Results 1 to 10 of about 192 (135)
Symplectic groupoids for Poisson integrators
Journal of Geometry and Physics, 2023 We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators.
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INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY [PDF]
Glasgow Mathematical Journal, 2007 AbstractQuaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space=Sp(n+1)/Sp(1) ×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry onmodelled on$(\mk{sp}_{n+1}, \mk{sp}_{1}\,{\times}\, \mk{sp}_{n})$.
Asadi, E., Sanders, J.A.
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A SYMPLECTIC INTEGRATOR FOR HILL'S EQUATIONS [PDF]
The Astronomical Journal, 2010 Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV and tested on some simple orbits.
Quinn, T. +3 more
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Anomaly in symplectic integrator [PDF]
Physics Letters A, 2007 6 pages, no ...
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Symplectic integrators for spin systems [PDF]
Physical Review E, 2014 We present a symplectic integrator, based on the canonical midpoint rule, for classical spin systems in which each spin is a unit vector in $\mathbb{R}^3$. Unlike splitting methods, it is defined for all Hamiltonians, and is $O(3)$-equivariant. It is a rare example of a generating function for symplectic maps of a noncanonical phase space. It yields an
Robert I. McLachlan +2 more
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Symplectic integration of magnetic systems [PDF]
Journal of Computational Physics, 2014 Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the dynamics for an approximate Hamiltonian.
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On the Nonlinear Stability of Symplectic Integrators [PDF]
BIT Numerical Mathematics, 2004 The paper deals mainly with the problem of achieving stability for symplectic numerical integrators by the mean of studying the topological equivalence of the level sets of the original Hamiltonian and those of the modified Hamiltonian associated to the numerical integrators.
Mclachlan, Robert Iain. +2 more
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Tuning Symplectic Integrators is Easy and Worthwhile [PDF]
Communications in Computational Physics, 2022 Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized algorithms and from choosing the best ordering of terms. The cost in programming and execution time is minimal, while the performance improvements can be large.
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Fourth-order symplectic integration [PDF]
Physica D: Nonlinear Phenomena, 1990 Die Autoren behandeln die Integration der Hamiltonschen Gleichungen unter Anwendung einer expliziten Vierter-Ordnung-Methode. Diese bewahrt die Eigenschaft, daß eine zeitliche Entwicklung eines solchen Systems eine kanonische Transformation aus den Anfangsbedingungen bis zum Endzustand erhält.
Forest, E., Ruth, R.D.
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Collective symplectic integrators [PDF]
Nonlinearity, 2014 We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge--Kutta methods. Their phase space is a symplectic vector space with a Hamiltonian action with momentum map $J$ whose range is the target Lie--Poisson manifold, and their Hamiltonian is collective, that is, it is the target ...
Robert I McLachlan +2 more
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