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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 ...
Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read ˙ = and ˙ =, where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
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 ).
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 ...
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.
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.