Abstract
This paper establishes the existence of symplectic structure indegeneratevariational problems, i.e. problems whose full development involves a hierarchy of equations of constraint as well as various equations of motion. Any variational problem, degenerate or otherwise, may be calledregularif the equations of the second variation provide a complete description of the infinitesimal relationships subsisting between any orbit and all its infinitesimal neighbour orbits. It is proved that Poincare’s conserved antisymmetric derived bilinear differential form in the orbit manifold of any regular degenerate problem admits no null vectors other than those which represent infinitesimal deviations due to indeterminacy in the evolution of the orbit. Conversely, it is shown how, given any continuous system of orbits endowed with a conserved antisymmetric closed bilinear differential form having this unique property of rank and nullity, one can construct at least one regular variational
Reference32 articles.
1. Abraham R. 1967 Foundations of mechanics. New York: Benjamin.
2. Allcock G. R. 1973 A simple Lagrangian formalism for Fermi-Dirac quantization. Contribution to Cooperative phenomena (ed. H. Haken & M. Wagner) pp. 350- 361. Heidelberg: Springer.
3. Invariant Lagrangian theory of the Poisson bracket for systems with constraints
4. Caratheodory G. 1965 Calculus of variations andpartial differential equations of thefirst order part 1. San Francisco: Holden-Day.
5. Gartan E. 1922 Lesons sur les invariants integraux. Paris: Hermann.
Cited by
19 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献