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 over . For instance, this is the case of non-autonomous mechanics.
An r-order differential equation on a fiber bundle is represented by a closed subbundle of a jet bundle of . A dynamic equation on is a differential equation which is algebraically solved for a higher-order derivatives.
In particular, a first-order dynamic equation on a fiber bundle is a kernel of the covariant differential of some connection on . Given bundle coordinates on and the adapted coordinates on a first-order jet manifold , a first-order dynamic equation reads
For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation
on is defined as a holonomic connection on a jet bundle . This equation also is represented by a connection on an affine jet bundle . Due to the canonical embedding , it is equivalent to a geodesic equation on the tangent bundle of . 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) ().