Hamiltonian field theory

✍ Scribed by Olga Krupková

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 318 KB
• Volume: 43
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(01)00087-0

✦ Synopsis

In this paper, a general Hamiltonian theory for Lagrangian systems on fibred manifolds is proposed. The concept of a Lepagean (n + 1)-form is defined (where n is the dimension of the base manifold), generalizing Krupka's concept of a Lepagean n-form. Lepagean (n + 1)-forms are used to study Lagrangian and Hamiltonian systems. Innovations and new results concern the following: a Lagrangian system is considered as an equivalence class of local Lagrangians (of all orders starting from a minimal one); a Hamiltonian system is associated with an Euler-Lagrange form (not with a particular Lagrangian); Hamilton equations are based upon a Lepagean (n + 1)-form, and cover Hamilton-De Donder equations (which are based upon the exterior derivative of the Poincaré-Cartan form) as a special case. First-order Hamiltonian systems, namely those carying higher-degree contact components of the corresponding Lepagean forms, are studied in detail. The presented geometric setting leads to a new (more general than the standard one) understanding of the concepts of regularity and Legendre transformation in the calculus of variations, relating them directly to the properties of the arising exterior differential systems. In this way, new regularity conditions and Legendre transformation formulas are obtained, depending on a Lepagean (n+1)-form, i.e., related with the corresponding Euler-Lagrange form.