Equivalence of variational problems of higher order Academic Article uri icon

abstract

  • We show that for n3 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the special class of rank 2 distributions associated with underdetermined ODEs z'=f(x,y,y',. .,y(n)); the equivalence problem for variational ODEs of order 2. n. This leads to new results such as the fundamental system of invariants for all these problems and the explicit description of the maximally symmetric models. The central role in all three equivalence problems is played by the geometry of self-dual curves in the projective space of odd dimension up to projective transformations via the linearization procedure (along the solutions of ODE or abnormal extremals of distributions). More precisely, we show that an object from one of the three equivalence problems is maximally symmetric if and only if all curves in projective spaces obtained by the linearization procedure are rational normal curves. 2010 Elsevier B.V.

published proceedings

  • Differential Geometry and its Applications

author list (cited authors)

  • Doubrov, B., & Zelenko, I.

citation count

  • 6

complete list of authors

  • Doubrov, Boris||Zelenko, Igor

publication date

  • March 2011