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Free motion equation

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A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a reference frame. In non-autonomous mechanics on a configuration space Q → R {\displaystyle Q\to \mathbb {R} } , a free motion equation is defined as a second order non-autonomous dynamic equation on Q → R {\displaystyle Q\to \mathbb {R} } which is brought into the form

q ¯ t t i = 0 {\displaystyle {\overline {q}}_{tt}^{i}=0}

with respect to some reference frame ( t , q ¯ i ) {\displaystyle (t,{\overline {q}}^{i})} on Q → R {\displaystyle Q\to \mathbb {R} } . Given an arbitrary reference frame ( t , q i ) {\displaystyle (t,q^{i})} on Q → R {\displaystyle Q\to \mathbb {R} } , a free motion equation reads

q t t i = d t Γ i + ∂ j Γ i ( q t j − Γ j ) − ∂ q i ∂ q ¯ m ∂ q ¯ m ∂ q j ∂ q k ( q t j − Γ j ) ( q t k − Γ k ) , {\displaystyle q_{tt}^{i}=d_{t}\Gamma ^{i}+\partial _{j}\Gamma ^{i}(q_{t}^{j}-\Gamma ^{j})-{\frac {\partial q^{i}}{\partial {\overline {q}}^{m}}}{\frac {\partial {\overline {q}}^{m}}{\partial q^{j}\partial q^{k}}}(q_{t}^{j}-\Gamma ^{j})(q_{t}^{k}-\Gamma ^{k}),}

where Γ i = ∂ t q i ( t , q ¯ j ) {\displaystyle \Gamma ^{i}=\partial _{t}q^{i}(t,{\overline {q}}^{j})} is a connection on Q → R {\displaystyle Q\to \mathbb {R} } associates with the initial reference frame ( t , q ¯ i ) {\displaystyle (t,{\overline {q}}^{i})} . The right-hand side of this equation is treated as an inertial force.

A free motion equation need not exist in general. It can be defined if and only if a configuration bundle Q → R {\displaystyle Q\to \mathbb {R} } of a mechanical system is a toroidal cylinder T m × R k {\displaystyle T^{m}\times \mathbb {R} ^{k}} .

See also

References

  • De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989).
  • Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 (arXiv:0911.0411).


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