The inverse problem in the calculus of variations: new developments
Thoan Do,Geoff Prince +1 more
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In this article, the existence and uniqueness of Lagrangians for systems of n second order ODEs is investigated. But the Lagrangian problem is not restricted to 2-dimensional systems.Abstract:
We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2. We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.read more
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Linear connections and shape maps for second order ODEs with and without constraints
TL;DR: GP and MFP thank Instituto de Ciencias Matematicas (ICMAT) for its warm hospitality as mentioned in this paper and acknowledge financial support from the FWO (Research Foundation - Flanders) under grant 1282021N.
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Almost every path structure is not variational
TL;DR: In this paper , it was shown that the Egorov structure is not pseudo-Riemannian metrizable in the class of Kropina pseudo-metrics.
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Linear connections and shape maps for second order ODEs\\ with and without constraints
TL;DR: In this paper, the construction of linear connections associated with second-order ODEs with and without first-order constraints is studied, using a novel method allowing glueing of submodule covariant derivatives to produce new, closed form expressions for the Massa-Pagani connection and their extension to the constrained case.
References
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Solution of the inverse problem of the calculus of variations
TL;DR: In this paper, a linear differential system e on which the inverse problem can be made to depend has been derived, based on the Riquier theory of systems of partial differential equations.
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The inverse problem in the calculus of variations and the geometry of the tangent bundle
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The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics
TL;DR: In this article, the problem of finding a multiplier matrix that can give to a prescribed system of second-order ordinary equations the structure of Euler-Lagrange equations was studied.
Book
The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations
Ian Anderson,Gerard Thompson +1 more
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A geometrical version of the Helmholtz conditions in time- dependent Lagrangian dynamics
TL;DR: In this article, a geometrical machinery for the study of time-dependent Lagrangian dynamics is developed, which is applied to the inverse problem of the calculus of variations, and a set of necessary and sufficient conditions for the existence of a Lagrangians are given, in terms of a 2-form with suitable properties, which are exactly equivalent to the Helmholtz conditions.