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 [math]\displaystyle{ Q\to \mathbb R }[/math] over [math]\displaystyle{ \mathbb R }[/math]. For instance, this is the case of non-autonomous mechanics. An r-order differential equation on a fiber bundle [math]\displaystyle{ Q\to \mathbb R }[/math] is represented by a closed subbundle of a jet bundle [math]\displaystyle{ J^rQ }[/math] of [math]\displaystyle{ Q\to \mathbb R }[/math]. A dynamic equation on [math]\displaystyle{ Q\to \mathbb R }[/math] is a differential equation which is algebraically solved for a higher-order derivatives.

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

[math]\displaystyle{ q^i_t=\Gamma (t,q^i). }[/math]

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

A second-order dynamic equation

[math]\displaystyle{ q^i_{tt}=\xi^i(t,q^j,q^j_t) }[/math]

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