List of equations in classical mechanics

Short description: none

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects.[1] 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.[2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space.[3]

Classical mechanics utilises many equations--as well as other mathematical concepts--which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory.[4] 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 and Hamiltonian mechanics).

## Contents

* 1 Classical mechanics
* 1.1 Mass and inertia
* 1.2 Derived kinematic quantities
* 1.3 Derived dynamic quantities
* 1.4 General energy definitions
* 1.5 Generalized mechanics
* 2 Kinematics
* 3 Dynamics
* 3.1 Precession
* 4 Energy
* 4.1 General work-energy theorem (translation and rotation)
* 4.2 Kinetic energy
* 4.3 Elastic potential energy
* 5 Euler's equations for rigid body dynamics
* 6 General planar motion
* 6.1 Central force motion
* 7 Equations of motion (constant acceleration)
* 8 Galilean frame transforms
* 9 Mechanical oscillators
* 10 See also
* 11 Notes
* 12 References

## Classical mechanics

### Mass and inertia

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Linear, surface, volumetric mass density  | λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume.  | m=∫λdℓ

m=∬σdS

m=∭ρdV

| kg m−n, n = 1, 2, 3  | M L−n
Moment of mass[5] | m (No common symbol)  | Point mass:

𝐦=𝐫m

Discrete masses about an axis xi: 𝐦=∑i=1N𝐫imi

Continuum of mass about an axis xi: 𝐦=∫ρ(𝐫)xid𝐫

| kg m  | M L
Center of mass | rcom

(Symbols vary)

| i-th moment of mass 𝐦i=𝐫imi

Discrete masses: 𝐫com=1M∑i𝐫imi=1M∑i𝐦i

Mass continuum: 𝐫com=1M∫d𝐦=1M∫𝐫dm=1M∫𝐫ρdV

| m  | L
2-Body reduced mass  | m12, μ Pair of masses = m1 and m2 | μ=m1m2m1+m2 | kg  | M
Moment of inertia (MOI)  | I | Discrete Masses:

I=∑i𝐦i⋅𝐫i=∑i|𝐫i|2m

Mass continuum: I=∫|𝐫|2dm=∫𝐫⋅d𝐦=∫|𝐫|2ρdV

| kg m2 | M L2

### Derived kinematic quantities

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Velocity | v | 𝐯=d𝐫dt | m s−1 | L T−1
Acceleration | a | 𝐚=d𝐯dt=d2𝐫dt2 | m s−2 | L T−2
Jerk | j | 𝐣=d𝐚dt=d3𝐫dt3 | m s−3 | L T−3
Jounce | s | 𝐬=d𝐣dt=d4𝐫dt4 | m s−4 | L T−4
Angular velocity | ω | ω=n^dθdt | rad s−1 | T−1
Angular Acceleration | α | α=dωdt=n^d2θdt2 | rad s−2 | T−2
Angular jerk | ζ | ζ=dαdt=n^d3θdt3 | rad s−3 | T−3

### Derived dynamic quantities

Angular momenta of a classical object.

Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point,

right: extrinsic orbital angular momentum L about an axis,

top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω)[6]

bottom: momentum p and its radial position r from the axis.

The total angular momentum (spin + orbital) is J. Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Momentum | p | 𝐩=m𝐯 | kg m s−1 | M L T−1
Force | F | 𝐅=d𝐩/dt | N = kg m s−2 | M L T−2
Impulse | J, Δp, I | 𝐉=Δ𝐩=∫t1t2𝐅dt | kg m s−1 | M L T−1
Angular momentum about a position point r0,  | L, J, S | 𝐋=(𝐫−𝐫0)×𝐩

Most of the time we can set r0 = 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 point r0,

Torque

| τ, M | τ=(𝐫−𝐫0)×𝐅=d𝐋dt | N m = kg m2 s−2 | M L2 T−2
Angular impulse | ΔL (no common symbol)  | Δ𝐋=∫t1t2τdt | kg m2 s−1 | M L2 T−1

### General energy definitions

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Mechanical work due to a Resultant Force  | W | W=∫C𝐅⋅d𝐫 | J = N m = kg m2 s−2 | M L2 T−2
Work done ON mechanical system, Work done BY  | WON, WBY | ΔWON=−ΔWBY | J = N m = kg m2 s−2 | M L2 T−2
Potential energy | φ, Φ, U, V, Ep | ΔW=−ΔV | J = N m = kg m2 s−2 | M L2 T−2
Mechanical power | P | P=dEdt | W = J s−1 | M L2 T−3

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

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Generalized coordinates | q, Q |  | varies with choice  | varies with choice
Generalized velocities  | q˙,Q˙ | q˙≡dq/dt | varies with choice  | varies with choice
Generalized momenta | p, P | p=∂L/∂q˙ | varies with choice  | varies with choice
Lagrangian | L | L(𝐪,q˙,t)=T(q˙)−V(𝐪,q˙,t)

where 𝐪=𝐪(t) and p = p(t) are vectors of the generalized coords and momenta, as functions of time

| J  | M L2 T−2
Hamiltonian | H | H(𝐩,𝐪,t)=𝐩⋅q˙−L(𝐪,q˙,t) | J  | M L2 T−2
Action, Hamilton's principal function  | S, 𝒮 | 𝒮=∫t1t2L(𝐪,q˙,t)dt | J s  | M L2 T−1

## 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

n^=e^r×e^θ

defines the axis of rotation, e^r = unit vector in direction of r, e^θ = unit vector tangential to the angle.

| Translation  | Rotation

|
|

Velocity | Average:

𝐯average=Δ𝐫Δt Instantaneous:

𝐯=d𝐫dt

| Angular velocityω=n^dθdtRotating rigid body:𝐯=ω×𝐫
Acceleration | Average:

𝐚average=Δ𝐯Δt

Instantaneous:

𝐚=d𝐯dt=d2𝐫dt2

| Angular acceleration

α=dωdt=n^d2θdt2

Rotating rigid body:

𝐚=α×𝐫+ω×𝐯

Jerk | Average:

𝐣average=Δ𝐚Δt

Instantaneous:

𝐣=d𝐚dt=d2𝐯dt2=d3𝐫dt3

| Angular jerk

ζ=dαdt=n^d2ωdt2=n^d3θdt3

Rotating rigid body:

𝐣=ζ×𝐫+α×𝐚

## Dynamics

| Translation  | Rotation

|
|

Momentum | Momentum is the "amount of translation"

𝐩=m𝐯

For a rotating rigid body:

𝐩=ω×𝐦

| Angular momentum

Angular momentum is the "amount of rotation":

𝐋=𝐫×𝐩=𝐈⋅ω

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.

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:

𝐅=d𝐩dt=d(m𝐯)dt=m𝐚+𝐯dmdt

For a number of particles, the equation of motion for one particle i is:[7]

d𝐩idt=𝐅E+∑i≠j𝐅ij

where pi = momentum of particle i, Fij = force on particle i by particle j, and FE = 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:[8]

τ=d𝐋dt=𝐫×𝐅=d(𝐈⋅ω)dt

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

τ=d𝐋dt=d(𝐈⋅ω)dt=d𝐈dt⋅ω+𝐈⋅α

Likewise, for a number of particles, the equation of motion for one particle i is:[9]

d𝐋idt=τE+∑i≠jτij

Yank  | Yank is rate of change of force:

𝐘=d𝐅dt=d2𝐩dt2=d2(m𝐯)dt2=m𝐣+𝟐𝐚dmdt+𝐯d2mdt2

For constant mass, it becomes; 𝐘=m𝐣

| Rotatum

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

P=dτdt=𝐫×𝐘=d(𝐈⋅α)dt

Impulse | Impulse is the change in momentum:

Δ𝐩=∫𝐅dt

For constant force F:

Δ𝐩=𝐅Δt

| Twirl/angular impulse is the change in angular momentum:

Δ𝐋=∫τdt

For constant torque τ:

Δ𝐋=τΔt

### Precession

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

Ω=wrIω

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:

W=ΔT=∫C(𝐅⋅d𝐫+τ⋅𝐧dθ)

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 v0 and later at speed v is: ΔEk=W=12m(v2−v02)

### Elastic potential energy

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

ΔEp=12k(r2−r1)2

where r2 and r1 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

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:[10]

𝐈⋅α+ω×(𝐈⋅ω)=τ

where I is the moment of inertia tensor.

## General planar motion

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,

𝐫=𝐫(t)=r𝐫^

the following general results apply to the particle.

Kinematics  | Dynamics

|

Position

𝐫=𝐫(r,θ,t)=r𝐫^

Velocity

𝐯=𝐫^drdt+rωθ^

| Momentum

𝐩=m(𝐫^drdt+rωθ^)

Angular momenta 𝐋=m𝐫×(𝐫^drdt+rωθ^)

Acceleration

𝐚=(d2rdt2−rω2)𝐫^+(rα+2ωdrdt)θ^

| The centripetal force is

𝐅⊥=−mω2R𝐫^=−ω2𝐦

where again m is the mass moment, and the Coriolis force is

𝐅c=2ωmdrdtθ^=2ωmvθ^

The Coriolis acceleration and force can also be written:

𝐅c=m𝐚c=−2mω×v

### 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:

d2dθ2(1𝐫)+1𝐫=−μ𝐫2𝐥2𝐅(𝐫)

## 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).

Linear motion  | Angular motion

|

𝐯−v𝟎=𝐚t | ω−ω0=αt
𝐱−x𝟎=12(v𝟎+𝐯)t | θ−θ0=12(ω0+ω)t
𝐱−x𝟎=𝐯0t+12𝐚t2 | θ−θ0=ω0t+12αt2
𝐱−x𝟎=𝐯t−12𝐚t2 | θ−θ0=ωt−12αt2
𝐱nth=𝐯0+𝐚(n−12) | θnth=ω0+α(n−12)
v2−v02=2𝐚(𝐱−x𝟎) | ω2−ω02=2α(θ−θ0)

## 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.

Motion of entities  | Inertial frames  | Accelerating frames

|
|

Translation

V = Constant relative velocity between two inertial frames F and F'.
A = (Variable) relative acceleration between two accelerating frames F and F'.

| Relative position

𝐫′=𝐫+𝐕t

Relative velocity 𝐯′=𝐯+𝐕

Equivalent accelerations 𝐚′=𝐚

| Relative accelerations

𝐚′=𝐚+𝐀

Apparent/fictitious forces 𝐅′=𝐅−𝐅app

Rotation

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

| Relative angular position

θ′=θ+Ωt Relative velocity ω′=ω+Ω

Equivalent accelerations α′=α

| Relative accelerations

α′=α+Λ

Apparent/fictitious torques τ′=τ−τapp

Transformation of any vector T to a rotating frame

d𝐓′dt=d𝐓dt−Ω×𝐓

## Mechanical oscillators

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

| d2xdt2=−ω2x

Solution: x=Asin⁡(ωt+ϕ)

| d2θdt2=−ω2θ

Solution: θ=Θsin⁡(ωt+ϕ)

Unforced DHM  |

* b = damping constant
* κ = torsion constant

| d2xdt2+bdxdt+ω2x=0

Solution (see below for ω'): x=Ae−bt/2mcos⁡(ω′)

Resonant frequency: ωres=ω2−(b4m)2

Damping rate: γ=b/m

Expected lifetime of excitation: τ=1/γ

| d2θdt2+bdθdt+ω2θ=0

Solution: θ=Θe−κt/2mcos⁡(ω)

Resonant frequency: ωres=ω2−(κ4m)2

Damping rate: γ=κ/m

Expected lifetime of excitation: τ=1/γ

Angular frequencies  Physical situation  | Nomenclature  | Equations

|
|

Linear undamped unforced SHO  |

* k = spring constant
* m = mass of oscillating bob

| ω=km
Linear unforced DHO  |

* k = spring constant
* b = Damping coefficient

| ω′=km−(b2m)2
Low amplitude angular SHO  |

* I = Moment of inertia about oscillating axis
* κ = torsion constant

| ω=κI
Low amplitude simple pendulum  |

* L = Length of pendulum
* g = Gravitational acceleration
* Θ = Angular amplitude

| Approximate value

ω=gL

Exact value can be shown to be: ω=gL[1+∑k=1∞∏n=1k(2n−1)∏n=1m(2n)sin2nΘ]

Energy in mechanical oscillations  Physical situation  | Nomenclature  | Equations

|
|

SHM energy  |

* T = kinetic energy
* U = potential energy
* E = total energy

| Potential energy

U=m2(x)2=m(ωA)22cos2(ωt+ϕ) Maximum value at x = A: Umax=m2(ωA)2

Kinetic energy T=ω2m2(dxdt)2=m(ωA)22sin2(ωt+ϕ)

Total energy E=T+U

DHM energy  |  | E=m(ωA)22e−bt/m

## See also

* 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

## Notes

1. ↑ Mayer, Sussman & Wisdom 2001, p. xiii
2. ↑ Berkshire & Kibble 2004, p. 1
3. ↑ Berkshire & Kibble 2004, p. 2
4. ↑ Arnold 1989, p. v
5. ↑ "Section: Moments and center of mass". http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm.
6. ↑ R.P. Feynman; R.B. Leighton; M. Sands (1964). Feynman's Lectures on Physics (volume 2). Addison-Wesley. pp. 31–7. ISBN 978-0-201-02117-2.
7. ↑ "Relativity, J.R. Forshaw 2009"
8. ↑ "Mechanics, D. Kleppner 2010"
9. ↑ "Relativity, J.R. Forshaw 2009"
10. ↑ "Relativity, J.R. Forshaw 2009"

## References

* Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics (2nd ed.), Springer, ISBN 978-0-387-96890-2, https://archive.org/details/mathematicalmeth0000arno
* 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

* v
* t
* e

Classical mechanics SI units

| Linear/translational quantities |  | Angular/rotational quantities

|
|

Dimensions | 1 | L | L2 | Dimensions | 1 | 1 | 1
T | time: t
s | absement: A
m s |  | T | time: t
s |  |
1 |  | distance: d, position: r, s, x, displacement
m | area: A
m2 | 1 |  | angle: θ, angular displacement: θ
rad | solid angle: Ω
rad2, sr
T−1 | frequency: f
s−1, Hz | speed: v, velocity: v
m s−1 | kinematic viscosity: ν,
specific angular momentum: h
m2 s−1 | T−1 | frequency: f
s−1, Hz | angular speed: ω, angular velocity: ω
0.1em"> s−1 |
T−2 |  | acceleration: a
m s−2 |  | T−2 |  | angular acceleration: α
rad s−2 |
T−3 |  | jerk: j
m s−3 |  | T−3 |  | angular jerk: ζ
rad s−3 |
M | mass: m
kg |  |  | ML2 | moment of inertia: I
kg m2 |  |
MT−1 |  | momentum: p, impulse: J
[[kilogram kg m s−1, Newton second|N s]] | action: 𝒮, [[Physics:Absement#applications|]] ℵ
kg m2 s−1, J s | ML2T−1 |  | angular momentum: L, angular impulse: ΔL
kg m2 s−1 | action: 𝒮, [[Physics:Absement#applications|]] ℵ
kg m2 s−1, J s
MT−2 |  | force: F, weight: Fg
kg  m s−2, N | energy: E, work: W, Lagrangian: L
kg m2 s−2, J | ML2T−2 |  | torque: τ, moment: M
kg m2 s−2, N m | energy: E, work: W, Lagrangian: L
kg m2 s−2, J
MT−3 |  | yank: Y
kg m s−3, N s−1 | power: P
kg m2 s−3, W | ML2T−3 |  | rotatum: P
kg m2 s−3, N m s−1 | power: P
kg m2 s−3, W

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