List of equations in classical mechanics explained

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. 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. 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.

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. 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).

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=\intλd\ell

m=\iint\sigmadS

m=\iiint\rhodV

kg m^-n, n = 1, 2, 3

M L^-n

Moment of mass^[1]

m (No common symbol)

Point mass:

\mathbf = \mathbfm

Discrete masses about an axis

x_i

\mathbf = \sum_^N \mathbf_i m_i

Continuum of mass about an axis

x_i

\mathbf = \int \rho \left (\mathbf \right) x_i \mathrm \mathbf

kg m

M L

r_com(Symbols vary)

i-th moment of mass

m_i=r_im_i

Discrete masses: $\mathbf_\mathrm = \frac \sum_i \mathbf_i m_i = \frac \sum_i \mathbf_i$

Mass continuum: $\mathbf_\mathrm = \frac \int \mathrm\mathbf = \frac \int \mathbf \, \mathrmm = \frac\int \mathbf \rho \, \mathrmV$

2-Body reduced mass

m₁₂, μ Pair of masses = m₁ and m₂

\mu=

	m₁m₂
	m₁+m₂

Moment of inertia (MOI)

Discrete Masses:

I = \sum_i \mathbf_i \cdot \mathbf_i = \sum_i \left

\mathbf_i \right

^2 m

Mass continuum: $I = \int \left$

\mathbf \right

^2 \mathrm m = \int \mathbf \cdot \mathrm \mathbf = \int \left

\mathbf \right

^2 \rho \, \mathrmV

kg m²

M L²

Derived kinematic quantities

Quantity (common name/s)

(Common) symbol/s

Defining equation

SI units

Dimension

	dr
	dt

m s⁻¹

L T⁻¹

	dv
	dt

	d²r
	dt²

m s⁻²

L T⁻²

	da
	dt

	d³r
	dt³

m s⁻³

L T⁻³

	dj
	dt

	d⁴r
	dt⁴

m s⁻⁴

L T⁻⁴

\boldsymbol{\omega}=\hat{n

} \frac

rad s⁻¹

T⁻¹

\boldsymbol{\alpha}=

	d\boldsymbol{\omega

} = \mathbf \frac

rad s⁻²

T⁻²

\boldsymbol{\zeta}=

	d\boldsymbol{\alpha

} = \mathbf \frac

rad s⁻³

T⁻³

Derived dynamic quantities

Quantity (common name/s)

(Common) symbol/s

Defining equation

SI units

Dimension

p=mv

kg m s⁻¹

M L T⁻¹

F=dp/dt

N = kg m s⁻²

M L T⁻²

J, Δp, I

J=\Deltap=

	t₂
\int
	t₁

Fdt

kg m s⁻¹

M L T⁻¹

Angular momentum about a position point r₀,

L, J, S

L=\left(r-r₀\right) x 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

\boldsymbol{\tau}=\left(r-r₀\right) x F=

	dL
	dt

N m = kg m² s⁻²

M L² T⁻²

Angular impulse

ΔL (no common symbol)

\DeltaL=

	t₂
\int
	t₁

\boldsymbol{\tau}dt

kg m² s⁻¹

M L² T⁻¹

General energy definitions

See main article: article and Mechanical energy.

Quantity (common name/s)

(Common) symbol/s

Defining equation

SI units

Dimension

Mechanical work due to a Resultant Force

W=\int_CF ⋅ dr

J = N m = kg m² s⁻²

M L² T⁻²

Work done ON mechanical system, Work done BY

W_ON, W_BY

\DeltaW_ON=-\DeltaW_BY

J = N m = kg m² s⁻²

M L² T⁻²

Potential energy

φ, Φ, U, V, E_p

\DeltaW=-\DeltaV

J = N m = kg m² s⁻²

M L² T⁻²

Mechanical power

	dE
	dt

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

See main article: article, Analytical mechanics, Lagrangian mechanics and Hamiltonian mechanics.

Quantity (common name/s)

(Common) symbol/s

Defining equation

SI units

Dimension

Generalized coordinates

q, Q

varies with choice

Generalized velocities

•
q	,

•

\equivdq/dt

varies with choice

Generalized momenta

p, P

p=\partialL/\partial

•

varies with choice

Lagrangian

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

M L² T⁻²

Hamiltonian

H(p,q,t)=

p ⋅

•

L(q,
•
q
,t)

M L² T⁻²

Action, Hamilton's principal function

\scriptstyle{l{S}}

l{S}=

	t₂
\int
	t₁

L(q,
•
q
,t)

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

$\mathbf = \mathbf_r\times\mathbf_\theta$

defines the axis of rotation,

\scriptstyle\hat{e

}_r = unit vector in direction of,

\scriptstyle\hat{e

}_\theta = unit vector tangential to the angle.

	Translation	Rotation
Velocity	Average: $\mathbf_ =$ Instantaneous: $\mathbf =$	Angular velocity $\boldsymbol = \mathbf\frac$ Rotating rigid body: $\mathbf = \boldsymbol \times \mathbf$
Acceleration	Average: $\mathbf_ = \frac$ Instantaneous: $\mathbf = \frac = \frac$	Angular acceleration $\boldsymbol = \frac = \mathbf\frac$ Rotating rigid body: $\mathbf = \boldsymbol \times \mathbf + \boldsymbol \times \mathbf$
Jerk	Average: $\mathbf_ = \frac$ Instantaneous: $\mathbf = \frac = \frac = \frac$	Angular jerk $\boldsymbol = \frac = \mathbf\frac = \mathbf\frac$ Rotating rigid body: $\mathbf = \boldsymbol \times \mathbf + \boldsymbol \times \mathbf$

Dynamics

	Translation	Rotation
Momentum	Momentum is the "amount of translation" $\mathbf = m\mathbf$ For a rotating rigid body: $\mathbf = \boldsymbol \times \mathbf$	Angular momentum Angular momentum is the "amount of rotation": $\mathbf = \mathbf \times \mathbf = \mathbf \cdot \boldsymbol$ 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: $\begin \mathbf & = \frac = \frac \\& = m\mathbf + \mathbf\frac \\\end$ For a number of particles, the equation of motion for one particle i is:^[2] $\frac = \mathbf_ + \sum_ \mathbf_$ where p_i = momentum of particle i, F_ij = force on particle i by particle j, and F_E = 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:^[3] $\boldsymbol = \frac = \mathbf\times\mathbf = \frac$ For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: $\begin \boldsymbol & = \frac = \frac \\& = \frac\cdot\boldsymbol + \mathbf\cdot\boldsymbol \\\end$ Likewise, for a number of particles, the equation of motion for one particle i is:^[4] $\frac = \boldsymbol_E + \sum_ \boldsymbol_$
-valign="top"	Yank	Yank is rate of change of force: $\begin \mathbf & = \frac = \frac = \frac \\[1ex]& = m\mathbf + \mathbf\frac + \mathbf\frac\end$ For constant mass, it becomes; $\mathbf = m\mathbf$	Rotatum Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank: $\boldsymbol = \frac = \mathbf\times\mathbf = \frac$
-valign="top"	Impulse	Impulse is the change in momentum: $\Delta \mathbf = \int \mathbf \, dt$ For constant force F: $\Delta \mathbf = \mathbf \Delta t$	Twirl/angular impulse is the change in angular momentum: $\Delta \mathbf = \int \boldsymbol \, dt$ For constant torque τ: $\Delta \mathbf = \boldsymbol \Delta t$

Precession

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

$\boldsymbol = \frac$

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:

$W = \Delta T = \int_C \left (\mathbf \cdot \mathrm \mathbf + \boldsymbol \cdot \mathbf \, \right)$

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

is:

\Delta E_k = W = \frac m(v^2 - ^2)

Elastic potential energy

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

$\Delta E_p = \frac k(r_2-r_1)^2$

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

See main article: article and 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:^[5]

$\mathbf \cdot \boldsymbol + \boldsymbol \times \left (\mathbf \cdot \boldsymbol \right) = \boldsymbol$

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,

$\mathbf = \mathbf(t) = r\hat\mathbf r$

the following general results apply to the particle.

Kinematics	Dynamics
Position $\mathbf =\mathbf\left (r,\theta, t \right) = r \hat\mathbf r$
Velocity $\mathbf = \hat\mathbf r \frac + r \omega \hat\mathbf\theta$	Momentum $\mathbf = m \left(\hat\mathbf r \frac + r \omega \hat\mathbf\theta \right)$ Angular momenta $\mathbf = m \mathbf\times \left(\hat\mathbf \frac + r\omega\hat\mathbf\theta\right)$
Acceleration $\mathbf =\left (\frac - r\omega^2\right)\hat\mathbf r + \left (r \alpha + 2 \omega \frac \right)\hat\mathbf\theta$	The centripetal force is $\mathbf_\bot = - m \omega^2 R \hat\mathbf r= - \omega^2 \mathbf$ where again m is the mass moment, and the Coriolis force is $\mathbf_c = 2\omega m \frac \hat\mathbf\theta = 2\omega m v \hat\mathbf\theta$ The Coriolis acceleration and force can also be written: $\mathbf_c = m\mathbf_c = -2 m \boldsymbol$

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:

$\frac\left(\frac\right) + \frac = -\frac\mathbf(\mathbf)$

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

\boldsymbol{\omega-\omega_0}=\boldsymbol\alphat

x-x₀

\tfrac{1}{2}(v_0+v)t

\boldsymbol{\theta-\theta_0}=\tfrac{1}{2}(\boldsymbol{\omega₀+\omega})t

x-x₀

	2
v
	0t+\tfrac{1}{2}at

\boldsymbol{\theta-\theta_0}=\boldsymbol\omega₀t+\tfrac{1}{2}\boldsymbol\alphat²

x
	n^th

=v_{0+a(n-\tfrac{1}{2})}

\boldsymbol\theta
	n^th

=\boldsymbol\omega_{0+\boldsymbol\alpha(n-\tfrac{1}{2})}

v²-

	2
v
	0

2a(x-x₀₎

\omega²-

	2
\omega
	0

=2\boldsymbol{\alpha(\theta-\theta_0)}

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 $\mathbf' = \mathbf + \mathbft$ Relative velocity $\mathbf' = \mathbf + \mathbf$ Equivalent accelerations $\mathbf' = \mathbf$	Relative accelerations $\mathbf' = \mathbf + \mathbf$ Apparent/fictitious forces $\mathbf' = \mathbf - \mathbf_\mathrm$
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 $\theta' = \theta + \Omega t$ Relative velocity $\boldsymbol' = \boldsymbol + \boldsymbol$ Equivalent accelerations $\boldsymbol' = \boldsymbol$	Relative accelerations $\boldsymbol' = \boldsymbol + \boldsymbol$ Apparent/fictitious torques $\boldsymbol' = \boldsymbol - \boldsymbol_\mathrm$
	Transformation of any vector T to a rotating frame $\frac = \frac - \boldsymbol \times \mathbf$

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	$\frac = - \omega^2 x$ Solution: $x = A \sin\left (\omega t + \phi \right)$	$\frac = - \omega^2 \theta$ Solution: $\theta = \Theta \sin\left (\omega t + \phi \right)$
Unforced DHM	b = damping constant κ = torsion constant	$\frac + b \frac + \omega^2 x = 0$ Solution (see below for ω): $x=Ae^\cos\left (\omega' \right)$ Resonant frequency: $\omega_\mathrm = \sqrt$ Damping rate: $\gamma = b/m$ Expected lifetime of excitation: $\tau = 1/\gamma$	$\frac + b \frac + \omega^2 \theta = 0$ Solution: $\theta=\Theta e^\cos\left (\omega \right)$ Resonant frequency: $\omega_\mathrm = \sqrt$ Damping rate: $\gamma = \kappa/m$ Expected lifetime of excitation: $\tau = 1/\gamma$

Angular frequencies

Physical situation

Nomenclature

Equations

Linear undamped unforced SHO

k = spring constant
m = mass of oscillating bob

\omega=\sqrt{

	k
	m

}

Linear unforced DHO

k = spring constant
b = Damping coefficient

\omega'=\sqrt{

	k
	m

-\left(

	b
	2m

\right)²}

Low amplitude angular SHO

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

\omega=\sqrt{

	\kappa
	I

}

Low amplitude simple pendulum

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

Approximate value

\omega = \sqrt

Exact value can be shown to be: $\omega = \sqrt \left [1 + \sum_{k=1}^\infty \frac{\prod_{n=1}^k \left (2n-1 \right)}{\prod_{n=1}^m \left (2n \right)} \sin^{2n} \Theta \right ]$

Energy in mechanical oscillations

Physical situation

Nomenclature

Equations

SHM energy

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

Potential energy

U = \frac \left (x \right)^2 = \frac \cos^2(\omega t + \phi)

Maximum value at x = A:

U_\mathrm \frac \left (\omega A \right)^2

Kinetic energy $T = \frac \left (\frac \right)^2 = \frac\sin^2\left (\omega t + \phi \right)$

Total energy $E = T + U$

DHM energy

	m\left(\omegaA\right)²
	2

e^-bt/m

Notes and References

Web site: Section: Moments and center of mass.
"Relativity, J.R. Forshaw 2009"
"Mechanics, D. Kleppner 2010"
"Relativity, J.R. Forshaw 2009"
"Relativity, J.R. Forshaw 2009"

List of equations in classical mechanics explained

Classical mechanics

Mass and inertia

Derived kinematic quantities

Derived dynamic quantities

General energy definitions

Generalized mechanics

Kinematics

Dynamics

Precession

Energy

General work-energy theorem (translation and rotation)

Kinetic energy

Elastic potential energy

Euler's equations for rigid body dynamics

General planar motion

Central force motion

Equations of motion (constant acceleration)

Galilean frame transforms

Mechanical oscillators

See also

Notes and References