In geometric mechanics a presymplectic form is a
closed differential 2-form of constant rank on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
.
However, some authors use different definitions. Recently, Hajduk and Walczak defined a presymplectic form as a closed differential 2-form of maximal rank on a manifold of odd dimension.
A
symplectic form
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
is a presymplectic form that is also
nondegenerate.
Lack of nondegeneracy, leading to presymplectic forms, occurs in dynamical systems with singular
Lagrangians,
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s with
constraints and
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
.
References
Dynamical systems
Differential geometry
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