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

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

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

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

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

A second-order dynamic equation

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

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