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**On the higher order Hamilton theory in fibered spaces.**
*(English)*
Zbl 0598.58022

Differential geometry and its applications, Proc. Conf., Nové Město na Moravě/Czech. 1983, Pt. 2, 167-183 (1984).

[For the entire collection see Zbl 0546.00021.]

The purpose of this paper is a brief discussion of the elements of the Hamilton-Cartan theory on the basis of Lepagean forms. These forms reflect all the main properties of the Cartan form, and are naturally becoming the fundamental element for geometrization of the higher order calculus of variations in fibered spaces. Basic properties of the Lepagean forms are recalled in Section 3. In Section 4, we introduce the Hamilton form, associated with a Lepagean form. The Hamilton form is precisely the Euler-Lagrange form of an ”extended” lagrangian, defined, in a canonical manner, by means of the initial lagrangian. Section 5 is concerned with the discussion of this form under a regularity assumption; the correspondence between extremals of the lagrangian and the extended lagrangian is studied. In Section 6 we discuss the Euler-Lagrange equations of the extended lagrangian, in the local and ”regular” case; these equations rewritten in the ”Legendre coordinates”, coincide with the canonical equations given by de Donder, Section 7 is devoted to some remarks on the theory of extremal fields.

The purpose of this paper is a brief discussion of the elements of the Hamilton-Cartan theory on the basis of Lepagean forms. These forms reflect all the main properties of the Cartan form, and are naturally becoming the fundamental element for geometrization of the higher order calculus of variations in fibered spaces. Basic properties of the Lepagean forms are recalled in Section 3. In Section 4, we introduce the Hamilton form, associated with a Lepagean form. The Hamilton form is precisely the Euler-Lagrange form of an ”extended” lagrangian, defined, in a canonical manner, by means of the initial lagrangian. Section 5 is concerned with the discussion of this form under a regularity assumption; the correspondence between extremals of the lagrangian and the extended lagrangian is studied. In Section 6 we discuss the Euler-Lagrange equations of the extended lagrangian, in the local and ”regular” case; these equations rewritten in the ”Legendre coordinates”, coincide with the canonical equations given by de Donder, Section 7 is devoted to some remarks on the theory of extremal fields.

### MSC:

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |