Classical mechanics is the branch ofphysics used to describe the motion ofmacroscopic objects.^[1] It is the most familiar of the theories of physics. The concepts it covers, such asmass,acceleration, andforce, are commonly used and known.^[2] The subject is based upon athree-dimensionalEuclidean space with fixed axes, called a frame of reference. The point ofconcurrency of the three axes is known as the origin of the particular space.^[3]

Classical mechanics utilises manyequations—as well as othermathematical concepts—which relate various physical quantities to one another. These includedifferential equations,manifolds,Lie groups, andergodic theory.^[4] This article gives a summary of the most important of these.

This article lists equations fromNewtonian mechanics, seeanalytical mechanics for the more general formulation of classical mechanics (which includesLagrangian andHamiltonian mechanics).

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Linear, surface, volumetric mass density	λ orμ (especially inacoustics, see below) for Linear,σ for surface,ρ for volume.	${\displaystyle m=\int \lambda \mathrm{d}\ell }$ ${\displaystyle m=\iint \sigma \mathrm{d}S}$ ${\displaystyle m=\iiint \rho \mathrm{d}V}$	kg m−n,n = 1, 2, 3	M L−n
Moment of mass[5]	m (No common symbol)	Point mass: $${\displaystyle \mathbf{m}=\mathbf{r}m}$$ Discrete masses about an axis${\displaystyle x_i}$:$${\displaystyle \mathbf{m}=\sum _{i=1}^N{\mathbf{r}}_i{m}_i}$$ Continuum of mass about an axis${\displaystyle x_i}$:$${\displaystyle \mathbf{m}=\int \rho \left(\mathbf{r}\right)x_i\mathrm{d}\mathbf{r}}$$	kg m	M L
Center of mass	rcom (Symbols vary)	i-th moment of mass${\displaystyle {\mathbf{m}}_i={\mathbf{r}}_i{m}_i}$ Discrete masses:$${\displaystyle {\mathbf{r}}_{\mathrm{c}\mathrm{o}\mathrm{m}}=\frac{1}{M}\sum _i{\mathbf{r}}_i{m}_i=\frac{1}{M}\sum _i{\mathbf{m}}_i}$$ Mass continuum:$${\displaystyle {\mathbf{r}}_{\mathrm{c}\mathrm{o}\mathrm{m}}=\frac{1}{M}\int \mathrm{d}\mathbf{m}=\frac{1}{M}\int \mathbf{r}\mathrm{d}m=\frac{1}{M}\int \mathbf{r}\rho \mathrm{d}V}$$	m	L
2-Body reduced mass	m12,μ Pair of masses =m1 andm2	${\displaystyle \mu =\frac{m_1{m}_2}{m_1+m_2}}$	kg	M
Moment of inertia (MOI)	I	Discrete Masses: $${\displaystyle I=\sum _i{\mathbf{m}}_i\cdot {\mathbf{r}}_i=\sum _i{\left|{\mathbf{r}}_i\right|}^{2}m}$$ Mass continuum:$${\displaystyle I=\int {\left|\mathbf{r}\right|}^2\mathrm{d}m=\int \mathbf{r}\cdot \mathrm{d}\mathbf{m}=\int {\left|\mathbf{r}\right|}^2\rho \mathrm{d}V}$$	kg m2	M L2

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Velocity	v	${\displaystyle \mathbf{v}=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}}$	m s−1	L T−1
Acceleration	a	${\displaystyle \mathbf{a}=\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=\frac{{\mathrm{d}}^2\mathbf{r}}{\mathrm{d}{t}^2}}$	m s−2	L T−2
Jerk	j	${\displaystyle \mathbf{j}=\frac{\mathrm{d}\mathbf{a}}{\mathrm{d}t}=\frac{{\mathrm{d}}^3\mathbf{r}}{\mathrm{d}{t}^3}}$	m s−3	L T−3
Jounce	s	${\displaystyle \mathbf{s}=\frac{\mathrm{d}\mathbf{j}}{\mathrm{d}t}=\frac{{\mathrm{d}}^4\mathbf{r}}{\mathrm{d}{t}^4}}$	m s−4	L T−4
Angular velocity	ω	${\displaystyle \mathit{\omega }=\hat{\mathbf{n}}\frac{\mathrm{d}\theta }{\mathrm{d}t}}$	rad s−1	T−1
Angular Acceleration	α	${\displaystyle \mathit{\alpha }=\frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}t}=\hat{\mathbf{n}}\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}}$	rad s−2	T−2
Angular jerk	ζ	${\displaystyle \mathit{\zeta }=\frac{\mathrm{d}\mathit{\alpha }}{\mathrm{d}t}=\hat{\mathbf{n}}\frac{{\mathrm{d}}^3\theta }{\mathrm{d}{t}^3}}$	rad s−3	T−3

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Momentum	p	${\displaystyle \mathbf{p}=m\mathbf{v}}$	kg m s−1	M L T−1
Force	F	${\displaystyle \mathbf{F}=\mathrm{d}\mathbf{p}/\mathrm{d}t}$	N = kg m s−2	M L T−2
Impulse	J, Δp,I	${\displaystyle \mathbf{J}=\mathrm{\Delta }\mathbf{p}={\int }_{t_1}^{t_2}\mathbf{F}\mathrm{d}t}$	kg m s−1	M L T−1
Angular momentum about a position pointr0,	L,J,S	${\displaystyle \mathbf{L}=\left(\mathbf{r}-{\mathbf{r}}_0\right)\times \mathbf{p}}$ Most of the time we can setr0 =0 if particles are orbiting about axes intersecting at a common point.	kg m2 s−1	M L2 T−1
Moment of a force about a position pointr0, Torque	τ,M	${\displaystyle \mathit{\tau }=\left(\mathbf{r}-{\mathbf{r}}_0\right)\times \mathbf{F}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}}$	N m = kg m2 s−2	M L2 T−2
Angular impulse	ΔL (no common symbol)	${\displaystyle \mathrm{\Delta }\mathbf{L}={\int }_{t_1}^{t_2}\mathit{\tau }\mathrm{d}t}$	kg m2 s−1	M L2 T−1

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Mechanical work due to a Resultant Force	W	${\displaystyle W={\int }_C\mathbf{F}\cdot \mathrm{d}\mathbf{r}}$	J = N m = kg m2 s−2	M L2 T−2
Work done ON mechanical system, Work done BY	WON,WBY	${\displaystyle \mathrm{\Delta }{W}_{\mathrm{O}\mathrm{N}}=-\mathrm{\Delta }{W}_{\mathrm{B}\mathrm{Y}}}$	J = N m = kg m2 s−2	M L2 T−2
Potential energy	φ, Φ,U,V,Ep	${\displaystyle \mathrm{\Delta }W=-\mathrm{\Delta }V}$	J = N m = kg m2 s−2	M L2 T−2
Mechanicalpower	P	${\displaystyle P=\frac{\mathrm{d}E}{\mathrm{d}t}}$	W = J s−1	M L2 T−3

Everyconservative force has apotential energy. By following two principles one can consistently assign a non-relative value toU:

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

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Generalized coordinates	q, Q		varies with choice	varies with choice
Generalized velocities	${\displaystyle \dot{q},\dot{Q}}$	${\displaystyle \dot{q}\equiv \mathrm{d}q/\mathrm{d}t}$	varies with choice	varies with choice
Generalized momenta	p, P	${\displaystyle p=\mathrm{\partial }L/\mathrm{\partial }\dot{q}}$	varies with choice	varies with choice
Lagrangian	L	${\displaystyle L(\mathbf{q},\dot{\mathbf{q}},t)=T(\dot{\mathbf{q}})-V(\mathbf{q},\dot{\mathbf{q}},t)}$ where${\displaystyle \mathbf{q}=\mathbf{q}(t)}$ andp =p(t) are vectors of the generalized coords and momenta, as functions of time	J	M L2 T−2
Hamiltonian	H	${\displaystyle H(\mathbf{p},\mathbf{q},t)=\mathbf{p}\cdot \dot{\mathbf{q}}-L(\mathbf{q},\dot{\mathbf{q}},t)}$	J	M L2 T−2
Action, Hamilton's principal function	S,${\displaystyle \mathcal{S}}$	${\displaystyle \mathcal{S}={\int }_{t_1}^{t_2}L(\mathbf{q},\dot{\mathbf{q}},t)\mathrm{d}t}$	J s	M L2 T−1

In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to useθ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

$${\displaystyle \hat{\mathbf{n}}={\hat{\mathbf{e}}}_r\times {\hat{\mathbf{e}}}_{\theta }}$$

defines the axis of rotation,${\displaystyle {\hat{\mathbf{e}}}_r}$ = unit vector in direction ofr,${\displaystyle {\hat{\mathbf{e}}}_{\theta }}$ = unit vector tangential to the angle.

	Translation	Rotation
Velocity	Average: $${\displaystyle {\mathbf{v}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}=\frac{\mathrm{\Delta }\mathbf{r}}{\mathrm{\Delta }t}}$$Instantaneous: $${\displaystyle \mathbf{v}=\frac{d\mathbf{r}}{dt}}$$	Angular velocity$${\displaystyle \mathit{\omega }=\hat{\mathbf{n}}\frac{\mathrm{d}\theta }{\mathrm{d}t}}$$Rotatingrigid body:$${\displaystyle \mathbf{v}=\mathit{\omega }\times \mathbf{r}}$$
Acceleration	Average: $${\displaystyle {\mathbf{a}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}=\frac{\mathrm{\Delta }\mathbf{v}}{\mathrm{\Delta }t}}$$ Instantaneous: $${\displaystyle \mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{d{t}^2}}$$	Angular acceleration $${\displaystyle \mathit{\alpha }=\frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}t}=\hat{\mathbf{n}}\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}}$$ Rotating rigid body: $${\displaystyle \mathbf{a}=\mathit{\alpha }\times \mathbf{r}+\mathit{\omega }\times \mathbf{v}}$$
Jerk	Average: $${\displaystyle {\mathbf{j}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}=\frac{\mathrm{\Delta }\mathbf{a}}{\mathrm{\Delta }t}}$$ Instantaneous: $${\displaystyle \mathbf{j}=\frac{d\mathbf{a}}{dt}=\frac{d^2\mathbf{v}}{d{t}^2}=\frac{d^3\mathbf{r}}{d{t}^3}}$$	Angular jerk $${\displaystyle \mathit{\zeta }=\frac{\mathrm{d}\mathit{\alpha }}{\mathrm{d}t}=\hat{\mathbf{n}}\frac{{\mathrm{d}}^2\omega }{\mathrm{d}{t}^2}=\hat{\mathbf{n}}\frac{{\mathrm{d}}^3\theta }{\mathrm{d}{t}^3}}$$ Rotating rigid body: $${\displaystyle \mathbf{j}=\mathit{\zeta }\times \mathbf{r}+\mathit{\alpha }\times \mathbf{a}}$$

The precession angular speed of aspinning top is given by:

$${\displaystyle \mathbf{\Omega }=\frac{wr}{I\mathit{\omega }}}$$

wherew is the weight of the spinning flywheel.

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

The work doneW by an external agent which exerts a forceF (atr) and torqueτ on an object along a curved pathC is:

$${\displaystyle W=\mathrm{\Delta }T={\int }_C\left(\mathbf{F}\cdot \mathrm{d}\mathbf{r}+\mathit{\tau }\cdot \mathbf{n}\mathrm{d}\theta \right)}$$

where θ is the angle of rotation about an axis defined by aunit vectorn.

The change inkinetic energy for an object initially traveling at speed${\displaystyle v_0}$ and later at speed${\displaystyle v}$ is:$${\displaystyle \mathrm{\Delta }{E}_k=W=\frac{1}{2}m(v^2-{v_0}^2)}$$

For a stretched spring fixed at one end obeyingHooke's law, theelastic potential energy is

$${\displaystyle \mathrm{\Delta }{E}_p=\frac{1}{2}k(r_2-r_1{)}^2}$$

wherer₂ andr₁ are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler also worked out analogous laws of motion to those of Newton, seeEuler'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:^[10]

$${\displaystyle \mathbf{I}\cdot \mathit{\alpha }+\mathit{\omega }\times \left(\mathbf{I}\cdot \mathit{\omega }\right)=\mathit{\tau }}$$

whereI is themoment of inertiatensor.

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,

$${\displaystyle \mathbf{r}=\mathbf{r}(t)=r\hat{\mathbf{r}}}$$

the following general results apply to the particle.

For a massive body moving in acentral 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:

$${\displaystyle \frac{d^2}{d{\theta }^2}\left(\frac{1}{\mathbf{r}}\right)+\frac{1}{\mathbf{r}}=-\frac{\mu {\mathbf{r}}^2}{{\mathbf{l}}^2}\mathbf{F}(\mathbf{r})}$$

These equations can be used only when acceleration is constant. If acceleration is not constant then the generalcalculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Linear motion	Angular motion
${\displaystyle \mathbf{v}\mathbf{-}{\mathbf{v}}_{\mathbf{0}}=\mathbf{a}t}$	${\displaystyle \mathit{\omega }\mathbf{-}{\mathit{\omega }}_{\mathbf{0}}=\mathit{\alpha }t}$
${\displaystyle \mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{0}}=\frac{1}{2}({\mathbf{v}}_{\mathbf{0}}\mathbf{+}\mathbf{v})t}$	${\displaystyle \mathit{\theta }\mathbf{-}{\mathit{\theta }}_{\mathbf{0}}=\frac{1}{2}({\mathit{\omega }}_{\mathbf{0}}\mathbf{+}\mathit{\omega })t}$
${\displaystyle \mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{0}}={\mathbf{v}}_{0}t+\frac{1}{2}\mathbf{a}{t}^2}$	${\displaystyle \mathit{\theta }\mathbf{-}{\mathit{\theta }}_{\mathbf{0}}={\mathit{\omega }}_{0}t+\frac{1}{2}\mathit{\alpha }{t}^2}$
${\displaystyle \mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{0}}=\mathbf{v}t-\frac{1}{2}\mathbf{a}{t}^2}$	${\displaystyle \mathit{\theta }\mathbf{-}{\mathit{\theta }}_{\mathbf{0}}=\mathit{\omega }t-\frac{1}{2}\mathit{\alpha }{t}^2}$
${\displaystyle {\mathbf{x}}_{n^{th}}={\mathbf{v}}_0+\mathbf{a}(n-\frac{1}{2})}$	${\displaystyle {\mathit{\theta }}_{n^{th}}={\mathit{\omega }}_0+\mathit{\alpha }(n-\frac{1}{2})}$
${\displaystyle v^2-v_0^2=2\mathbf{a}\mathbf{(}\mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{0}}\mathbf{)}}$	${\displaystyle {\omega }^2-{\omega }_0^2=2\mathit{\alpha }\mathbf{(}\mathit{\theta }\mathbf{-}{\mathit{\theta }}_{\mathbf{0}}\mathbf{)}}$

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 velocityV or angular velocityΩ relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

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

Equations of motion

Physical situation	Nomenclature	Translational equations	Angular equations
SHM	x = Transverse displacement θ = Angular displacement A = Transverse amplitude Θ = Angular amplitude	$${\displaystyle \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}=-{\omega }^{2}x}$$ Solution:$${\displaystyle x=A\sin \left(\omega t+\varphi \right)}$$	$${\displaystyle \frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}=-{\omega }^2\theta }$$ Solution:$${\displaystyle \theta =\mathrm{\Theta }\sin \left(\omega t+\varphi \right)}$$
Unforced DHM	b = damping constant κ = torsion constant	$${\displaystyle \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+b\frac{\mathrm{d}x}{\mathrm{d}t}+{\omega }^{2}x=0}$$ Solution (see below forω'):$${\displaystyle x=A{e}^{-bt/2m}\cos \left({\omega }^{\prime }\right)}$$ Resonant frequency:$${\displaystyle {\omega }_{\mathrm{r}\mathrm{e}\mathrm{s}}=\sqrt{{\omega }^2-{\left(\frac{b}{4m}\right)}^2}}$$ Damping rate:$${\displaystyle \gamma =b/m}$$ Expected lifetime of excitation:$${\displaystyle \tau =1/\gamma }$$	$${\displaystyle \frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}+b\frac{\mathrm{d}\theta }{\mathrm{d}t}+{\omega }^2\theta =0}$$ Solution:$${\displaystyle \theta =\mathrm{\Theta }{e}^{-\kappa t/2m}\cos \left(\omega \right)}$$ Resonant frequency:$${\displaystyle {\omega }_{\mathrm{r}\mathrm{e}\mathrm{s}}=\sqrt{{\omega }^2-{\left(\frac{\kappa }{4m}\right)}^2}}$$ Damping rate:$${\displaystyle \gamma =\kappa /m}$$ Expected lifetime of excitation:$${\displaystyle \tau =1/\gamma }$$

Angular frequencies

Physical situation	Nomenclature	Equations
Linear undamped unforced SHO	k = spring constant m = mass of oscillating bob	${\displaystyle \omega =\sqrt{\frac{k}{m}}}$
Linear unforced DHO	k = spring constant b = Damping coefficient	${\displaystyle {\omega }^{\prime }=\sqrt{\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2}}$
Low amplitude angular SHO	I = Moment of inertia about oscillating axis κ = torsion constant	${\displaystyle \omega =\sqrt{\frac{\kappa }{I}}}$
Low amplitude simple pendulum	L = Length of pendulum g = Gravitational acceleration Θ = Angular amplitude	Approximate value $${\displaystyle \omega =\sqrt{\frac{g}{L}}}$$ Exact value can be shown to be:$${\displaystyle \omega =\sqrt{\frac{g}{L}}\left[1+\sum _{k=1}^{\mathrm{\infty }}\frac{\prod _{n=1}^k\left(2n-1\right)}{\prod _{n=1}^m\left(2n\right)}{\sin }^{2n}\mathrm{\Theta }\right]}$$

Energy in mechanical oscillations

Physical situation	Nomenclature	Equations
SHM energy	T = kinetic energy U = potential energy E = total energy	Potential energy $${\displaystyle U=\frac{m}{2}{\left(x\right)}^2=\frac{m{\left(\omega A\right)}^2}{2}{\cos }^2(\omega t+\varphi )}$$Maximum value atx =A:$${\displaystyle U_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{m}{2}{\left(\omega A\right)}^2}$$ Kinetic energy$${\displaystyle T=\frac{{\omega }^{2}m}{2}{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)}^2=\frac{m{\left(\omega A\right)}^2}{2}{\sin }^2\left(\omega t+\varphi \right)}$$ Total energy$${\displaystyle E=T+U}$$
DHM energy		${\displaystyle E=\frac{m{\left(\omega A\right)}^2}{2}{e}^{-bt/m}}$

• Arnold, Vladimir I. (1989),Mathematical Methods of Classical Mechanics (2nd ed.), Springer,ISBN 978-0-387-96890-2
• Berkshire, Frank H.;Kibble, T. W. B. (2004),Classical Mechanics (5th ed.), Imperial College Press,ISBN 978-1-86094-435-2
• Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001),Structure and Interpretation of Classical Mechanics, MIT Press,ISBN 978-0-262-19455-6

• Classical mechanics
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