Mathematics – Differential Geometry
Scientific paper
2009-08-12
Journal of Geometry and Physics, vol. 60, no 11 (2010), 1710-1725
Mathematics
Differential Geometry
Scientific paper
10.1016/j.geomphys.2010.06.016
We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Fr\"olicher-Nijenhuis theory on the first jet bundle, $J^1\pi$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, $\Lambda^1_S(J^1\pi)$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $\Lambda^1_S(J^1\pi)$ of semi-basic 1-forms. Each such class contains the Poincar\'e-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in $\Lambda^1_S(J^1\pi)$, which is not a Poincar\'e-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.
Bucataru Ioan
Constantinescu Oana
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