Non-autonomous system (mathematics)

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In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-autonomous mechanics.

An r-order differential equation on a fiber bundle is represented by a closed subbundle of a jet bundle of . A dynamic equation on is a differential equation which is algebraically solved for a higher-order derivatives.

In particular, a first-order dynamic equation on a fiber bundle is a kernel of the covariant differential of some connection on . Given bundle coordinates on and the adapted coordinates on a first-order jet manifold , a first-order dynamic equation reads

For instance, this is the case of Hamiltonian non-autonomous mechanics.

A second-order dynamic equation

on is defined as a holonomic connection on a jet bundle . This equation also is represented by a connection on an affine jet bundle . Due to the canonical imbedding , it is equivalent to a geodesic equation on the tangent bundle of . A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.


  • 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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