Collective Lie-Poisson integrators on R3

McLachlan, Robert I.; Modin, Klas; Verdier, Olivier

doi:10.1093/imanum/dru013

Cited by 13 publications

(33 citation statements)

References 12 publications

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“…because the second part 2 maps S 3 √ R ⊂ C 2 to S 2 R ⊂ R 3 for any R > 0; note also that we identified so(3) * with so(3) here. Therefore, the assertion follows upon the identification of so ( R is a conserved quantity of the (canonical) Hamiltonian system (14) or (15), and that F is collective, i.e., there exists f : se(3) * → R such that f • M = F . Then one easily sees that f is a conserved quantity of the heavy top dynamics.…”

Section: 4mentioning

confidence: 99%

“…and again by Noether's Theorem this is a conserved quantity of the Hamiltonian system (14). However, because ψ 2 is conserved, we may define an alternative conserved quantity…”

Section: Its Cotangent Lift Ismentioning

confidence: 99%

“…Again, f 3 is a well-known conserved quantity of the Lagrange top. [13]; it is implemented for the free rigid body in [14].…”

Section: Its Cotangent Lift Ismentioning

confidence: 99%

“…Our focus here is to compare the behaviors of these four conserved quantities for several heavy top integrators. Figure 1 shows the time evolutions of the four conserved quantities along the numerical solutions obtained by applying the explicit and implicit midpoint rules to the canonical Hamiltonian system (14) as well as that obtained by applying the explicit midpoint rule directly to the heavy top equations (1). The time step ∆t is 1/50 in all the cases.…”

Section: Numericalmentioning

confidence: 99%

“…We note that the implicit midpoint rule is a symplectic integrator for canonical Hamiltonian systems (see, e.g., Hairer et al [5,Theorem VI.3.5] and Leimkuhler and Reich [9, Section 4.1]). So the implicit midpoint rule applied to (14) gives a collective Lie-Poisson integrator for the heavy top equations (1) via the map M, and is our main focus here.…”

Section: Numericalmentioning

confidence: 99%

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Collective Heavy Top Dynamics

Ohsawa¹

2019

SIGMA

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We construct a Poisson map M : T * C 2 → se(3) * with respect to the canonical Poisson bracket on T * C 2 ∼ = T * R 4 and the (−)-Lie-Poisson bracket on the dual se(3) * of the Lie algebra of the special Euclidean group SE(3). The essential part of this map is the momentum map associated with the cotangent lift of the natural right action of the semidirect product Lie group SU(2) C 2 on C 2 . This Poisson map gives rise to a canonical Hamiltonian system on T * C 2 whose solutions are mapped by M to solutions of the heavy top equations. We show that the Casimirs of the heavy top dynamics and the additional conserved quantity of the Lagrange top correspond to the Noether conserved quantities associated with certain symmetries of the canonical Hamiltonian system. We also construct a Lie-Poisson integrator for the heavy top dynamics by combining the Poisson map M with a simple symplectic integrator, and demonstrate that the integrator exhibits either exact or near conservation of the conserved quantities of the Kovalevskaya top.

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Section: 4mentioning

confidence: 99%

“…and again by Noether's Theorem this is a conserved quantity of the Hamiltonian system (14). However, because ψ 2 is conserved, we may define an alternative conserved quantity…”

Section: Its Cotangent Lift Ismentioning

confidence: 99%

“…Again, f 3 is a well-known conserved quantity of the Lagrange top. [13]; it is implemented for the free rigid body in [14].…”

Section: Its Cotangent Lift Ismentioning

confidence: 99%

Section: Numericalmentioning

confidence: 99%

Section: Numericalmentioning

confidence: 99%

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Collective Heavy Top Dynamics

Ohsawa¹

2019

SIGMA

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Full Affine Equivariance and Weak Natural Transformations in Numerical Analysis—The Case of B-Series

Verdier

2018

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

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Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant constructions. There is, however, a crucial ingredient missing: most algorithms are in fact defined regardless of the underlying dimension. As a result, they are also invariant with respect to non-invertible affine transformation from spaces of different dimensions. We formulate this property precisely: these algorithms fall short of being natural transformations between affine functors. We give a precise definition of what we call a weak natural transformation between functors, and illustrate the point using examples coming from numerical analysis, in particular B-Series.

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Geometry of Discrete-Time Spin Systems

2016

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Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space (S 2 ) n . In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity.The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on T * R 2n for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.

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Collective Lie-Poisson integrators on R3

Cited by 13 publications

References 12 publications

Collective Heavy Top Dynamics

Collective Heavy Top Dynamics

Full Affine Equivariance and Weak Natural Transformations in Numerical Analysis—The Case of B-Series

Geometry of Discrete-Time Spin Systems

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