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Hamiltonian field theory. (English) Zbl 1016.37033
This interesting paper is devoted to the study of a Hamiltonian theory for Lagrangian systems of arbitrary order on fibred manifolds, from the Lepagean viewpoint. Special care is devoted to the concepts of regularity and equivalent Lagrangians.
The motivation for the introduction of a new Hamilton theory is justified by the author from the effects of nonuniqueness in higher-order theory, problems with equivalent Lagrangians (it is known that equivalent Lagrangians can be regular or singular with respect to the usual definition of regularity, so their Hamilton equations are essentially different), or the fact that most important physical fields are singular. Within the new setting, many singular Lagrangians (e.g. the Dirac field, the electromagnetic field, the Yang-Mills field or gravity) are no longer singular, so there is no need to apply constraint techniques for obtaining Hamilton equations for them.
The approach of the author concerning the differential geometric setting for Hamiltonian field theory in fibred manifolds is different from the usual one. The key concept of the formulation is that of Lepagean \((n+1)\)-form (where \(n\) is the dimension of the base manifold). If Lepagean \(n\)-forms are counterparts of Lagrangians, Lepagean \((n+1)\)-forms in this paper are counterparts of Euler-Lagrange forms. Then, a Lagrangian system is introduced as an equivalence class of Lepagean \((n+1)\)-forms. Thus, a Lagrangian system is meant to be the family of all equivalent Lagrangians (i.e., Lagrangians whose Euler-Lagrange forms coincide).
Within this framework, a Hamiltonian system is associated with an Euler-Lagrange form, not with a particular Lagrangian. Hamilton equations become equations for integral sections of a Hamilton exterior differential system arising from a Lepagean \((n+1)\)-form, and they become a counterpart of the Euler-Lagrange equations (not a particular Lagrangian). Moreover, they depend not only upon the Lagrangian system itself, but also upon higher degree contact components of the corresponding Lepagean \((n+1)\)-form.
Regularity and Legendre transformation for a Hamiltonian system are defined to be properties of the corresponding Hamilton exterior differential system. The generalized setting for the Hamilton theory suggests a new understanding of the role of regularity, Legendre transformation, and Hamilton equations in the calculus of variations. Namely, higher-degree contact terms which appear in the generalized Hamilton equations can be considered as parameters so one has the possibility to search for appropriate Hamilton equations (regular and admitting Legendre transformation) for a given variational problem. From this point of view, regularizations of some interesting Lagrangians (namely, Lagrangians affine or quadratic in the first derivatives, or affine in the second derivatives) are studied.

MSC:
37J60 Nonholonomic dynamical systems
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
58A20 Jets in global analysis
58E30 Variational principles in infinite-dimensional spaces
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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