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 $\mathrm{\Gamma }$ 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

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

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).