Poisson Brackets

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• Poisson manifold — In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra. Since their introduction by André… …   Wikipedia

• Poisson bracket — 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. In a more general… …   Wikipedia

• Screened Poisson equation — In mathematics, the screened Poisson equation is the following partial differential equation::left [ abla^2 lambda^2 ight] u(mathbf{r}) = f(mathbf{r})where ² is the Laplace operator, λ is a constant, f is an arbitrary function of position (known… …   Wikipedia

• Laplace–Runge–Lenz vector — Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively; for example, left| mathbf{A} ight| = A. In classical mechanics, the Laplace–Runge–Lenz vector (or simply the LRL vector) is a vector… …   Wikipedia

• Dirac bracket — The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to correctly treat systems with second class constraints in Hamiltonian mechanics and canonical quantization. It is an important part of Dirac s development of… …   Wikipedia

• Matrix mechanics — Quantum mechanics Uncertainty principle …   Wikipedia

• Lagrange bracket — Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange in 1808–1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have… …   Wikipedia

• First class constraint — In Hamiltonian mechanics, consider a symplectic manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite dimensional). Poisson bracketsSuppose we have some constraints : f i(x)=0, for n smooth functions :{ f i } {i=… …   Wikipedia

• Hasegawa-Mima equation — The Hasegawa Mima equation, named after Akira Hasegawa and Kunioki Mima, is an equation that describes a certain regime of plasma, where the time scales are very fast, and the distance scale in the direction of the magnetic field is long. In… …   Wikipedia

• Integrable system — In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In… …   Wikipedia

• Second class constraints — In a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second… …   Wikipedia