Geodesic
Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
Lobachevskii journal of mathematics, Tome 23 (2006), pp. 95-150

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The paper is a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The subject of the paper are systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in fibered manifolds. A geometric setting for Euler–Lagrange and Hamilton equations, based on the concept of Lepage class is presented. A constraint is modeled in the underlying fibered manifold as a fibered submanifold endowed with a distribution (the canonical distribution). A constrained system is defined by means of a Lepage class on the constraint submanifold. Constrained EulerЧ-Lagrange equations and constrained Hamilton equations, and properties of the corresponding exterior differential systems, such as regularity, canonical form, or existence of a constraint Legendre transformation, are presented. The case of mechanics (ODEТs) and field theory (PDEТs) are investigated separately, however, stress is put on a unified exposition, so that a direct comparison of results and formulas is at hand.
Keywords: jet bundles, non-holonomic constraints, semiholonomic constraints, holonomic constraints, constrained Lagrangian systems, constrained Euler–Lagrange equations, Hamilton–De Donder equations, regularity of constrained systems, Hamiltonian
Mots-clés : momenta, Legendre transformation. 
@article{LJM_2006_23_a4,
     author = {O. Krupkov\'a and P. Voln\'y},
     title = {Differential equations with constraints in jet bundles: {Lagrangian} and {Hamiltonian} systems},
     journal = {Lobachevskii journal of mathematics},
     pages = {95--150},
     publisher = {mathdoc},
     volume = {23},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/}
}
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O. Krupková; P. Volný. Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems. Lobachevskii journal of mathematics, Tome 23 (2006), pp. 95-150. http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/