Editing List of equations in classical mechanics

{{Short description|none}}
[[Classical mechanics]] is the branch of [[physics]] used to describe the motion of [[macroscopic]] objects.<ref>{{Harvnb|Mayer|Sussman|Wisdom|2001|p=xiii}}</ref> It is the most familiar of the theories of physics. The concepts it covers, such as [[mass]], [[acceleration]], and [[force]], are commonly used and known.<ref>{{Harvnb|Berkshire|Kibble|2004|p=1}}</ref> The subject is based upon a [[three-dimensional space|three-dimensional]] [[Euclidean space]] with fixed axes, called a frame of reference. The point of [[concurrent lines|concurrency]] of the three axes is known as the origin of the particular space.<ref>{{Harvnb|Berkshire|Kibble|2004|p=2}}</ref>

Classical mechanics utilises many [[equation]]s&mdash;as well as other [[mathematics|mathematical]] concepts&mdash;which relate various physical quantities to one another. These include [[differential equations]], [[manifold]]s, [[Lie group]]s, and [[ergodic theory]].<ref>{{Harvnb|Arnold|1989|p=v}}</ref> This article gives a summary of the most important of these.

This article lists equations from [[Newtonian mechanics]], see [[analytical mechanics]] for the more general formulation of classical mechanics (which includes [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics]]).

==Classical mechanics==

===Mass and inertia===

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension

|-
| Linear, surface, volumetric mass density
| ''λ'' or ''μ'' (especially in [[acoustics]], see below) for Linear, ''σ'' for surface, ''ρ'' for volume.
| <math> m = \int \lambda \, \mathrm{d} \ell</math>
<math> m = \iint \sigma \, \mathrm{d} S </math>

<math> m = \iiint \rho \, \mathrm{d} V </math>
| kg m<sup>&minus;''n''</sup>, ''n'' = 1, 2, 3
| M L<sup>&minus;''n''</sup>
|-
| Moment of mass{{Anchor|Moment of mass}}<!-- linked from redirect [[Moment of Mass]] --><ref>{{cite web| url = http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm| title = ''Section: Moments and center of mass''}}</ref>
| '''m''' (No common symbol)
| Point mass:
<math display="block"> \mathbf{m} = \mathbf{r}m </math>

Discrete masses about an axis <math> x_i </math>:
<math display="block"> \mathbf{m} = \sum_{i=1}^N \mathbf{r}_i m_i </math>

Continuum of mass about an axis <math> x_i </math>:
<math display="block"> \mathbf{m} = \int \rho \left ( \mathbf{r} \right ) x_i \mathrm{d} \mathbf{r} </math>
|| kg m
|| M L
|-
| [[Center of mass]] || '''r'''<sub>com</sub>
(Symbols vary)
|| ''i''-th moment of mass <math> \mathbf{m}_i = \mathbf{r}_i m_i </math>

Discrete masses:
<math display="block"> \mathbf{r}_\mathrm{com} = \frac{1}{M} \sum_i \mathbf{r}_i m_i = \frac{1}{M} \sum_i \mathbf{m}_i </math>

Mass continuum:
<math display="block"> \mathbf{r}_\mathrm{com} = \frac{1}{M} \int \mathrm{d}\mathbf{m} = \frac{1}{M} \int \mathbf{r} \, \mathrm{d}m = \frac{1}{M}\int \mathbf{r} \rho \, \mathrm{d}V </math>
|| m
|| L
|-
| 2-Body reduced mass
|| ''m''<sub>12</sub>, ''μ'' Pair of masses = ''m''<sub>1</sub> and ''m''<sub>2</sub>
|| <math> \mu = \frac{m_1 m_2}{m_1 + m_2} </math>
|| kg
|| M
|-
| Moment of inertia (MOI)
|| ''I''
|| Discrete Masses:
<math display="block"> I = \sum_i \mathbf{m}_i \cdot \mathbf{r}_i = \sum_i \left | \mathbf{r}_i \right | ^2 m </math>

Mass continuum:
<math display="block"> I = \int \left | \mathbf{r} \right | ^2 \mathrm{d} m = \int \mathbf{r} \cdot \mathrm{d} \mathbf{m}  = \int \left | \mathbf{r} \right | ^2 \rho \, \mathrm{d}V </math>
|| kg m<sup>2</sup>
|| M L<sup>2</sup>
|}

===Derived kinematic quantities===

[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass ''m'', position '''r''', velocity '''v''', acceleration '''a'''.]]

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension
|-
| [[Velocity]] || '''v''' || <math> \mathbf{v} = \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} t} </math>|| m s<sup>−1</sup> || L T<sup>−1</sup>
|-
| [[Acceleration]] || '''a''' || <math> \mathbf{a} = \frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t} = \frac{\mathrm{d}^2 \mathbf{r}}{\mathrm{d} t^2 } </math>|| m s<sup>−2</sup> || L T<sup>−2</sup>
|-
| [[Jerk (physics)|Jerk]] || '''j''' || <math> \mathbf{j} = \frac{\mathrm{d} \mathbf{a}}{\mathrm{d} t} = \frac{\mathrm{d}^3 \mathbf{r}}{\mathrm{d} t^3} </math>|| m s<sup>−3</sup> || L T<sup>−3</sup>
|-
| [[Jounce]] || '''s''' || <math> \mathbf{s} = \frac{\mathrm{d} \mathbf{j}}{\mathrm{d} t} = \frac{\mathrm{d}^4 \mathbf{r}}{\mathrm{d} t^4}
</math>|| m s<sup>−4</sup> || L T<sup>−4</sup>
|-
| [[Angular velocity]] || '''ω''' || <math> \boldsymbol{\omega} = \mathbf{\hat{n}} \frac{ \mathrm{d} \theta }{\mathrm{d} t} </math>|| rad s<sup>−1</sup> || T<sup>−1</sup>
|-
| [[Angular acceleration|Angular Acceleration]] || '''α''' || <math> \boldsymbol{\alpha} = \frac{\mathrm{d} \boldsymbol{\omega}}{\mathrm{d} t} = \mathbf{\hat{n}} \frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2} </math>|| rad s<sup>−2</sup> || T<sup>−2</sup>
|-
| [[Angular jerk]] || '''ζ''' || <math> \boldsymbol{\zeta} = \frac{\mathrm{d} \boldsymbol{\alpha}}{\mathrm{d} t} = \mathbf{\hat{n}} \frac{ \mathrm{d}^3 \theta}{\mathrm{d} t^3} </math>|| rad s<sup>−3</sup> || T<sup>−3</sup>
|}

===Derived dynamic quantities===

[[File:Classical angular momentum.svg|350px|thumb|Angular momenta of a classical object.<br/><br/>'''Left:''' intrinsic "spin" angular momentum '''S''' is really orbital angular momentum of the object at every point,<br/><br/>'''right:''' extrinsic orbital angular momentum '''L''' about an axis,<br/><br/>'''top:''' the [[moment of inertia tensor]] '''I''' and angular velocity '''ω''' ('''L''' is not always parallel to '''ω''')<ref>{{cite book | title=Feynman's Lectures on Physics (volume 2) | author1=R.P. Feynman | author2=R.B. Leighton | author3=M. Sands | publisher=Addison-Wesley | year=1964 | pages=31–7 | isbn=978-0-201-02117-2}}</ref><br/><br/>'''bottom:''' momentum '''p''' and its radial position '''r''' from the axis.<br/><br/>The total angular momentum (spin + orbital) is '''J'''.]]

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension
|-
| [[Momentum]] || '''p''' || <math> \mathbf{p} = m\mathbf{v} </math> || kg m s<sup>−1</sup> || M L T<sup>−1</sup>
|-
| [[Force]] || '''F''' || <math> \mathbf{F} = \mathrm{d} \mathbf{p}/\mathrm{d} t </math>
|| N = kg m s<sup>−2</sup> || M L T<sup>−2</sup>
|-
| [[Impulse (physics)|Impulse]] || '''J''', Δ'''p''', '''I''' || <math> \mathbf{J} = \Delta \mathbf{p} = \int_{t_1}^{t_2} \mathbf{F} \, \mathrm{d} t </math>|| kg m s<sup>−1</sup> || M L T<sup>−1</sup>
|-
| [[Angular momentum]] about a position point '''r'''<sub>0</sub>,
|| '''L''', '''J''', '''S''' || <math> \mathbf{L} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{p} </math>

Most of the time we can set '''r'''<sub>0</sub> = '''0''' if particles are orbiting about axes intersecting at a common point.
|| kg m<sup>2</sup> s<sup>−1</sup> || M L<sup>2</sup> T<sup>−1</sup>
|-
| Moment of a force about a position point '''r'''<sub>0</sub>,
[[Torque]]
|| '''τ''', '''M''' || <math> \boldsymbol{\tau} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{F} = \frac{\mathrm{d} \mathbf{L}}{\mathrm{d} t} </math>|| N m = kg m<sup>2</sup> s<sup>−2</sup> || M L<sup>2</sup> T<sup>−2</sup>
|-
| Angular impulse || Δ'''L''' (no common symbol)
|| <math> \Delta \mathbf{L} = \int_{t_1}^{t_2} \boldsymbol{\tau} \, \mathrm{d} t </math>|| kg m<sup>2</sup> s<sup>−1</sup> || M L<sup>2</sup> T<sup>−1</sup>
|}

===General energy definitions===

{{Main article|Mechanical energy}}

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension
|-
| [[Work (physics)|Mechanical work]] due to a Resultant Force
|| ''W'' || <math> W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{r} </math> || J = N m = kg m<sup>2</sup> s<sup>−2</sup> || M L<sup>2</sup> T<sup>−2</sup>
|-
| Work done ON mechanical system, Work done BY
|| ''W''<sub>ON</sub>, ''W''<sub>BY</sub> || <math> \Delta W_\mathrm{ON} = - \Delta W_\mathrm{BY} </math> || J = N m = kg m<sup>2</sup> s<sup>−2</sup> ||  M  L<sup>2</sup> T<sup>−2</sup>
|-
| [[Potential energy]]|| ''φ'', Φ, ''U'', ''V'', ''E<sub>p</sub>'' || <math> \Delta W = - \Delta V </math> || J = N m = kg m<sup>2</sup> s<sup>−2</sup> ||  M  L<sup>2</sup> T<sup>−2</sup>
|-
| Mechanical [[Power (physics)|power]]
|| ''P'' || <math> P = \frac{\mathrm{d}E}{\mathrm{d}t} </math>|| W = J s<sup>−1</sup> ||  M  L<sup>2</sup> T<sup>−3</sup>
|}

Every [[conservative force]] has a [[potential energy]]. By following two principles one can consistently assign a non-relative value to ''U'':

* Wherever the force is zero, its potential energy is defined to be zero as well.
* Whenever the force does work, potential energy is lost.

===Generalized mechanics===
{{main article|Analytical mechanics|Lagrangian mechanics|Hamiltonian mechanics|}}

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension
|-
|[[Generalized coordinates]]
|| ''q, Q''
||
|| varies with choice
|| varies with choice
|-
|[[Generalized velocities]]
|| <math>\dot{q},\dot{Q} </math>
|| <math>\dot{q}\equiv \mathrm{d}q/\mathrm{d}t </math>
|| varies with choice
|| varies with choice
|-
|[[Canonical coordinates|Generalized momenta]]
|| ''p, P''
||<math> p = \partial L /\partial \dot{q} </math>
|| varies with choice
|| varies with choice
|-
| [[Lagrangian mechanics|Lagrangian]]
|| ''L''
|| <math> L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{\dot{q}}) - V(\mathbf{q},\mathbf{\dot{q}},t) </math>

where <math> \mathbf{q} = \mathbf{q}(t) </math> and '''p''' = '''p'''(''t'') are vectors of the generalized coords and momenta, as functions of time
|| J
||  M  L<sup>2</sup> T<sup>−2</sup>
|-
| [[Hamiltonian mechanics|Hamiltonian]]
|| ''H''
|| <math> H(\mathbf{p},\mathbf{q},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t) </math>
|| J
||  M  L<sup>2</sup> T<sup>−2</sup>
|-
| [[Action (physics)|Action]], Hamilton's principal function
|| ''S'', <math> \scriptstyle{\mathcal{S}} </math>
|| <math> \mathcal{S} = \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) \mathrm{d}t </math>
|| J s
||  M  L<sup>2</sup> T<sup>−1</sup>
|}

==Kinematics==

In the following 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 unit axial vector

<math display="block">\mathbf{\hat{n}} = \mathbf{\hat{e}}_r\times\mathbf{\hat{e}}_\theta </math>

defines the axis of rotation, <math> \scriptstyle \mathbf{\hat{e}}_r </math> = unit vector in direction of {{math|'''r'''}}, <math> \scriptstyle \mathbf{\hat{e}}_\theta </math> = unit vector tangential to the angle.

{| class="wikitable"
|-
!
! Translation
! Rotation
|- valign="top"
![[Velocity]]
|Average:
<math display="block">\mathbf{v}_{\mathrm{average}} = {\Delta \mathbf{r} \over \Delta t}</math>
Instantaneous:

<math display="block">\mathbf{v} = {d\mathbf{r} \over dt}</math>
|[[Angular velocity]]<math display="block"> \boldsymbol{\omega} = \mathbf{\hat{n}}\frac{{\rm d} \theta}{{\rm d} t}</math>Rotating [[rigid body]]:<math display="block"> \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} </math>
|- valign="top"
![[Acceleration]]
|Average:
<math display="block">\mathbf{a}_{\mathrm{average}} = \frac{\Delta\mathbf{v}}{\Delta t} </math>

Instantaneous:

<math display="block">\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2} </math>

|[[Angular acceleration]]

<math display="block">\boldsymbol{\alpha} = \frac{{\rm d} \boldsymbol{\omega}}{{\rm d} t} = \mathbf{\hat{n}}\frac{{\rm d}^2 \theta}{{\rm d} t^2} </math>

Rotating rigid body:

<math display="block"> \mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r} + \boldsymbol{\omega} \times \mathbf{v} </math>

|- valign="top"
![[Jerk (physics)|Jerk]]
|Average:
<math display="block">\mathbf{j}_{\mathrm{average}} = \frac{\Delta\mathbf{a}}{\Delta t} </math>

Instantaneous:

<math display="block">\mathbf{j} = \frac{d\mathbf{a}}{dt} = \frac{d^2\mathbf{v}}{dt^2} = \frac{d^3\mathbf{r}}{dt^3} </math>
|[[Angular jerk]]

<math display="block">\boldsymbol{\zeta} = \frac{{\rm d} \boldsymbol{\alpha}}{{\rm d} t} = \mathbf{\hat{n}}\frac{{\rm d}^2 \omega}{{\rm d} t^2} = \mathbf{\hat{n}}\frac{{\rm d}^3 \theta}{{\rm d} t^3} </math>

Rotating rigid body:

<math display="block"> \mathbf{j} = \boldsymbol{\zeta} \times \mathbf{r} + \boldsymbol{\alpha} \times \mathbf{a} </math>
|}

==Dynamics==

{| class="wikitable"
|-
!
! scope="col" style="width:450px;"| Translation
! scope="col" style="width:450px;"| Rotation
|- valign="top"
![[Momentum]]
|Momentum is the "amount of translation"

<math display="block">\mathbf{p} = m\mathbf{v}</math>

For a rotating rigid body:

<math display="block"> \mathbf{p} = \boldsymbol{\omega} \times \mathbf{m} </math>
|[[Angular momentum]]

Angular momentum is the "amount of rotation":

<math display="block"> \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{I} \cdot \boldsymbol{\omega} </math>

and the cross-product is a [[pseudovector]] i.e. if '''r''' and '''p''' are reversed in direction (negative), '''L''' is not.

In general '''I''' is an order-2 [[tensor]], see above for its components. The dot '''·''' indicates [[tensor contraction]].
|- valign="top"
![[Force]] and [[Newton's 2nd law]]
|Resultant force acts on a system at the center of mass, equal to the rate of change of momentum:

<math display="block"> \begin{align} \mathbf{F} & = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} \\
& = m\mathbf{a} + \mathbf{v}\frac{{\rm d}m}{{\rm d}t} \\
\end{align} </math>

For a number of particles, the equation of motion for one particle ''i'' is:<ref>"Relativity, J.R. Forshaw 2009"</ref>

<math display="block"> \frac{\mathrm{d}\mathbf{p}_i}{\mathrm{d}t} = \mathbf{F}_{E} + \sum_{i \neq j} \mathbf{F}_{ij} </math>

where '''p'''<sub>''i''</sub> = momentum of particle ''i'', '''F'''<sub>''ij''</sub> = force '''''on''''' particle ''i'' '''''by''''' particle ''j'', and '''F'''<sub>''E''</sub> = resultant external force (due to any agent not part of system). Particle ''i'' does not exert a force on itself.
|[[Torque]]

Torque '''τ''' is also called moment of a force, because it is the rotational analogue to force:<ref>"Mechanics, D. Kleppner 2010"</ref>

<math display="block"> \boldsymbol{\tau} = \frac{{\rm d}\mathbf{L}}{{\rm d}t} = \mathbf{r}\times\mathbf{F} = \frac{{\rm d}(\mathbf{I} \cdot \boldsymbol{\omega})}{{\rm d}t} </math>

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

<math display="block"> \begin{align} 
\boldsymbol{\tau} & = \frac{{\rm d}\mathbf{L}}{{\rm d}t} = \frac{{\rm d}(\mathbf{I}\cdot\boldsymbol{\omega})}{{\rm d}t} \\
& = \frac{{\rm d}\mathbf{I}}{{\rm d}t}\cdot\boldsymbol{\omega} + \mathbf{I}\cdot\boldsymbol{\alpha} \\
\end{align} </math>

Likewise, for a number of particles, the equation of motion for one particle ''i'' is:<ref>"Relativity, J.R. Forshaw 2009"</ref>

<math display="block"> \frac{\mathrm{d}\mathbf{L}_i}{\mathrm{d}t} = \boldsymbol{\tau}_E + \sum_{i \neq j} \boldsymbol{\tau}_{ij} </math>
|- valign="top"|-valign="top"
![[Yank (physics)|Yank]]
|Yank is rate of change of force:

<math display="block" display="block"> \begin{align} \mathbf{Y} & = \frac{d\mathbf{F}}{dt}  = \frac{d^2\mathbf{p}}{dt^2} = \frac{d^2(m\mathbf{v})}{dt^2} \\[1ex]
& = m\mathbf{j} + \mathbf{2a}\frac{{\rm d}m}{{\rm d}t} + \mathbf{v}\frac{{\rm d^2}m}{{\rm d}t^2}
\end{align} </math>

For constant mass, it becomes;
<math display="block">\mathbf{Y} = m\mathbf{j}</math>
|[[Rotatum]]

Rotatum '''Ρ''' is also called moment of a Yank, because it is the rotational analogue to yank:

<math display="block"> \boldsymbol{\Rho} = \frac{{\rm d}\boldsymbol{\tau}}{{\rm d}t} = \mathbf{r}\times\mathbf{Y} = \frac{{\rm d}(\mathbf{I} \cdot \boldsymbol{\alpha})}{{\rm d}t} </math>
|- valign="top"|-valign="top"
![[Impulse (physics)|Impulse]]

|Impulse is the change in momentum:

<math display="block"> \Delta \mathbf{p} = \int \mathbf{F} \, dt </math>

For constant force '''F''':

<math display="block"> \Delta \mathbf{p} = \mathbf{F} \Delta t </math>
|Twirl/angular impulse is the change in angular momentum:

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

For constant torque '''τ''':

<math display="block"> \Delta \mathbf{L} = \boldsymbol{\tau} \Delta t </math>
|}

=== Precession ===

The precession angular speed of a [[spinning top]] is given by:

<math display="block"> \boldsymbol{\Omega} = \frac{wr}{I\boldsymbol{\omega}} </math>

where ''w'' is the weight of the spinning flywheel.

== Energy ==

The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

=== General [[work-energy theorem]] (translation and rotation) ===

The work done ''W'' by an external agent which exerts a force '''F''' (at '''r''') and torque '''τ''' on an object along a curved path ''C'' is:

<math display="block">  W = \Delta T = \int_C \left ( \mathbf{F} \cdot \mathrm{d} \mathbf{r} + \boldsymbol{\tau} \cdot \mathbf{n} \, {\mathrm{d} \theta} \right ) </math>

where θ is the angle of rotation about an axis defined by a [[unit vector]] '''n'''.

=== Kinetic energy ===
The change in [[kinetic energy]] for an object initially traveling at speed <math>v_0</math> and later at speed <math>v</math> is:
<math display="block"> \Delta E_k = W = \frac{1}{2} m(v^2 - {v_0}^2) </math>

=== Elastic potential energy ===

For a stretched spring fixed at one end obeying [[Hooke's law]], the [[elastic potential energy]] is

<math display="block"> \Delta E_p =  \frac{1}{2} k(r_2-r_1)^2 </math>

where ''r''<sub>2</sub> and ''r''<sub>1</sub> are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

==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. A new equation Euler formulated is:<ref>"Relativity, J.R. Forshaw 2009"</ref>

<math display="block"> \mathbf{I} \cdot \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left ( \mathbf{I} \cdot \boldsymbol{\omega} \right ) = \boldsymbol{\tau} </math>

where '''I''' is the [[moment of inertia]] [[tensor]].

==General planar motion==

{{see also|Polar coordinate system#Vector calculus|label 1=Polar coordinate system (section: vector calculus)}}

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,

<math display="block"> \mathbf{r} = \mathbf{r}(t) = r\hat\mathbf r  </math>

the following general results apply to the particle.

{| class="wikitable"
|-
! Kinematics
! Dynamics
|-
| Position
<math display="block"> \mathbf{r} =\mathbf{r}\left ( r,\theta, t \right ) = r \hat\mathbf r </math>
|
|-
| Velocity

<math display="block"> \mathbf{v} = \hat\mathbf r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \hat\mathbf\theta </math>
| Momentum
<math display="block"> \mathbf{p} = m \left(\hat\mathbf r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \hat\mathbf\theta \right) </math>

Angular momenta
<math display="block">\mathbf{L} = m \mathbf{r}\times \left(\hat\mathbf{r} \frac{\mathrm{d} r}{\mathrm{d}t} + r\omega\hat\mathbf\theta\right) </math>
|-
| Acceleration

<math display="block"> \mathbf{a} =\left ( \frac{\mathrm{d}^2 r}{\mathrm{d}t^2} - r\omega^2\right )\hat\mathbf r + \left ( r \alpha + 2 \omega \frac{\mathrm{d}r}{{\rm d}t} \right )\hat\mathbf\theta </math>
| The [[centripetal force]] is

<math display="block"> \mathbf{F}_\bot = - m \omega^2 R \hat\mathbf r= - \omega^2 \mathbf{m} </math>

where again '''m''' is the mass moment, and the [[Coriolis force]] is

<math display="block"> \mathbf{F}_c = 2\omega m \frac{{\rm d}r}{{\rm d}t} \hat\mathbf\theta = 2\omega m v \hat\mathbf\theta </math>

The [[Coriolis effect|Coriolis acceleration and force]] can also be written:

<math display="block">\mathbf{F}_c = m\mathbf{a}_c = -2 m \boldsymbol{ \omega \times v}</math>
|}

=== Central force motion ===

For a massive body moving in a [[central potential]] due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:

<math display="block">\frac{d^2}{d\theta^2}\left(\frac{1}{\mathbf{r}}\right) + \frac{1}{\mathbf{r}} = -\frac{\mu\mathbf{r}^2}{\mathbf{l}^2}\mathbf{F}(\mathbf{r})</math>

== Equations of motion (constant acceleration) ==
These equations can be used only when acceleration is constant. If acceleration is not constant then the general [[calculus]] equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

{| class="wikitable"
|-
!Linear motion
!Angular motion
|-
|<math>\mathbf{v-v_0}=\mathbf at </math>
|<math> \boldsymbol{\omega - \omega_0} = \boldsymbol\alpha t </math>
|-
|<math>\mathbf{x - x_0} = \tfrac{1}{2}(\mathbf{v_0+v})t </math>
|<math> \boldsymbol{\theta - \theta_0} = \tfrac{1}{2}(\boldsymbol{\omega_0 + \omega})t</math>
|-
|<math>\mathbf{x - x_0} = \mathbf v_0t+\tfrac{1}{2}\mathbf at^2 </math>
|<math> \boldsymbol{\theta - \theta_0} = \boldsymbol\omega _0 t + \tfrac{1}{2} \boldsymbol\alpha t^2</math>
|-
|<math> \mathbf x_{n^{th}} = \mathbf v_0+\mathbf a(n-\tfrac{1}{2})</math>
|<math> \boldsymbol\theta_{n^{th}} =\boldsymbol\omega_0+\boldsymbol\alpha(n-\tfrac{1}{2})</math>
|-
|<math>v^2 - v_0^2 = 2\mathbf{a(x-x_0)} </math>
|<math> \omega^2 - \omega_0^2 = 2\boldsymbol{\alpha(\theta-\theta_0)}</math>
|}

{{see also|Linear motion#Analogy between linear and rotational motion}}

==Galilean frame transforms==

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity '''V''' or angular velocity '''Ω''' relative to F. Conversely F moves at velocity (—'''V''' or —'''Ω''') relative to F'. The situation is similar for relative accelerations.

{| class="wikitable"
|-
! scope="col" style="width:250px;"| Motion of entities
! scope="col" style="width:200px;"| Inertial frames
! scope="col" style="width:200px;"| Accelerating frames
|-
|'''Translation'''

'''V''' = Constant relative velocity between two inertial frames F and F'.<br />
'''A''' = (Variable) relative acceleration between two accelerating frames F and F'.<br />
|Relative position
<math display="block"> \mathbf{r}' = \mathbf{r} + \mathbf{V}t </math>

Relative velocity
<math display="block"> \mathbf{v}' = \mathbf{v} + \mathbf{V} </math>

Equivalent accelerations
<math display="block"> \mathbf{a}' = \mathbf{a} </math>
|Relative accelerations
<math display="block"> \mathbf{a}' = \mathbf{a} + \mathbf{A} </math>

Apparent/fictitious forces
<math display="block"> \mathbf{F}' = \mathbf{F} - \mathbf{F}_\mathrm{app} </math>
|-
|rowspan="2" |'''Rotation'''

'''Ω''' = Constant relative angular velocity between two frames F and F'.<br />
'''Λ''' = (Variable) relative angular acceleration between two accelerating frames F and F'.

|Relative angular position
<math display="block"> \theta' = \theta + \Omega t </math>
Relative velocity
<math display="block"> \boldsymbol{\omega}' = \boldsymbol{\omega} + \boldsymbol{\Omega} </math>

Equivalent accelerations
<math display="block"> \boldsymbol{\alpha}' = \boldsymbol{\alpha} </math>

| Relative accelerations
<math display="block"> \boldsymbol{\alpha}' = \boldsymbol{\alpha} + \boldsymbol{\Lambda} </math>

Apparent/fictitious torques
<math display="block"> \boldsymbol{\tau}' = \boldsymbol{\tau} - \boldsymbol{\tau}_\mathrm{app} </math>
|-
|colspan="2"| Transformation of any vector '''T''' to a rotating frame
<math display="block"> \frac{{\rm d}\mathbf{T}'}{{\rm d}t} = \frac{{\rm d}\mathbf{T}}{{\rm d}t} - \boldsymbol{\Omega} \times \mathbf{T} </math>
|}

==Mechanical oscillators==

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

{| class="wikitable"
|+ Equations of motion
|-
! scope="col" style="width:100px;"| Physical situation
! scope="col" style="width:250px;"| Nomenclature
! scope="col" style="width:10px;"| Translational equations
! scope="col" style="width:10px;"| Angular equations
|-
! scope="row" | SHM
| {{plainlist}}
* ''x'' = Transverse displacement
* ''θ'' = Angular displacement
* ''A'' = Transverse amplitude
* Θ = Angular amplitude
{{endplainlist}}
| <math display="block">\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - \omega^2 x </math>

Solution:
<math display="block"> x = A \sin\left ( \omega t + \phi \right ) </math>
| <math display="block">\frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2} = - \omega^2 \theta </math>

Solution:
<math display="block"> \theta = \Theta \sin\left ( \omega t + \phi \right ) </math>
|-
! scope="row" | Unforced DHM
| {{plainlist}}
* ''b'' = damping constant
* ''κ'' = torsion constant
{{endplainlist}}
| <math display="block">\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + b \frac{\mathrm{d}x}{\mathrm{d}t} + \omega^2 x = 0 </math>

Solution (see below for ''ω'''):
<math display="block">x=Ae^{-bt/2m}\cos\left ( \omega' \right )</math>

Resonant frequency:
<math display="block">\omega_\mathrm{res} = \sqrt{\omega^2 - \left ( \frac{b}{4m} \right )^2 } </math>

Damping rate:
<math display="block">\gamma = b/m </math>

Expected lifetime of excitation:
<math display="block">\tau = 1/\gamma</math>
| <math display="block">\frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2} + b \frac{\mathrm{d}\theta}{\mathrm{d}t} + \omega^2 \theta = 0 </math>

Solution:
<math display="block">\theta=\Theta e^{-\kappa t/2m}\cos\left ( \omega \right )</math>

Resonant frequency:
<math display="block">\omega_\mathrm{res} = \sqrt{\omega^2 - \left ( \frac{\kappa}{4m} \right )^2 } </math>

Damping rate:
<math display="block">\gamma = \kappa/m </math>

Expected lifetime of excitation:
<math display="block">\tau = 1/\gamma</math>
|}
{| class="wikitable"
|+ Angular frequencies
|-
! scope="col" style="width:100px;"| Physical situation
! scope="col" style="width:250px;"| Nomenclature
! scope="col" style="width:10px;"| Equations
|-
! scope="row" | Linear undamped unforced SHO
| {{plainlist}}
* ''k'' = spring constant
* ''m'' = mass of oscillating bob
{{endplainlist}}
| <math>\omega = \sqrt{\frac{k}{m}} </math>
|-
! scope="row" | Linear unforced DHO
| {{plainlist}}
* ''k'' = spring constant
* ''b'' = Damping coefficient
{{endplainlist}}
| <math>\omega' = \sqrt{\frac{k}{m}-\left ( \frac{b}{2m} \right )^2 } </math>
|-
! scope="row" | Low amplitude angular SHO
| {{plainlist}}
* ''I'' = Moment of inertia about oscillating axis
* ''κ'' = torsion constant
{{endplainlist}}
| <math>\omega = \sqrt{\frac{\kappa}{I}}</math>
|-
! scope="row" | Low amplitude simple pendulum
| {{plainlist}}
* ''L'' = Length of pendulum
* ''g'' = Gravitational acceleration
* Θ = Angular amplitude
{{endplainlist}}
| Approximate value
<math display="block">\omega = \sqrt{\frac{g}{L}}</math>

Exact value can be shown to be:
<math display="block">\omega = \sqrt{\frac{g}{L}} \left [ 1 + \sum_{k=1}^\infty \frac{\prod_{n=1}^k \left ( 2n-1 \right )}{\prod_{n=1}^m \left ( 2n \right )} \sin^{2n} \Theta \right ]</math>
|}
{| class="wikitable"
|+ Energy in mechanical oscillations
|-
! scope="col" style="width:100px;"| Physical situation
! scope="col" style="width:250px;"| Nomenclature
! scope="col" style="width:10px;"| Equations
|-
! scope="row" | SHM energy
| {{plainlist}}
* ''T'' = kinetic energy
* ''U'' = potential energy
* ''E'' = total energy
{{endplainlist}}
| Potential energy
<math display="block">U = \frac{m}{2} \left ( x \right )^2 = \frac{m \left( \omega A \right )^2}{2} \cos^2(\omega t + \phi)</math>
Maximum value at ''x'' = ''A'':
<math display="block">U_\mathrm{max} = \frac{m}{2} \left ( \omega A \right )^2  </math>

Kinetic energy
<math display="block">T = \frac{\omega^2 m}{2} \left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )^2 = \frac{m \left ( \omega A \right )^2}{2}\sin^2\left ( \omega t + \phi \right )</math>

Total energy
<math display="block">E = T + U </math>
|-
! scope="row" | DHM energy
|
| <math>E = \frac{m \left ( \omega A \right )^2}{2}e^{-bt/m} </math>
|}

==See also==
{{div col}}
*[[List of physics formulae]]
*[[Defining equation (physical chemistry)]]
*[[Constitutive equation]]
*[[Mechanics]]
*[[Optics]]
*[[Electromagnetism]]
*[[Thermodynamics]]
*[[Acoustics]]
*[[Isaac Newton]]
*[[List of equations in wave theory]]
*[[List of relativistic equations]]
*[[List of equations in fluid mechanics]]
*[[List of equations in gravitation]]
*[[List of electromagnetism equations]]
*[[List of photonics equations]]
*[[List of equations in quantum mechanics]]
*[[List of equations in nuclear and particle physics]]
{{div col end}}

==Notes==
{{reflist}}

==References==
*{{citation|title=Mathematical Methods of Classical Mechanics|last=Arnold|first=Vladimir I.|publisher=Springer|year=1989|isbn=978-0-387-96890-2|edition=2nd|url-access=registration|url=https://archive.org/details/mathematicalmeth0000arno}}
*{{citation|title=[[Classical Mechanics (Kibble and Berkshire)|Classical Mechanics]]|last1=Berkshire|last2=Kibble|first1=Frank H.|first2=T. W. B.|author1-link=Frank H. Berkshire|author2-link=Tom Kibble|edition=5th|publisher=Imperial College Press|year=2004|isbn=978-1-86094-435-2}}
*{{citation|title=Structure and Interpretation of Classical Mechanics|last1=Mayer|last2=Sussman|last3=Wisdom|first1=Meinhard E.|first2=Gerard J.|first3=Jack|publisher=MIT Press|year=2001|isbn=978-0-262-19455-6}}

{{Classical mechanics derived SI units}}

{{DEFAULTSORT:Equations In Classical Mechanics}}
[[Category:Classical mechanics]]
[[Category:Lists of physics equations|Classical Mechanics]]