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Integrals of motion and semi-regular Lepagean forms in higher-order mechanics. (English) Zbl 0533.58017

The author discusses two ways of describing the dynamics of a (higher- order) mechanical system: one as critical section with respect to a Lepagean 1-form \(\theta\) on a suitable jet-extension of a fibred manifold, the other one as critical section with respect to what he calls a canonical set of integrals of motion. He proves that both views are essentially equivalent, thereby relying on Darboux’s theorem for the canonical structure of \(d\theta\). He further recalls how the Lepagean form picture gives rise to Euler-Lagrange equations and deduces from the above equivalence an expression for the Lagrangian in terms of a canonical set of integrals of the motion. Invariance transformations of \(d\theta\) are shown to be related to first integrals just as in the more familiar case of first-order mechanics. Finally, an appropriate generalization is presented of the Liouville theorem concerning the complete integrability of a system with known integrals in involution.
Reviewer: W.Sarlet

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H03 Lagrange’s equations
49Q99 Manifolds and measure-geometric topics
58A15 Exterior differential systems (Cartan theory)
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