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Non-autonomous system (mathematics)

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

An r-order differential equation on a fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } is represented by a closed subbundle of a jet bundle J r Q {\displaystyle J^{r}Q} of Q → R {\displaystyle Q\to \mathbb {R} } . A dynamic equation on Q → R {\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 Q → R {\displaystyle Q\to \mathbb {R} } is a kernel of the covariant differential of some connection Γ {\displaystyle \Gamma } on Q → R {\displaystyle Q\to \mathbb {R} } . Given bundle coordinates ( t , q i ) {\displaystyle (t,q^{i})} on Q {\displaystyle Q} and the adapted coordinates ( t , q i , q t i ) {\displaystyle (t,q^{i},q_{t}^{i})} on a first-order jet manifold J 1 Q {\displaystyle J^{1}Q} , a first-order dynamic equation reads

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

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

A second-order dynamic equation

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

on Q → R {\displaystyle Q\to \mathbb {R} } is defined as a holonomic connection ξ {\displaystyle \xi } on a jet bundle J 1 Q → R {\displaystyle J^{1}Q\to \mathbb {R} } . This equation also is represented by a connection on an affine jet bundle J 1 Q → Q {\displaystyle J^{1}Q\to Q} . Due to the canonical embedding J 1 Q → T Q {\displaystyle J^{1}Q\to TQ} , it is equivalent to a geodesic equation on the tangent bundle T Q {\displaystyle TQ} of Q {\displaystyle Q} . A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.

See also

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).
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