In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by and , respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.
In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.
All of these objects are named in honor of Siméon Denis Poisson.
Given two functions f and g that depend on phase space and time, their Poisson bracket is another function that depends on phase space and time. The following rules hold for any three functions of phase space and time:
Anticommutativity
Bilinearity
Leibniz's rule
Jacobi identity
Also, if a function is constant over phase space (but may depend on time), then for any .
In canonical coordinates (also known as Darboux coordinates) on the phase space, given two functions and , the Poisson bracket takes the form
The Poisson brackets of the canonical coordinates are
where is the Kronecker delta.
Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket.
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Présentation des méthodes de la mécanique analytique (équations de Lagrange et de Hamilton) et introduction aux notions de modes normaux et de stabilité.
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.
The aim of this course is to acquire the basic knowledge on specific dynamical phenomena related to the origin, equilibrium, and evolution of star
clusters, galaxies, and galaxy clusters.
Explores canonical transformations, their properties, and applications in Hamiltonian mechanics, emphasizing their role in simplifying the analysis of complex systems.
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Paul Dirac. In the meantime this idea was independently introduced in 1946 by Hip Groenewold. The Moyal bracket is a way of describing the commutator of observables in the phase space formulation of quantum mechanics when these observables are described as functions on phase space.
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.
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