="Physics 1991"/>
<math display="block"> \frac{d\mathbf{p}_i}{dt} = \mathbf{F}_{E} + \sum_{i \neq j} \mathbf{F}_{ij} </math>

where {{math|'''p'''<sub>''i''</sub>}} is the momentum of particle {{math|''i''}}, {{math|'''F'''<sub>''ij''</sub>}} is the force on particle {{math|''i''}} by particle {{math|''j''}}, and {{math|'''F'''<sub>''E''</sub>}} is the resultant external force due to any agent not part of system. Particle {{math|''i''}} does not exert a force on itself.

[[Euler's laws of motion]] are similar to Newton's laws, but they are applied specifically to the motion of [[rigid body|rigid bodies]]. The [[Newton–Euler equations]] combine the forces and torques acting on a rigid body into a single equation.

Newton's second law for rotation takes a similar form to the translational case,<ref name="Mechanics, D. Kleppner 2010"/>

<math display="block">\boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} \,, </math>

by equating the [[torque]] acting on the body to the rate of change of its [[angular momentum]] {{math|'''L'''}}. Analogous to mass times acceleration, the [[moment of inertia]] [[tensor]] {{math|'''I'''}} depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity,

<math display="block"> \boldsymbol{\tau} = \mathbf{I} \boldsymbol{\alpha}.</math>

Again, these equations apply to point like particles, or at each point of a rigid body.

Likewise, for a number of particles, the equation of motion for one particle {{math|''i''}} is<ref name="Relativity"/>

<math display="block"> \frac{d\mathbf{L}_i}{dt} = \boldsymbol{\tau}_E + \sum_{i \neq j} \boldsymbol{\tau}_{ij} \,,</math>

where {{math|'''L'''<sub>''i''</sub>}} is the angular momentum of particle {{math|''i''}}, {{math|'''τ'''<sub>''ij''</sub>}} the torque on particle {{math|''i''}} by particle {{math|''j''}}, and {{math|'''τ'''<sub>''E''</sub>}} is resultant external torque (due to any agent not part of system). Particle {{math|''i''}} does not exert a torque on itself.

===Applications===

Some examples<ref name="Waves 1983">{{cite book | last = Pain | first = H. J. | url = https://www.worldcat.org/oclc/9392845 | title = The Physics of Vibrations and Waves | date = 1983 | publisher = Wiley | isbn = 0-471-90182-2 | edition = 3rd | location = Chichester [Sussex] | oclc = 9392845}}</ref> of Newton's law include describing the motion of a [[simple pendulum]],

<math display="block"> - mg\sin\theta = m\frac{d^2 (\ell\theta)}{dt^2} \quad \Rightarrow \quad  \frac{d^2 \theta}{dt^2} = - \frac{g}{\ell}\sin\theta  \,,</math>

and a [[Harmonic oscillator#Sinusoidal driving force|damped, sinusoidally driven harmonic oscillator]],

<math display="block"> F_0 \sin(\omega t) = m\left(\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x \right)\,.</math>

For describing the motion of masses due to gravity, [[Newton's law of gravity]] can be combined with Newton's second law. For two examples, a ball of mass {{mvar|m}} thrown in the air, in air currents (such as wind) described by a vector field of resistive forces {{math|'''R''' {{=}} '''R'''('''r''', ''t'')}},

<math display="block"> - \frac{GmM}{|\mathbf{r}|^2} \mathbf{\hat{e}}_r + \mathbf{R} = m\frac{d^2 \mathbf{r}}{d t^2} + 0 \quad \Rightarrow \quad \frac{d^2 \mathbf{r}}{d t^2} = - \frac{GM}{|\mathbf{r}|^2} \mathbf{\hat{e}}_r + \mathbf{A} </math>

where {{math|''G''}} is the [[gravitational constant]], {{math|''M''}} the mass of the Earth, and {{math|'''A''' {{=}} {{sfrac|'''R'''|''m''}}}} is the acceleration of the projectile due to the air currents at position {{math|'''r'''}} and time {{mvar|t}}.

The classical [[many body problem|{{mvar|N}}-body problem]] for {{mvar|N}} particles each interacting with each other due to gravity is a set of {{mvar|N}} nonlinear coupled second order ODEs,

<math display="block">\frac{d^2\mathbf{r}_i}{dt^2} = G\sum_{i\neq j}\frac{m_j}{|\mathbf{r}_j - \mathbf{r}_i|^3} (\mathbf{r}_j - \mathbf{r}_i)</math>

where {{math|''i'' {{=}} 1, 2, ..., ''N''}} labels the quantities (mass, position, etc.) associated with each particle.