The theory of problems in the calculus of variations whose Lagrangian function involves second order derivatives: A new approach |
| |
Authors: | Hanno Rund |
| |
Institution: | (1) Durban, South-Africa |
| |
Abstract: | Summary Variational principles whose Lagrangian functions involve higher order derivatives have, in the past, been applied to certain
aspects of the theory of elementary particles. The corresponding Lagrangian functions must satisfy certain conditions if consistency
with the classical electromagnetic interaction terms is sought, and it is found that these conditions are closely related
to the requirement that the action integral be invariant under a parameter transformation. If, however, the latter condition
is accepted, the usual expression for the Hamiltonian function vanishes identically, resulting in a complete break-down of
the canonical equations. Thus an alternative approach to the theory of parameter-invariant problems in the calculus of variations
whose Lagrangians depend on second order derivatives is developed. A general Finsler metric is introduced in a natural manner,
which provides a geometrical background to the theory as well as useful analytical techniques. It is possible to define an
alternative Hamiltonian function corresponding to which a canonical formalism is developed. The method of equivalent integrals
is generalised, giving rise to a new and rigorous derivation of theEuler-Lagrange equations, which in turn leads to a generalisation of the so-called excess-function and the analogue of the well-known condition
of Weierstrass in the calculus of variations.
To Enrico Bompiani on his scientific Jubilee. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|