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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 ...
For the cotangent bundle of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the free loop space of M (proofs of various versions of this statement are due to Viterbo, Salamon–Weber, Abbondandolo ...
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 ...
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 ...
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 ...
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 position of a single particle moving in ordinary Euclidean 3-space is defined by the vector = (,,), and therefore its configuration space is =.It is conventional to use the symbol for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics, and in Lagrangian mechanics.
SnapPea inputs data in a variety of formats. Given a link diagram, SnapPea can ideally triangulate the link complement. It then performs a sequence of simplifications to find a locally minimal ideal triangulation. Once a suitable ideal triangulation is found, SnapPea can try to find a hyperbolic structure.