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List of equations in classical mechanics

From Wikipedia, the free encyclopedia

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).

m = ∬ σ d S {\displaystyle m=\iint \sigma \,\mathrm {d} S}

m = ∭ ρ d V {\displaystyle m=\iiint \rho \,\mathrm {d} V}

m = r m {\displaystyle \mathbf {m} =\mathbf {r} m}

Discrete masses about an axis x i {\displaystyle x_{i}} : m = ∑ i = 1 N r i m i {\displaystyle \mathbf {m} =\sum _{i=1}^{N}\mathbf {r} _{i}m_{i}}

Continuum of mass about an axis x i {\displaystyle x_{i}} : m = ∫ ρ ( r ) x i d r {\displaystyle \mathbf {m} =\int \rho \left(\mathbf {r} \right)x_{i}\mathrm {d} \mathbf {r} }

(Symbols vary)

Discrete masses: r c o m = 1 M ∑ i r i m i = 1 M ∑ i m i {\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}={\frac {1}{M}}\sum _{i}\mathbf {m} _{i}}

Mass continuum: r c o m = 1 M ∫ d m = 1 M ∫ r d m = 1 M ∫ r ρ d V {\displaystyle \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}

I = ∑ i m i ⋅ r i = ∑ i | r i | 2 m {\displaystyle I=\sum _{i}\mathbf {m} _{i}\cdot \mathbf {r} _{i}=\sum _{i}\left|\mathbf {r} _{i}\right|^{2}m}

Mass continuum: I = ∫ | r | 2 d m = ∫ r ⋅ d m = ∫ | r | 2 ρ d V {\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}

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.

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.

where q = q ( t ) {\displaystyle \mathbf {q} =\mathbf {q} (t)} and p = p(t) are vectors of the generalized coords and momenta, as functions of time

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

defines the axis of rotation, e ^ r {\displaystyle \scriptstyle \mathbf {\hat {e}} _{r}} = unit vector in direction of r, e ^ θ {\displaystyle \scriptstyle \mathbf {\hat {e}} _{\theta }} = unit vector tangential to the angle.

v a v e r a g e = Δ r Δ t {\displaystyle \mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}} Instantaneous:

v = d r d t {\displaystyle \mathbf {v} ={d\mathbf {r} \over dt}}

a a v e r a g e = Δ v Δ t {\displaystyle \mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}}

Instantaneous:

a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}

α = d ω d t = n ^ d 2 θ d t 2 {\displaystyle {\boldsymbol {\alpha }}={\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}}

Rotating rigid body:

a = α × r + ω × v {\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} }

j a v e r a g e = Δ a Δ t {\displaystyle \mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}}

Instantaneous:

j = d a d t = d 2 v d t 2 = d 3 r d t 3 {\displaystyle \mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {r} }{dt^{3}}}}

ζ = d α d t = n ^ d 2 ω d t 2 = n ^ d 3 θ d t 3 {\displaystyle {\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}}}}

Rotating rigid body:

j = ζ × r + α × a {\displaystyle \mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a} }

p = m v {\displaystyle \mathbf {p} =m\mathbf {v} }

For a rotating rigid body:

p = ω × m {\displaystyle \mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} }

Angular momentum is the "amount of rotation":

L = r × p = I ⋅ ω {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}}

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.

F = d p d t = d ( m v ) d t = m a + v d m d t {\displaystyle {\begin{aligned}\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{aligned}}}

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

d p i d t = F E + ∑ i ≠ j F i j {\displaystyle {\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}}

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 τ is also called moment of a force, because it is the rotational analogue to force:^[8]

τ = d L d t = r × F = d ( I ⋅ ω ) d t {\displaystyle {\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}}}

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

τ = d L d t = d ( I ⋅ ω ) d t = d I d t ⋅ ω + I ⋅ α {\displaystyle {\begin{aligned}{\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{aligned}}}

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

d L i d t = τ E + ∑ i ≠ j τ i j {\displaystyle {\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}}

Y = d F d t = d 2 p d t 2 = d 2 ( m v ) d t 2 = m j + 2 a d m d t + v d 2 m d t 2 {\displaystyle {\begin{aligned}\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{aligned}}}

For constant mass, it becomes; Y = m j {\displaystyle \mathbf {Y} =m\mathbf {j} }

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

P = d τ d t = r × Y = d ( I ⋅ α ) d t {\displaystyle {\boldsymbol {\mathrm {P} }}={\frac {{\rm {d}}{\boldsymbol {\tau }}}{{\rm {d}}t}}=\mathbf {r} \times \mathbf {Y} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\alpha }})}{{\rm {d}}t}}}

Δ p = ∫ F d t {\displaystyle \Delta \mathbf {p} =\int \mathbf {F} \,dt}

For constant force F:

Δ p = F Δ t {\displaystyle \Delta \mathbf {p} =\mathbf {F} \Delta t}

Δ L = ∫ τ d t {\displaystyle \Delta \mathbf {L} =\int {\boldsymbol {\tau }}\,dt}

For constant torque τ:

Δ L = τ Δ t {\displaystyle \Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t}

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

Ω = w r I ω {\displaystyle {\boldsymbol {\Omega }}={\frac {wr}{I{\boldsymbol {\omega }}}}}

where w 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 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 = Δ T = ∫ C ( F ⋅ d r + τ ⋅ n d θ ) {\displaystyle W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\boldsymbol {\tau }}\cdot \mathbf {n} \,{\mathrm {d} \theta }\right)}

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

The change in kinetic energy for an object initially traveling at speed v 0 {\displaystyle v_{0}} and later at speed v {\displaystyle v} is: Δ E k = W = 1 2 m ( v 2 − v 0 2 ) {\displaystyle \Delta E_{k}=W={\frac {1}{2}}m(v^{2}-{v_{0}}^{2})}

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

Δ E p = 1 2 k ( r 2 − r 1 ) 2 {\displaystyle \Delta E_{p}={\frac {1}{2}}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 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 {\displaystyle \mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}}

where I is the moment of inertia tensor.

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

the following general results apply to the particle.

r = r ( r , θ , t ) = r r ^ {\displaystyle \mathbf {r} =\mathbf {r} \left(r,\theta ,t\right)=r{\hat {\mathbf {r} }}}

v = r ^ d r d t + r ω θ ^ {\displaystyle \mathbf {v} ={\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}}

p = m ( r ^ d r d t + r ω θ ^ ) {\displaystyle \mathbf {p} =m\left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}

Angular momenta L = m r × ( r ^ d r d t + r ω θ ^ ) {\displaystyle \mathbf {L} =m\mathbf {r} \times \left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}

a = ( d 2 r d t 2 − r ω 2 ) r ^ + ( r α + 2 ω d r d t ) θ ^ {\displaystyle \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 } }}}

F ⊥ = − m ω 2 R r ^ = − ω 2 m {\displaystyle \mathbf {F} _{\bot }=-m\omega ^{2}R{\hat {\mathbf {r} }}=-\omega ^{2}\mathbf {m} }

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

F c = 2 ω m d r d t θ ^ = 2 ω m v θ ^ {\displaystyle \mathbf {F} _{c}=2\omega m{\frac {{\rm {d}}r}{{\rm {d}}t}}{\hat {\mathbf {\theta } }}=2\omega mv{\hat {\mathbf {\theta } }}}

The Coriolis acceleration and force can also be written:

F c = m a c = − 2 m ω × v {\displaystyle \mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}}

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:

d 2 d θ 2 ( 1 r ) + 1 r = − μ r 2 l 2 F ( r ) {\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 general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

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.

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

r ′ = r + V t {\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t}

Relative velocity v ′ = v + V {\displaystyle \mathbf {v} '=\mathbf {v} +\mathbf {V} }

Equivalent accelerations a ′ = a {\displaystyle \mathbf {a} '=\mathbf {a} }

a ′ = a + A {\displaystyle \mathbf {a} '=\mathbf {a} +\mathbf {A} }

Apparent/fictitious forces F ′ = F − F a p p {\displaystyle \mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }}

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

θ ′ = θ + Ω t {\displaystyle \theta '=\theta +\Omega t} Relative velocity ω ′ = ω + Ω {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}}

Equivalent accelerations α ′ = α {\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}}

α ′ = α + Λ {\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}}

Apparent/fictitious torques τ ′ = τ − τ a p p {\displaystyle {\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }}

d T ′ d t = d T d t − Ω × T {\displaystyle {\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T} }

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

• x = Transverse displacement
• θ = Angular displacement
• A = Transverse amplitude
• Θ = Angular amplitude

Solution: x = A sin ⁡ ( ω t + ϕ ) {\displaystyle x=A\sin \left(\omega t+\phi \right)}

Solution: θ = Θ sin ⁡ ( ω t + ϕ ) {\displaystyle \theta =\Theta \sin \left(\omega t+\phi \right)}

• b = damping constant
• κ = torsion constant

Solution (see below for ω'): x = A e − b t / 2 m cos ⁡ ( ω ′ ) {\displaystyle x=Ae^{-bt/2m}\cos \left(\omega '\right)}

Resonant frequency: ω r e s = ω 2 − ( b 4 m ) 2 {\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}}

Damping rate: γ = b / m {\displaystyle \gamma =b/m}

Expected lifetime of excitation: τ = 1 / γ {\displaystyle \tau =1/\gamma }

Solution: θ = Θ e − κ t / 2 m cos ⁡ ( ω ) {\displaystyle \theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)}

Resonant frequency: ω r e s = ω 2 − ( κ 4 m ) 2 {\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}}

Damping rate: γ = κ / m {\displaystyle \gamma =\kappa /m}

Expected lifetime of excitation: τ = 1 / γ {\displaystyle \tau =1/\gamma }

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

U = m 2 ( x ) 2 = m ( ω A ) 2 2 cos 2 ⁡ ( ω t + ϕ ) {\displaystyle U={\frac {m}{2}}\left(x\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\cos ^{2}(\omega t+\phi )} Maximum value at x = A: U m a x = m 2 ( ω A ) 2 {\displaystyle U_{\mathrm {max} }={\frac {m}{2}}\left(\omega A\right)^{2}}

Kinetic energy T = ω 2 m 2 ( d x d t ) 2 = m ( ω A ) 2 2 sin 2 ⁡ ( ω t + ϕ ) {\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+\phi \right)}

Total energy E = T + U {\displaystyle E=T+U}

• 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

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