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List of equations in classical mechanics
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<h2 id="classical-mechanics">Classical mechanics</h2> <h3>Mass and inertia</h3> <table><tbody><tr><th scope="col">Quantity (common name/s)</th><th scope="col">(Common) symbol/s</th><th scope="col">Defining equation</th><th scope="col">SI units</th><th scope="col">Dimension</th></tr><tr><td>Linear, surface, volumetric mass density</td><td><i>λ</i> or <i>μ</i> (especially in <a href="/facts/Acoustics/2GpQUymy">acoustics</a>, see below) for Linear, <i>σ</i> for surface, <i>ρ</i> for volume.</td><td> m = ∫ λ d ℓ {\displaystyle m=\int \lambda \,\mathrm {d} \ell } <p> m = ∬ σ d S {\displaystyle m=\iint \sigma \,\mathrm {d} S} </p><p> m = ∭ ρ d V {\displaystyle m=\iiint \rho \,\mathrm {d} V} </p></td><td>kg m−<i>n</i>, <i>n</i> = 1, 2, 3</td><td>M L−<i>n</i></td></tr><tr><td>Moment of mass<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></td><td>m (No common symbol)</td><td>Point mass:<p> m = r m {\displaystyle \mathbf {m} =\mathbf {r} m} </p><p>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}} </p><p>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} } </p></td><td>kg m</td><td>M L</td></tr><tr><td><a href="/facts/Center_of_mass/n4Lr9GAz">Center of mass</a></td><td>rcom<p>(Symbols vary)</p></td><td><i>i</i>-th moment of mass m i = r i m i {\displaystyle \mathbf {m} _{i}=\mathbf {r} _{i}m_{i}} <p>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}} </p><p>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} </p></td><td>m</td><td>L</td></tr><tr><td>2-Body reduced mass</td><td><i>m</i>12, <i>μ</i> Pair of masses = <i>m</i>1 and <i>m</i>2</td><td> μ = m 1 m 2 m 1 + m 2 {\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}} </td><td>kg</td><td>M</td></tr><tr><td>Moment of inertia (MOI)</td><td><i>I</i></td><td>Discrete Masses:<p> 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} </p><p>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} </p></td><td>kg m2</td><td>M L2</td></tr></tbody></table> <h3>Derived kinematic quantities</h3> <table><tbody><tr><th scope="col">Quantity (common name/s)</th><th scope="col">(Common) symbol/s</th><th scope="col">Defining equation</th><th scope="col">SI units</th><th scope="col">Dimension</th></tr><tr><td><a href="/facts/Velocity/Q3o1LVDe">Velocity</a></td><td>v</td><td> v = d r d t {\displaystyle \mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}} </td><td>m s−1</td><td>L T−1</td></tr><tr><td><a href="/facts/Acceleration/UVDmjYvR">Acceleration</a></td><td>a</td><td> a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}} </td><td>m s−2</td><td>L T−2</td></tr><tr><td><a href="/facts/Jerk_(physics)/8T5H2g0C">Jerk</a></td><td>j</td><td> j = d a d t = d 3 r d t 3 {\displaystyle \mathbf {j} ={\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{3}\mathbf {r} }{\mathrm {d} t^{3}}}} </td><td>m s−3</td><td>L T−3</td></tr><tr><td><a href="/facts/Jounce/t8AhXSsc">Jounce</a></td><td>s</td><td> s = d j d t = d 4 r d t 4 {\displaystyle \mathbf {s} ={\frac {\mathrm {d} \mathbf {j} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{4}\mathbf {r} }{\mathrm {d} t^{4}}}} </td><td>m s−4</td><td>L T−4</td></tr><tr><td><a href="/facts/Angular_velocity/2n8JljAT">Angular velocity</a></td><td>ω</td><td> ω = n ^ d θ d t {\displaystyle {\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {\mathrm {d} \theta }{\mathrm {d} t}}} </td><td>rad s−1</td><td>T−1</td></tr><tr><td><a href="/facts/Angular_acceleration/jDv9uzAS">Angular Acceleration</a></td><td>α</td><td> α = d ω d t = n ^ d 2 θ d t 2 {\displaystyle {\boldsymbol {\alpha }}={\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}} </td><td>rad s−2</td><td>T−2</td></tr><tr><td><a href="/facts/Angular_jerk/8T5H2g0C">Angular jerk</a></td><td>ζ</td><td> ζ = d α d t = n ^ d 3 θ d t 3 {\displaystyle {\boldsymbol {\zeta }}={\frac {\mathrm {d} {\boldsymbol {\alpha }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{3}\theta }{\mathrm {d} t^{3}}}} </td><td>rad s−3</td><td>T−3</td></tr></tbody></table> <h3>Derived dynamic quantities</h3> <table><tbody><tr><th scope="col">Quantity (common name/s)</th><th scope="col">(Common) symbol/s</th><th scope="col">Defining equation</th><th scope="col">SI units</th><th scope="col">Dimension</th></tr><tr><td><a href="/facts/Momentum/TTkGv5eY">Momentum</a></td><td>p</td><td> p = m v {\displaystyle \mathbf {p} =m\mathbf {v} } </td><td>kg m s−1</td><td>M L T−1</td></tr><tr><td><a href="/facts/Force/533Q1uzq">Force</a></td><td>F</td><td> F = d p / d t {\displaystyle \mathbf {F} =\mathrm {d} \mathbf {p} /\mathrm {d} t} </td><td>N = kg m s−2</td><td>M L T−2</td></tr><tr><td><a href="/facts/Impulse_(physics)/maePm9Ua">Impulse</a></td><td>J, Δp, I</td><td> J = Δ p = ∫ t 1 t 2 F d t {\displaystyle \mathbf {J} =\Delta \mathbf {p} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t} </td><td>kg m s−1</td><td>M L T−1</td></tr><tr><td><a href="/facts/Angular_momentum/7TuSVFxq">Angular momentum</a> about a position point r0,</td><td>L, J, S</td><td> L = ( r − r 0 ) × p {\displaystyle \mathbf {L} =\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {p} } <p>Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point.</p></td><td>kg m2 s−1</td><td>M L2 T−1</td></tr><tr><td>Moment of a force about a position point r0,<p><a href="/facts/Torque/KcrcvNtV">Torque</a></p></td><td>τ, M</td><td> τ = ( r − r 0 ) × F = d L d t {\displaystyle {\boldsymbol {\tau }}=\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {F} ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}} </td><td>N m = kg m2 s−2</td><td>M L2 T−2</td></tr><tr><td>Angular impulse</td><td>ΔL (no common symbol)</td><td> Δ L = ∫ t 1 t 2 τ d t {\displaystyle \Delta \mathbf {L} =\int _{t_{1}}^{t_{2}}{\boldsymbol {\tau }}\,\mathrm {d} t} </td><td>kg m2 s−1</td><td>M L2 T−1</td></tr></tbody></table> <h3>General energy definitions</h3> <p class="note">Main article: <a href="/facts/Mechanical_energy/gI4oMzoY">Mechanical energy</a></p> <table><tbody><tr><th scope="col">Quantity (common name/s)</th><th scope="col">(Common) symbol/s</th><th scope="col">Defining equation</th><th scope="col">SI units</th><th scope="col">Dimension</th></tr><tr><td><a href="/facts/Work_(physics)/mbclFPBs">Mechanical work</a> due to a Resultant Force</td><td><i>W</i></td><td> W = ∫ C F ⋅ d r {\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r} } </td><td>J = N m = kg m2 s−2</td><td>M L2 T−2</td></tr><tr><td>Work done ON mechanical system, Work done BY</td><td><i>W</i>ON, <i>W</i>BY</td><td> Δ W O N = − Δ W B Y {\displaystyle \Delta W_{\mathrm {ON} }=-\Delta W_{\mathrm {BY} }} </td><td>J = N m = kg m2 s−2</td><td>M L2 T−2</td></tr><tr><td><a href="/facts/Potential_energy/I03KLbmn">Potential energy</a></td><td><i>φ</i>, Φ, <i>U</i>, <i>V</i>, <i>Ep</i></td><td> Δ W = − Δ V {\displaystyle \Delta W=-\Delta V} </td><td>J = N m = kg m2 s−2</td><td>M L2 T−2</td></tr><tr><td>Mechanical <a href="/facts/Power_(physics)/mkaEyRnn">power</a></td><td><i>P</i></td><td> P = d E d t {\displaystyle P={\frac {\mathrm {d} E}{\mathrm {d} t}}} </td><td>W = J s−1</td><td>M L2 T−3</td></tr></tbody></table> <p>Every <a href="/facts/Conservative_force/xY5QmC4q">conservative force</a> has a <a href="/facts/Potential_energy/I03KLbmn">potential energy</a>. By following two principles one can consistently assign a non-relative value to <i>U</i>: </p> <ul><li>Wherever the force is zero, its potential energy is defined to be zero as well.</li> <li>Whenever the force does work, potential energy is lost.</li></ul> <h3>Generalized mechanics</h3> <p class="note">Main articles: <a href="/facts/Analytical_mechanics/aotB0n4t">Analytical mechanics</a>, <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian mechanics</a>, and <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a></p> <table><tbody><tr><th scope="col">Quantity (common name/s)</th><th scope="col">(Common) symbol/s</th><th scope="col">Defining equation</th><th scope="col">SI units</th><th scope="col">Dimension</th></tr><tr><td><a href="/facts/Generalized_coordinates/xrhApGk2">Generalized coordinates</a></td><td><i>q, Q</i></td><td></td><td>varies with choice</td><td>varies with choice</td></tr><tr><td><a href="/facts/Generalized_velocities/xrhApGk2">Generalized velocities</a></td><td> q ˙ , Q ˙ {\displaystyle {\dot {q}},{\dot {Q}}} </td><td> q ˙ ≡ d q / d t {\displaystyle {\dot {q}}\equiv \mathrm {d} q/\mathrm {d} t} </td><td>varies with choice</td><td>varies with choice</td></tr><tr><td><a href="/facts/Canonical_coordinates/9g2SFFRz">Generalized momenta</a></td><td><i>p, P</i></td><td> p = ∂ L / ∂ q ˙ {\displaystyle p=\partial L/\partial {\dot {q}}} </td><td>varies with choice</td><td>varies with choice</td></tr><tr><td><a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian</a></td><td><i>L</i></td><td> L ( q , q ˙ , t ) = T ( q ˙ ) − V ( q , q ˙ , t ) {\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} )-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)} <p>where q = q ( t ) {\displaystyle \mathbf {q} =\mathbf {q} (t)} and p = p(<i>t</i>) are vectors of the generalized coords and momenta, as functions of time</p></td><td>J</td><td>M L2 T−2</td></tr><tr><td><a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian</a></td><td><i>H</i></td><td> H ( p , q , t ) = p ⋅ q ˙ − L ( q , q ˙ , t ) {\displaystyle H(\mathbf {p} ,\mathbf {q} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)} </td><td>J</td><td>M L2 T−2</td></tr><tr><td><a href="/facts/Action_(physics)/NhWS2Mts">Action</a>, Hamilton's principal function</td><td><i>S</i>, S {\displaystyle \scriptstyle {\mathcal {S}}} </td><td> S = ∫ t 1 t 2 L ( q , q ˙ , t ) d t {\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\mathrm {d} t} </td><td>J s</td><td>M L2 T−1</td></tr></tbody></table> <h2 id="kinematics">Kinematics</h2> <p>In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use <i>θ</i>, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector </p><p> n ^ = e ^ r × e ^ θ {\displaystyle \mathbf {\hat {n}} =\mathbf {\hat {e}} _{r}\times \mathbf {\hat {e}} _{\theta }} </p><p>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. </p> <table><tbody><tr><th></th><th>Translation</th><th>Rotation</th></tr><tr><th><a href="/facts/Velocity/Q3o1LVDe">Velocity</a></th><td>Average:<p> v a v e r a g e = Δ r Δ t {\displaystyle \mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}} Instantaneous:</p><p> v = d r d t {\displaystyle \mathbf {v} ={d\mathbf {r} \over dt}} </p></td><td><a href="/facts/Angular_velocity/2n8JljAT">Angular velocity</a> ω = n ^ d θ d t {\displaystyle {\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {{\rm {d}}\theta }{{\rm {d}}t}}} Rotating <a href="/facts/Rigid_body/THezBlq2">rigid body</a>: v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } </td></tr><tr><th><a href="/facts/Acceleration/UVDmjYvR">Acceleration</a></th><td>Average:<p> a a v e r a g e = Δ v Δ t {\displaystyle \mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}} </p><p>Instantaneous:</p><p> 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}}}} </p></td><td><a href="/facts/Angular_acceleration/jDv9uzAS">Angular acceleration</a><p> 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}}}} </p><p>Rotating rigid body:</p><p> a = α × r + ω × v {\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} } </p></td></tr><tr><th><a href="/facts/Jerk_(physics)/8T5H2g0C">Jerk</a></th><td>Average:<p> j a v e r a g e = Δ a Δ t {\displaystyle \mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}} </p><p>Instantaneous:</p><p> 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}}}} </p></td><td><a href="/facts/Angular_jerk/8T5H2g0C">Angular jerk</a><p> 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}}}} </p><p>Rotating rigid body:</p><p> j = ζ × r + α × a {\displaystyle \mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a} } </p></td></tr></tbody></table> <h2 id="dynamics">Dynamics</h2> <table><tbody><tr><th></th><th scope="col">Translation</th><th scope="col">Rotation</th></tr><tr><th><a href="/facts/Momentum/TTkGv5eY">Momentum</a></th><td>Momentum is the "amount of translation"<p> p = m v {\displaystyle \mathbf {p} =m\mathbf {v} } </p><p>For a rotating rigid body:</p><p> p = ω × m {\displaystyle \mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} } </p></td><td><a href="/facts/Angular_momentum/7TuSVFxq">Angular momentum</a><p>Angular momentum is the "amount of rotation":</p><p> L = r × p = I ⋅ ω {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}} </p><p>and the cross-product is a <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a> i.e. if r and p are reversed in direction (negative), L is not.</p><p>In general I is an order-2 <a href="/facts/Tensor/pNjFo5tV">tensor</a>, see above for its components. The dot · indicates <a href="/facts/Tensor_contraction/fqz0zvWp">tensor contraction</a>.</p></td></tr><tr><th><a href="/facts/Force/533Q1uzq">Force</a> and <a href="/facts/Newton%2527s_2nd_law/XKzGTXK1">Newton's 2nd law</a></th><td>Resultant force acts on a system at the center of mass, equal to the rate of change of momentum:<p> 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}}} </p><p>For a number of particles, the equation of motion for one particle <i>i</i> is:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></p><p> 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}} </p><p>where p<i>i</i> = momentum of particle <i>i</i>, F<i>ij</i> = force <i>on</i> particle <i>i</i> <i>by</i> particle <i>j</i>, and F<i>E</i> = resultant external force (due to any agent not part of system). Particle <i>i</i> does not exert a force on itself.</p></td><td><a href="/facts/Torque/KcrcvNtV">Torque</a><p>Torque τ is also called moment of a force, because it is the rotational analogue to force:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></p><p> 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}}} </p><p>For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation:</p><p> 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}}} </p><p>Likewise, for a number of particles, the equation of motion for one particle <i>i</i> is:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></p><p> 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}} </p></td></tr><tr><th><a href="/facts/Yank_(physics)/TIf4zMcJ">Yank</a></th><td>Yank is rate of change of force:<p> 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}}} </p><p>For constant mass, it becomes; Y = m j {\displaystyle \mathbf {Y} =m\mathbf {j} } </p></td><td><a href="/facts/Rotatum/dSm2Kz18">Rotatum</a><p>Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank:</p><p> 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}}} </p></td></tr><tr><th><a href="/facts/Impulse_(physics)/maePm9Ua">Impulse</a></th><td>Impulse is the change in momentum:<p> Δ p = ∫ F d t {\displaystyle \Delta \mathbf {p} =\int \mathbf {F} \,dt} </p><p>For constant force F:</p><p> Δ p = F Δ t {\displaystyle \Delta \mathbf {p} =\mathbf {F} \Delta t} </p></td><td>Twirl/angular impulse is the change in angular momentum:<p> Δ L = ∫ τ d t {\displaystyle \Delta \mathbf {L} =\int {\boldsymbol {\tau }}\,dt} </p><p>For constant torque τ:</p><p> Δ L = τ Δ t {\displaystyle \Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t} </p></td></tr></tbody></table> <h3>Precession</h3> <p>The precession angular speed of a <a href="/facts/Spinning_top/u50YUBP6">spinning top</a> is given by: </p><p> Ω = w r I ω {\displaystyle {\boldsymbol {\Omega }}={\frac {wr}{I{\boldsymbol {\omega }}}}} </p><p>where <i>w</i> is the weight of the spinning flywheel. </p> <h2 id="energy">Energy</h2> <p>The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system: </p> <h3>General <a href="/facts/Work-energy_theorem/mbclFPBs">work-energy theorem</a> (translation and rotation)</h3> <p>The work done <i>W</i> by an external agent which exerts a force F (at r) and torque τ on an object along a curved path <i>C</i> is: </p><p> 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)} </p><p>where θ is the angle of rotation about an axis defined by a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> n. </p> <h3>Kinetic energy</h3> <p>The change in <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a> 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})} </p> <h3>Elastic potential energy</h3> <p>For a stretched spring fixed at one end obeying <a href="/facts/Hooke%2527s_law/7hTUcKUm">Hooke's law</a>, the <a href="/facts/Elastic_potential_energy/dI07J3PI">elastic potential energy</a> is </p><p> Δ E p = 1 2 k ( r 2 − r 1 ) 2 {\displaystyle \Delta E_{p}={\frac {1}{2}}k(r_{2}-r_{1})^{2}} </p><p>where <i>r</i>2 and <i>r</i>1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant. </p> <h2 id="eulers-equations-for-rigid-body-dynamics">Euler's equations for rigid body dynamics</h2> <p class="note">Main article: <a href="/facts/Euler%2527s_equations_(rigid_body_dynamics)/1jo8Ta2E">Euler's equations (rigid body dynamics)</a></p> <p><a href="/facts/Euler/z2fsbjgj">Euler</a> also worked out analogous laws of motion to those of Newton, see <a href="/facts/Euler%2527s_laws_of_motion/2KymtP2P">Euler's laws of motion</a>. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p> I ⋅ α + ω × ( I ⋅ ω ) = τ {\displaystyle \mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}} </p><p>where I is the <a href="/facts/Moment_of_inertia/TpLLAVl7">moment of inertia</a> <a href="/facts/Tensor/pNjFo5tV">tensor</a>. </p> <h2 id="general-planar-motion">General planar motion</h2> <p class="note">See also: <a href="/facts/Polar_coordinate_system/HBko4YoN">Polar coordinate system (section: vector calculus)</a></p> <p>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, </p><p> r = r ( t ) = r r ^ {\displaystyle \mathbf {r} =\mathbf {r} (t)=r{\hat {\mathbf {r} }}} </p><p>the following general results apply to the particle. </p> <table><tbody><tr><th>Kinematics</th><th>Dynamics</th></tr><tr><td>Position<p> r = r ( r , θ , t ) = r r ^ {\displaystyle \mathbf {r} =\mathbf {r} \left(r,\theta ,t\right)=r{\hat {\mathbf {r} }}} </p></td><td></td></tr><tr><td>Velocity<p> 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></td><td>Momentum<p> 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)} </p><p>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)} </p></td></tr><tr><td>Acceleration<p> 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 } }}} </p></td><td>The <a href="/facts/Centripetal_force/VWBADgMM">centripetal force</a> is<p> F ⊥ = − m ω 2 R r ^ = − ω 2 m {\displaystyle \mathbf {F} _{\bot }=-m\omega ^{2}R{\hat {\mathbf {r} }}=-\omega ^{2}\mathbf {m} } </p><p>where again m is the mass moment, and the <a href="/facts/Coriolis_force/SqCyUY6d">Coriolis force</a> is</p><p> 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 } }}} </p><p>The <a href="/facts/Coriolis_effect/SqCyUY6d">Coriolis acceleration and force</a> can also be written:</p><p> F c = m a c = − 2 m ω × v {\displaystyle \mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}} </p></td></tr></tbody></table> <h3>Central force motion</h3> <p>For a massive body moving in a <a href="/facts/Central_potential/WT5WWNgq">central potential</a> 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: </p><p> 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} )} </p> <h2 id="equations-of-motion-constant-acceleration">Equations of motion (constant acceleration)</h2> <p>These equations can be used only when acceleration is constant. If acceleration is not constant then the general <a href="/facts/Calculus/MEmUTYoz">calculus</a> equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above). </p> <table><tbody><tr><th>Linear motion</th><th>Angular motion</th></tr><tr><td> v − v 0 = a t {\displaystyle \mathbf {v-v_{0}} =\mathbf {a} t} </td><td> ω − ω 0 = α t {\displaystyle {\boldsymbol {\omega -\omega _{0}}}={\boldsymbol {\alpha }}t} </td></tr><tr><td> x − x 0 = 1 2 ( v 0 + v ) t {\displaystyle \mathbf {x-x_{0}} ={\tfrac {1}{2}}(\mathbf {v_{0}+v} )t} </td><td> θ − θ 0 = 1 2 ( ω 0 + ω ) t {\displaystyle {\boldsymbol {\theta -\theta _{0}}}={\tfrac {1}{2}}({\boldsymbol {\omega _{0}+\omega }})t} </td></tr><tr><td> x − x 0 = v 0 t + 1 2 a t 2 {\displaystyle \mathbf {x-x_{0}} =\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}} </td><td> θ − θ 0 = ω 0 t + 1 2 α t 2 {\displaystyle {\boldsymbol {\theta -\theta _{0}}}={\boldsymbol {\omega }}_{0}t+{\tfrac {1}{2}}{\boldsymbol {\alpha }}t^{2}} </td></tr><tr><td> x n t h = v 0 + a ( n − 1 2 ) {\displaystyle \mathbf {x} _{n^{th}}=\mathbf {v} _{0}+\mathbf {a} (n-{\tfrac {1}{2}})} </td><td> θ n t h = ω 0 + α ( n − 1 2 ) {\displaystyle {\boldsymbol {\theta }}_{n^{th}}={\boldsymbol {\omega }}_{0}+{\boldsymbol {\alpha }}(n-{\tfrac {1}{2}})} </td></tr><tr><td> v 2 − v 0 2 = 2 a ( x − x 0 ) {\displaystyle v^{2}-v_{0}^{2}=2\mathbf {a(x-x_{0})} } </td><td> ω 2 − ω 0 2 = 2 α ( θ − θ 0 ) {\displaystyle \omega ^{2}-\omega _{0}^{2}=2{\boldsymbol {\alpha (\theta -\theta _{0})}}} </td></tr></tbody></table> <p class="note">See also: <a href="/facts/Linear_motion/Vom0QwWe">Linear motion § Analogy between linear and rotational motion</a></p> <h2 id="galilean-frame-transforms">Galilean frame transforms</h2> <p>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. </p><p>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. </p> <table><tbody><tr><th scope="col">Motion of entities</th><th scope="col">Inertial frames</th><th scope="col">Accelerating frames</th></tr><tr><td>Translation<p>V = Constant relative velocity between two inertial frames F and F'.A = (Variable) relative acceleration between two accelerating frames F and F'.</p></td><td>Relative position<p> r ′ = r + V t {\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t} </p><p>Relative velocity v ′ = v + V {\displaystyle \mathbf {v} '=\mathbf {v} +\mathbf {V} } </p><p>Equivalent accelerations a ′ = a {\displaystyle \mathbf {a} '=\mathbf {a} } </p></td><td>Relative accelerations<p> a ′ = a + A {\displaystyle \mathbf {a} '=\mathbf {a} +\mathbf {A} } </p><p>Apparent/fictitious forces F ′ = F − F a p p {\displaystyle \mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }} </p></td></tr><tr><td rowspan="2">Rotation<p>Ω = Constant relative angular velocity between two frames F and F'.Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.</p></td><td>Relative angular position<p> θ ′ = θ + Ω t {\displaystyle \theta '=\theta +\Omega t} Relative velocity ω ′ = ω + Ω {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}} </p><p>Equivalent accelerations α ′ = α {\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}} </p></td><td>Relative accelerations<p> α ′ = α + Λ {\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}} </p><p>Apparent/fictitious torques τ ′ = τ − τ a p p {\displaystyle {\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }} </p></td></tr><tr><td colspan="2">Transformation of any vector T to a rotating frame<p> 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} } </p></td></tr></tbody></table> <h2 id="mechanical-oscillators">Mechanical oscillators</h2> <p>SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively. </p> Equations of motion<table><tbody><tr><th scope="col">Physical situation</th><th scope="col">Nomenclature</th><th scope="col">Translational equations</th><th scope="col">Angular equations</th></tr><tr><th scope="row">SHM</th><td><ul><li><i>x</i> = Transverse displacement</li><li><i>θ</i> = Angular displacement</li><li><i>A</i> = Transverse amplitude</li><li>Θ = Angular amplitude</li></ul></td><td> d 2 x d t 2 = − ω 2 x {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x} <p>Solution: x = A sin ( ω t + ϕ ) {\displaystyle x=A\sin \left(\omega t+\phi \right)} </p></td><td> d 2 θ d t 2 = − ω 2 θ {\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta } <p>Solution: θ = Θ sin ( ω t + ϕ ) {\displaystyle \theta =\Theta \sin \left(\omega t+\phi \right)} </p></td></tr><tr><th scope="row">Unforced DHM</th><td><ul><li><i>b</i> = damping constant</li><li><i>κ</i> = torsion constant</li></ul></td><td> d 2 x d t 2 + b d x d t + ω 2 x = 0 {\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0} <p>Solution (see below for <i>ω'</i>): x = A e − b t / 2 m cos ( ω ′ ) {\displaystyle x=Ae^{-bt/2m}\cos \left(\omega '\right)} </p><p>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}}}} </p><p>Damping rate: γ = b / m {\displaystyle \gamma =b/m} </p><p>Expected lifetime of excitation: τ = 1 / γ {\displaystyle \tau =1/\gamma } </p></td><td> d 2 θ d t 2 + b d θ d t + ω 2 θ = 0 {\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0} <p>Solution: θ = Θ e − κ t / 2 m cos ( ω ) {\displaystyle \theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)} </p><p>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}}}} </p><p>Damping rate: γ = κ / m {\displaystyle \gamma =\kappa /m} </p><p>Expected lifetime of excitation: τ = 1 / γ {\displaystyle \tau =1/\gamma } </p></td></tr></tbody></table> Angular frequencies<table><tbody><tr><th scope="col">Physical situation</th><th scope="col">Nomenclature</th><th scope="col">Equations</th></tr><tr><th scope="row">Linear undamped unforced SHO</th><td><ul><li><i>k</i> = spring constant</li><li><i>m</i> = mass of oscillating bob</li></ul></td><td> ω = k m {\displaystyle \omega ={\sqrt {\frac {k}{m}}}} </td></tr><tr><th scope="row">Linear unforced DHO</th><td><ul><li><i>k</i> = spring constant</li><li><i>b</i> = Damping coefficient</li></ul></td><td> ω ′ = k m − ( b 2 m ) 2 {\displaystyle \omega '={\sqrt {{\frac {k}{m}}-\left({\frac {b}{2m}}\right)^{2}}}} </td></tr><tr><th scope="row">Low amplitude angular SHO</th><td><ul><li><i>I</i> = Moment of inertia about oscillating axis</li><li><i>κ</i> = torsion constant</li></ul></td><td> ω = κ I {\displaystyle \omega ={\sqrt {\frac {\kappa }{I}}}} </td></tr><tr><th scope="row">Low amplitude simple pendulum</th><td><ul><li><i>L</i> = Length of pendulum</li><li><i>g</i> = Gravitational acceleration</li><li>Θ = Angular amplitude</li></ul></td><td>Approximate value<p> ω = g L {\displaystyle \omega ={\sqrt {\frac {g}{L}}}} </p><p>Exact value can be shown to be: ω = g L [ 1 + ∑ k = 1 ∞ ∏ n = 1 k ( 2 n − 1 ) ∏ n = 1 m ( 2 n ) sin 2 n Θ ] {\displaystyle \omega ={\sqrt {\frac {g}{L}}}\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]} </p></td></tr></tbody></table> Energy in mechanical oscillations<table><tbody><tr><th scope="col">Physical situation</th><th scope="col">Nomenclature</th><th scope="col">Equations</th></tr><tr><th scope="row">SHM energy</th><td><ul><li><i>T</i> = kinetic energy</li><li><i>U</i> = potential energy</li><li><i>E</i> = total energy</li></ul></td><td>Potential energy<p> 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 <i>x</i> = <i>A</i>: U m a x = m 2 ( ω A ) 2 {\displaystyle U_{\mathrm {max} }={\frac {m}{2}}\left(\omega A\right)^{2}} </p><p>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)} </p><p>Total energy E = T + U {\displaystyle E=T+U} </p></td></tr><tr><th scope="row">DHM energy</th><td></td><td> E = m ( ω A ) 2 2 e − b t / m {\displaystyle E={\frac {m\left(\omega A\right)^{2}}{2}}e^{-bt/m}} </td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/List_of_physics_formulae/FtL8Dgtz">List of physics formulae</a></li> <li><a href="/facts/Defining_equation_(physical_chemistry)/AY7qGdJn">Defining equation (physical chemistry)</a></li> <li><a href="/facts/Constitutive_equation/IXTjEAbL">Constitutive equation</a></li> <li><a href="/facts/Mechanics/A3yHLQA5">Mechanics</a></li> <li><a href="/facts/Optics/p0vq4S1T">Optics</a></li> <li><a href="/facts/Electromagnetism/oAkmALHT">Electromagnetism</a></li> <li><a href="/facts/Thermodynamics/q4hIx3yP">Thermodynamics</a></li> <li><a href="/facts/Acoustics/2GpQUymy">Acoustics</a></li> <li><a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a></li> <li><a href="/facts/List_of_equations_in_wave_theory/XR5gMdgS">List of equations in wave theory</a></li> <li><a href="/facts/List_of_relativistic_equations/9HNtjaU0">List of relativistic equations</a></li> <li><a href="/facts/List_of_equations_in_fluid_mechanics/EwdLuErt">List of equations in fluid mechanics</a></li> <li><a href="/facts/List_of_equations_in_gravitation/aLfPQT7F">List of equations in gravitation</a></li> <li><a href="/facts/List_of_electromagnetism_equations/gK8bT9Tc">List of electromagnetism equations</a></li> <li><a href="/facts/List_of_photonics_equations/ss1YUgx1">List of photonics equations</a></li> <li><a href="/facts/List_of_equations_in_quantum_mechanics/M9zAsApQ">List of equations in quantum mechanics</a></li> <li><a href="/facts/List_of_equations_in_nuclear_and_particle_physics/vTdwmoxo">List of equations in nuclear and particle physics</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Arnold, Vladimir I. (1989), <a href="https://archive.org/details/mathematicalmeth0000arno"><i>Mathematical Methods of Classical Mechanics</i></a> (2nd ed.), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96890-2</li> <li><a href="/facts/Frank_H._Berkshire/KExUeTUp">Berkshire, Frank H.</a>; <a href="/facts/Tom_Kibble/3D1YbVel">Kibble, T. W. B.</a> (2004), <i><a href="/facts/Classical_Mechanics_(Kibble_and_Berkshire)/0AhTbH1W">Classical Mechanics</a></i> (5th ed.), Imperial College Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-86094-435-2</li> <li>Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001), <i>Structure and Interpretation of Classical Mechanics</i>, MIT Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-262-19455-6</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mayer, Sussman & Wisdom 2001, p. xiii - Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001), Structure and Interpretation of Classical Mechanics, MIT Press, ISBN 978-0-262-19455-6 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Berkshire & Kibble 2004, p. 1 - Berkshire, Frank H.; Kibble, T. W. B. (2004), Classical Mechanics (5th ed.), Imperial College Press, ISBN 978-1-86094-435-2 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Berkshire & Kibble 2004, p. 2 - Berkshire, Frank H.; Kibble, T. W. B. (2004), Classical Mechanics (5th ed.), Imperial College Press, ISBN 978-1-86094-435-2 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Arnold 1989, p. v - Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics (2nd ed.), Springer, ISBN 978-0-387-96890-2 <a href="https://archive.org/details/mathematicalmeth0000arno" target="_blank">https://archive.org/details/mathematicalmeth0000arno</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Section: Moments and center of mass". <a href="http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm" target="_blank">http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Relativity, J.R. Forshaw 2009" <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"Mechanics, D. Kleppner 2010" <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Relativity, J.R. Forshaw 2009" <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>"Relativity, J.R. Forshaw 2009" <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>