The fundamental Lepage form in variational theory for submanifolds

Abstract

A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrange equations for extremal submanifolds and the Noether theorem for invariant variational functionals are proved. The theory is illustrated on the variational functional for minimal submanifolds.2000 Mathematics Subject Classification. 58E30; 58A20; 58D19; 53A10.

On the Carathéodory Form in Higher-Order Variational Field Theory

“…With respect to a fibered chart (V, ψ), Z λ has an expression 7) is known as the fundamental Lepage form [7,8]), and it is characterized by the equivalence: Z λ is closed if and only if λ is trivial (i.e., the Euler-Lagrange expressions associated with λ vanish identically). Recently, the form (7) was generalized in [21], and studied for variational problems for submanifolds in [5], as well as applied for studying symmetries and conservation laws in [22].…”

“…In this note, we describe a generalization of the Carathéodory form [1] (cf. [2][3][4][5]) of the calculus of variations in second-order and, for specific Lagrangians, in higher-order field theory.…”

The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility

“…See also Gotay [10], Goldschimdt and Sternberg [11], Rund [12], Dedecker [13], Horák and Kolář [14], Krupka [6], Krupka and Štěpánková [15], Saunders [16], and Sniatycki [17]. Further recent attempts of generalization and study of the fundamental Lepage equivalent for first-and second-order Lagrangians also include [18][19][20]. For a review of basic properties and results, see Krupka, Krupková, and Saunders [21].…”

On the closure property of Lepage equivalents of Lagrangians

“…-First order Lagrangians. In this case, a Lepage equivalent with the desired feature, called the fundamental Lepage equivalent ρ λ , was introduced by Krupka, [8] and re-discovered by Bethounes, [1]; it adds to the Lagrangian λ contact components up to degree k = min(m, n); for homogeneous Lagrangians, a similar notion was introduced by Urban and Brajercik, [14].…”