Non-autonomous system (mathematics)

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In mathematics, 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 $J^{r}Q$ 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 $(t,q^{i},q_{t}^{i})$ on a first-order jet manifold $J^{1}Q$ , a first-order dynamic equation reads

$q_{t}^{i}=\Gamma (t,q^{i}).$

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

A second-order dynamic equation

$q_{{tt}}^{i}=\xi ^{i}(t,q^{j},q_{t}^{j})$

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

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).

Non-autonomous system (mathematics)

References

See also