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The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field. The Hamiltonian vector field induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of ...
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. [ 1 ] The term "symplectic", introduced by Hermann Weyl , [ 2 ] is a calque of "complex"; previously, the "symplectic group" had been called the "line ...
Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a Lie group on a coadjoint orbit.
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle of a manifold. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the manifold ).
The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted () where represents the coordinates of the origin of the frame attached to the body, and () represents the rotation matrices that define the orientation of this frame relative to a ground ...
Conversely, given any contact manifold M, the product M×R has a natural structure of a symplectic manifold. If α is a contact form on M, then ω = d(e t α) is a symplectic form on M×R, where t denotes the variable in the R-direction. This new manifold is called the symplectization (sometimes symplectification in the literature) of the ...
An -action on a symplectic manifold (,) is called Hamiltonian if it is symplectic and if there exists a momentum map. A momentum map is often also required to be G {\displaystyle G} -equivariant , where G {\displaystyle G} acts on g ∗ {\displaystyle {\mathfrak {g}}^{*}} via the coadjoint action , and sometimes this requirement is included in ...