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The inverse problem of Lagrangian dynamics for higher-order differential equations: A geometrical approach. (English) Zbl 0754.53025

Given a manifold \(Q\), a \(k\)th order Lagrangian on \(Q\) means a real-valued function on \(T^ kQ=J^ k_ 0(R,Q)\). The authors first present a geometric construction transforming a regular \(k\)th order Lagrangian on \(Q\) into the Euler-Lagrange vector field on \(T^{2k-1}Q\). Then they prove that a necessary and sufficient condition for a \(2k\)th order differential equation on \(T^{2k-1}Q\) to be a regular Euler-Lagrange vector field is the existence of a certain symplectic form on \(T^{2k- 1}Q\). For the case \(k=1\) some results by M. Crampin [J. Phys. A 14, 2567-2575 (1981; Zbl 0475.70022)] are reduced in such a way.
Reviewer: I.Kolář (Brno)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70H03 Lagrange’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0475.70022
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