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However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold. The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such ...
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian.Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics.
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 ...
Symplectic manifolds are special cases of a Poisson manifold. A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. [5] A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued (+)-form; it is utilized in Hamiltonian field theory. [6]
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 ...
The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.
If the first Betti number of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and symplectic isotopy of symplectomorphisms coincide. It can be shown that the equations for a geodesic may be formulated as a Hamiltonian flow, see Geodesics as Hamiltonian flows.
In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem.In Hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6n-dimensional Lebesgue measure).