Non-autonomous system (mathematics) explained

Q\to R over R . For instance, this is the case of non-autonomous mechanics.

An r-order differential equation on a fiber bundle

Q\to R is represented by a closed subbundle of a jet bundle J ^{rQ} of Q\to R . A dynamic equation on Q\to 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 R is a kernel of the covariant differential of some connection \Gamma on Q\to R . Given bundle coordinates (t,q ^{i)} on Q and the adapted coordinates (t,q ^{i,q} ^{i} ^{t)} on a first-order jet manifold J ^{1Q} , a first-order dynamic equation readsq ^{i} _{t = \Gamma} (t,q ^{i).}

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

A second-order dynamic equation

q ^{i} _{tt} = \xi ^{i(t,q} ^{j,q} ^{j} ^{t)}

on

Q\to R is defined as a holonomicconnection \xi on a jet bundle J ^{1Q\toR} . Thisequation also is represented by a connection on an affine jet bundle J ^{1Q\to} Q . Due to the canonicalembedding J ^{1Q\to} TQ , it is equivalent to a geodesic equationon 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) .