Hamiltonian mechanics Hamiltonian mechanics In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of or with the x 's or q 's denoting the coordinates on the underlying manifold and the p 's denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point q in the manifold. This article defines the canonical coordinates as they appear in classical physics. A closely related concept also appears in quantum mechanics; see the Stone-von Neumann theorem and canonical commutation relations for details. In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tangent vectors produce real numbers. ...more on Wikipedia about "Canonical coordinates"
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". ...more on Wikipedia about "Geodesic"
In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. This contrasts with Hamilton's equations of motion in that the HJE is a single differential equation of one variable for each coordinate, where Hamilton's equations are a system of first order equations, two for each coordinate. The HJE can be used to solve several problems elegantly, such as the Kepler problem. If we have a Hamiltonian of the form then the HJE for that system is ...more on Wikipedia about "Hamilton-Jacobi equations"
Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. It arose from Lagrangian mechanics, another re-formulation of classical mechanics, introduced by Joseph Louis Lagrange in 1788. It can however be formulated without recourse to Lagrangian mechanics, using symplectic spaces. See the section on its mathematical formulation for this. ...more on Wikipedia about "Hamiltonian mechanics"
In mathematics and physics, a Hamiltonian vector field is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. The integral curves of the symplectic vector field are solutions to the Hamilton-Jacobi equations of motion. The vector field, taken together with the symplectic manifold and the symplectic form on the manifold, comprise a Hamiltonian system. ...more on Wikipedia about "Hamiltonian vector field"
In mathematical physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system - that is that the density of system points in the vicinity of a given system point travelling through phase-space is constant with time.. ...more on Wikipedia about "Liouville's theorem (Hamiltonian)"
In mathematics, specifically in symplectic geometry, the moment map (or momentum map) is a tool used to glean information about the action of a Lie group on a symplectic manifold. The moment map generalizes the classical notions of linear and angular momentum. It is used in various constructions of symplectic manifolds, including symplectic quotients. ...more on Wikipedia about "Moment map"
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In mathematics, Nambu dynamics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu-Poisson manifolds with more than one Hamiltonian. ...more on Wikipedia about "Nambu mechanics"
In mathematics and physics, phase space is the space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. A plot of position and momentum variables as a function of time is sometimes called a phase diagram. ...more on Wikipedia about "Phase space"
In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. It is named after Siméon-Denis Poisson. ...more on Wikipedia about "Poisson bracket"
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related to classical mechanics. ...more on Wikipedia about "Symplectic integrator"
In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. The study of symplectic manifolds is called symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g. in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field: The set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. ...more on Wikipedia about "Symplectic manifold"
In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ...more on Wikipedia about "Symplectomorphism"
In mathematics, the tautological one-form is a special 1-form defined on symplectic manifolds that plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. In local coordinates, the canonical symplectic form is exact; the tautological one-form is the one-form whose differential is (minus) the symplectic form on the symplectic manifold. The tautological one-form is sometimes also called the canonical one-form or the symplectic potential. ...more on Wikipedia about "Tautological one-form" Please inform your friends about shortopedia
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