A presymplectic form is a closed two-form, commonly assumed to have constant rank, that need not be nondegenerate. A symplectic form is its nondegenerate special case. Its kernel distribution is integrable: for kernel fields , the identity gives . If the characteristic leaf quotient is smooth and the form descends, that quotient has a symplectic form. A regular Hamiltonian energy hypersurface carries such a form with kernel spanned by its Hamiltonian vector field.
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The presymplectic form is a concept from differential geometry and mathematical physics, particularly in the study of Hamiltonian dynamics and the theory of differential forms. It generalizes the notion of a symplectic form, which is a closed, non-degenerate 2-form defined on an even-dimensional manifold. In more detail: 1. **Definition**: A presymplectic form on a smooth manifold \( M \) is a closed 2-form \( \omega \) (i.e.