Lepagean forms in higher order variational theory. (English) Zbl 0572.58003
Modern developments in analytical mechanics, Vol. I: Geometrical dynamics, Proc. IUTAM-ISIMM Symp., Torino/Italy 1982, 197-238 (1983).
[For the entire collection see Zbl 0559.00013.]
A Lepagean equivalent of a higher order Lagrangian defined on a fibered manifold is a differential form with the following two properties: (1) It defines the same variational function as the Lagrangian, and (2) its exterior derivative is equal (modulo certain contact forms) to the Euler- Lagrange form of the Lagrangian. In the present paper it has been shown that the concept of a Lepagean equivalent unifies the theory of Hamilton- Poincaré-Cartan fundamental forms. The theory of Lepagean forms is applied to the higher order inverse problem of the calculus of variations; the proof of local formulas is given by means of an appropriate Poincaré lemma.
For further developments in the theory of Lepagean forms and the Hamiltonian-de Donder theory one should consult [W. F. Shadwick, Lett. Math. Phys. 6, 409-416 (1982; Zbl 0514.58013); I. Kolář, J. Geom. Phys. 1, No.2, 127-137 (1984); the author, Differential geometry and its applications, Proc. Conf., Nové Město na Moravě/Czech. 1983, Pt. 2, 167-183 (1984)].
A Lepagean equivalent of a higher order Lagrangian defined on a fibered manifold is a differential form with the following two properties: (1) It defines the same variational function as the Lagrangian, and (2) its exterior derivative is equal (modulo certain contact forms) to the Euler- Lagrange form of the Lagrangian. In the present paper it has been shown that the concept of a Lepagean equivalent unifies the theory of Hamilton- Poincaré-Cartan fundamental forms. The theory of Lepagean forms is applied to the higher order inverse problem of the calculus of variations; the proof of local formulas is given by means of an appropriate Poincaré lemma.
For further developments in the theory of Lepagean forms and the Hamiltonian-de Donder theory one should consult [W. F. Shadwick, Lett. Math. Phys. 6, 409-416 (1982; Zbl 0514.58013); I. Kolář, J. Geom. Phys. 1, No.2, 127-137 (1984); the author, Differential geometry and its applications, Proc. Conf., Nové Město na Moravě/Czech. 1983, Pt. 2, 167-183 (1984)].
MSC:
| 58A99 | General theory of differentiable manifolds |
| 58E30 | Variational principles in infinite-dimensional spaces |
