Symplectic integration of Hamiltonian systems

Channell, Paul J.; Scovel, Clint

doi:10.1088/0951-7715/3/2/001

Cited by 395 publications

(285 citation statements)

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“…Therefore the integration of equations (20) over one time step τ , which evolves the initial coordinate vector x(t) to its final state x(t + τ ), is represented by the action on x(t) of a symplectic map S produced by the composition of products of e ciτ LA and e diτ LB . In this context several symplectic integrators of different orders have been developed by various researchers [23,24].…”

Section: Symplectic Integratorsmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

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We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the 'tangent map (TM) method', a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map S, while the corresponding tangent map T S, is used for the integration of the variational equations. A simple and systematic technique to construct T S is also presented.

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Section: Symplectic Integratorsmentioning

confidence: 99%

Numerical integration of variational equations

Skokos

Gerlach

2010

Phys. Rev. E

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“…We remark that many symplecticmomentum integrators have been based on the use of generating functions. We refer to Channell and Scovel [1990] and Ge [1991a] for surveys. They are based on the (standard) fact that if S : Q × Q → R defines a diffeomorphism (q 0 , p 0 ) → (q, p) implicitly by p = ∂S ∂q and…”

Section: Generalities On Integration Algorithmsmentioning

confidence: 99%

“…The overviews of symplectic integrators in Channell and Scovel [1990], SanzSerna [1991] and McLachlan and Scovel [1996] provide background and references. References related to the work in this paper are Reich [1993], , McLachlan and Scovel [1995], and Jay [1996].…”

Section: Introductionmentioning

confidence: 99%

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Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

Marsden

Wendlandt

1997

Current and Future Directions in Applied Mathematics

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This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov.The general idea for these variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators.

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“…The first instances of such schemes are the symplectic integrators arising in Hamiltonian mechanics, and the related energy conserving methods, [7,21,30]. The design of symmetry-based numerical approximation schemes for differential equations has been studied by various authors, including Shokin, [29], Dorodnitsyn, [10,11], Axford and Jaegers, [19], and Budd and Collins, [2].…”

Section: Introductionmentioning

confidence: 99%

Geometric Foundations of Numerical Algorithms and Symmetry

Olver

2001

Applicable Algebra in Engineering, Communication and Computing

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Abstract. This paper outlines a new general construction, named "multi-space", that forms the proper geometrical foundation for the numerical analysis of differential equations -in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations. Application of the theory of moving frames leads to a general framework for constructing symmetry-preserving numerical approximations to differential invariants and invariant differential equations. †

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Symplectic integration of Hamiltonian systems

Cited by 395 publications

References 10 publications

Numerical integration of variational equations

Numerical integration of variational equations

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

Geometric Foundations of Numerical Algorithms and Symmetry

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