Non-autonomous system (mathematics)

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 $J^{r}Q$ 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 embedding $J^{1}Q\to TQ$, 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.

See also

• Autonomous system (mathematics)
• Non-autonomous mechanics
• Free motion equation
• Relativistic system (mathematics)

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).