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 Q\to \mathbb R over \mathbb R. For instance, this is the case of non-autonomous mechanics.

An r-order differential equation on a fiber bundle Q\to \mathbb R is represented by a closed subbundle of a jet bundle J^rQ of Q\to \mathbb R. A dynamic equation on Q\to \mathbb R is a differential equation which is algebraically solved for a higher-order derivatives.

In particular, a first-order dynamic equation on a fiber bundle Q\to \mathbb R is a kernel of the covariant differential of some connection \Gamma on Q\to \mathbb R. Given bundle coordinates (t,q^i) on Q and the adapted coordinates (t,q^i,q^i_t) on a first-order jet manifold J^1Q, a first-order dynamic equation reads

q^i_t=\Gamma (t,q^i).

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

A second-order dynamic equation


on Q\to\mathbb R is defined as a holonomic connection \xi on a jet bundle J^1Q\to\mathbb R. This equation also is represented by a connection on an affine jet bundle J^1Q\to Q. Due to the canonical imbedding J^1Q\to TQ, it is equivalent to a geodesic equation on the tangent bundle TQ of Q. 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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