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 threedimensional 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 page 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. 

kg m^{−n}, n = 1, 2, 3  [M][L]^{−n}  
Moment of mass^{[5]}  m (No common symbol) 
Point mass:
Discrete masses about an axis
Continuum of mass about an axis

kg m  [M][L]  
Centre of mass 
r_{com}
(Symbols vary) 
i^{th} moment of mass
Discrete masses: Mass continuum: 
m  [L]  
2Body reduced mass  m_{12}, μ Pair of masses = m_{1} and m_{2} 

kg  [M]  
Moment of inertia (MOI)  I 
Discrete Masses:
Mass continuum: 
kg m^{2}  [M][L]^{2} 
Derived kinematic quantities
Quantity (common name/s)  (Common) symbol/s  Defining equation  SI units  Dimension  

Velocity  v 

m s^{−1}  [L][T]^{−1}  
Acceleration  a 

m s^{−2}  [L][T]^{−2}  
Jerk  j 

m s^{−3}  [L][T]^{−3}  
Angular velocity  ω 

rad s^{−1}  [T]^{−1}  
Angular Acceleration  α 

rad s^{−2}  [T]^{−2} 
Derived dynamic quantities
Quantity (common name/s)  (Common) symbol/s  Defining equation  SI units  Dimension  

Momentum  p 

kg m s^{−1}  [M][L][T]^{−1}  
Force  F 

N = kg m s^{−2}  [M][L][T]^{−2}  
Impulse  Δp, I 

kg m s^{−1}  [M][L][T]^{−1}  
Angular momentum about a position point r_{0},  L, J, S 
Most of the time we can set r_{0} = 0 if particles are orbiting about axes intersecting at a common point. 
kg m^{2} s^{−1}  [M][L]^{2}[T]^{−1}  
Moment of a force about a position point r_{0},  τ, M 

N m = kg m^{2} s^{−2}  [M][L]^{2}[T]^{−2}  
Angular impulse  ΔL (no common symbol) 

kg m^{2} s^{−1}  [M][L]^{2}[T]^{−1} 
General energy definitions
Main article: Mechanical energy
Quantity (common name/s)  (Common) symbol/s  Defining equation  SI units  Dimension  

Mechanical work due
to a Resultant Force 
W 

J = N m = kg m^{2} s^{−2}  [M][L]^{2}[T]^{−2}  
Work done ON mechanical
system, Work done BY 
W_{ON}, W_{BY} 

J = N m = kg m^{2} s^{−2}  [M][L]^{2}[T]^{−2}  
Potential energy  φ, Φ, U, V, E_{p} 

J = N m = kg m^{2} s^{−2}  [M][L]^{2}[T]^{−2}  
Mechanical power  P 

W = J s^{−1}  [M][L]^{2}[T]^{−3} 
Every conservative force has a potential energy. By following two principles one can consistently assign a nonrelative 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 


varies with choice  varies with choice  
Generalized momenta  p, P 

varies with choice  varies with choice  
Lagrangian  L 
where 
J  [M][L]^{2}[T]^{−2}  
Hamiltonian  H 

J  [M][L]^{2}[T]^{−2}  
Action, Hamilton's principle function 
S,


J s  [M][L]^{2}[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
defines the axis of rotation,
Translation  Rotation  

Velocity 
Average:
Instantaneous: 
Angular velocity
Rotating rigid body: 

Acceleration 
Average:
Instantaneous: 
Angular acceleration
Rotating rigid body: 

Jerk 
Average:
Instantaneous: 
Angular jerk
Rotating rigid body: 
Dynamics
Translation  Rotation  

Momentum 
Momentum is the "amount of translation"
For a rotating rigid body: 
Angular momentum
Angular momentum is the "amount of rotation": and the crossproduct is a pseudovector i.e. if r and p are reversed in direction (negative), L is not. In general I is an order2 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:
For a number of particles, the equation of motion for one particle i is:^{[7]} 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:^{[8]} For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: Likewise, for a number of particles, the equation of motion for one particle i is:^{[9]} 

Yank 
Yank is rate of change of force:
For constant mass, it becomes; 
Rotatum
Rotatum Ρ is also called moment of a Yank, becuause it is the rotational analogue to yank: 

Impulse 
Impulse is the change in momentum:
For constant force F: 
Angular impulse is the change in angular momentum:
For constant torque τ: 
Precession
The precession angular speed of a spinning top is given by:
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 workenergy 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:
where θ is the angle of rotation about an axis defined by a unit vector n.
For a stretched spring fixed at one end obeying Hooke's law:
where r_{2} and r_{1} 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:^{[10]}
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,
the following general results apply to the particle.
Kinematics  Dynamics  

Position


Velocity

Momentum
Angular momenta 

Acceleration

The centripetal force is
where again m is the mass moment, and the coriolis force is The Coriolis acceleration and force can also be written: 
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 centres of masses of the two objects, the equation of motion is:
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  















Galilean frame transforms
For classical (GalileoNewtonian) 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'. 
Relative position
Relative velocity Equivalent accelerations 
Relative accelerations
Apparent/fictitious forces 

Rotation
Ω = Constant relative angular velocity between two frames F and F'. 
Relative angular position
Relative velocity Equivalent accelerations 
Relative accelerations
Apparent/fictitious torques 

Transformation of any vector T to a rotating frame

Mechanical oscillators
SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.
Physical situation  Nomenclature  Translational equations  Angular equations  

SHM 

Solution: 
Solution: 

Unforced DHM 

Solution (see below for ω'): Resonant frequency: Damping rate: Expected lifetime of excitation:

Solution: Resonant frequency: Damping rate: Expected lifetime of excitation:

Physical situation  Nomenclature  Equations  

Linear undamped unforced SHO 



Linear unforced DHO 



Low amplitude angular SHO 



Low amplitude simple pendulum 

Approximate value
Exact value can be shown to be: 
Physical situation  Nomenclature  Equations  

SHM energy 

Potential energy
Kinetic energy Total energy 

DHM energy 

See also
Notes
 ^ Mayer, Sussman & Wisdom 2001, p. xiii
 ^ Berkshire & Kibble 2004, p. 1
 ^ Berkshire & Kibble 2004, p. 2
 ^ Arnold 1989, p. v
 ^ http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm, Section: Moments and center of mass
 ^ R.P. Feynman, R.B. Leighton, M. Sands (1964). Feynman's Lectures on Physics (volume 2). AddisonWesley. pp. 31–7. ISBN 9780201021172.
 ^ "Relativity, J.R. Forshaw 2009"
 ^ "Mechanics, D. Kleppner 2010"
 ^ "Relativity, J.R. Forshaw 2009"
 ^ "Relativity, J.R. Forshaw 2009"
References
 Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics (2nd ed.), Springer, ISBN 9780387968902
 Berkshire, Frank H.; Kibble, T. W. B. (2004), Classical Mechanics (5th ed.), Imperial College Press, ISBN 9781860944352
 Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001), Structure and Interpretation of Classical Mechanics, MIT Press, ISBN 9780262194556