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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 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 complex ...
Thus, the time evolution of a function on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian.
In this case, one says that the manifold is the sphere, i.e. =. For n disconnected, non-interacting point particles, the configuration space is R 3 n {\displaystyle \mathbb {R} ^{3n}} . In general, however, one is interested in the case where the particles interact: for example, they are specific locations in some assembly of gears, pulleys ...
When the first-class secondary constraints are added into the Hamiltonian with arbitrary as the first-class primary constraints are added to arrive at the total Hamiltonian, then one obtains the extended Hamiltonian. The extended Hamiltonian gives the most general possible time evolution for any gauge-dependent quantities, and may actually ...