Presymplectic form

In mathematical physics, especially geometric mechanics, a presymplectic form is a geometric structure on differentiable manifolds. It is a generalization of symplectic form.

Given a differentiable manifold, a symplectic form over it is differential 2-form that is closed and nondegenerate. A presymplectic form relaxes the requirement for nondegeneracy. Instead, it is merely required to be closed and have constant rank at all points on the manifold.^[1] Note that a symplectic form, by virtue of nondegeneracy, necessarily have rank equaling the dimension of the underlying manifold, so it has constant rank.^[2]

The definition is not standardized. Recently, Hajduk and Walczak defined a presymplectic form as a closed, differential 2-form, of maximal rank on a manifold of odd dimension.^[3] This may be motivated thus: A symplectic form necessarily exists over a manifold of even dimension, so a manifold of odd dimension cannot have a symplectic form. However, it can at least attempt to reach a rank as high as possible, since a sympletic form, by virtue of nondegeneracy, necessarily have rank equaling the dimension of the underlying manifold, which is the maximal rank possible on the manifold.