Numerical integration scheme for Hamiltonian systems
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.
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In mathematics, a symplecticintegrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplecticintegrators form the subclass of geometric...
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds...
Because of its time-reversibility, and because it is a symplecticintegrator, leapfrog integration is also used in Hamiltonian Monte Carlo, a method for...
\omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally...
lemma for the numerical method to solve differential equations, see Symplecticintegrator Split (disambiguation) Splitter (disambiguation) This disambiguation...
Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics...
conserving properties of the simulated Hamiltonian dynamic when using a symplecticintegrator. The reduced correlation means fewer Markov chain samples are needed...
simple collocation method, and, applied to Hamiltonian dynamics, a symplecticintegrator. Note that the modified Euler method can refer to Heun's method...
time step. The energy computed from the modified Hamiltonian of a symplecticintegrator is O ( Δ t p ) {\displaystyle {\mathcal {O}}\left(\Delta t^{p}\right)}...
Hamilton's equations that preserves the symplectic structure Variational integrator — symplecticintegrators derived using the underlying variational...
multisymplectic integrator is a numerical method for solving multisymplectic PDEs whose numerical solution conserves a discrete form of symplecticity. One example...
odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by...
mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact...
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action...
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism...
equations. geometric integration methods are especially designed for special classes of ODEs (for example, symplecticintegrators for the solution of Hamiltonian...
symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization of a square-integrable functions...
Symplectic Geometry. Methods and Applications (2nd ed.). Gordon and Breach. ISBN 978-2-88124-901-3. Fomenko, A.T.; Bolsinov, A.V. (2003). Integrable Hamiltonian...
of a discretized Hamilton's principle. Variational integrators are momentum-preserving and symplectic. Consider a mechanical system with a single particle...
it is a symplectic form. An almost symplectic manifold is an Sp-structure; requiring ω {\displaystyle \omega } to be closed is an integrability condition...
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