The Intrinsic Properties of Rank and Nullity of the Lagrange Bracket in the One Dimensional Calculus of Variations

This paper establishes the existence of symplectic structure in degenerate variational 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 called regular if 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