The Moser–Veselov equation

Cardoso, João R.; Leite, F. Silva

doi:10.1016/s0024-3795(02)00450-0

Cited by 20 publications

(24 citation statements)

References 10 publications

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“…For details see [3,5,16] and also [6]. We now show that in the more general situation where G is one of the quadratic groups (2.1), there are at most two possible solutions for U k when Σ D is invertible.…”

Section: This Gives Us the Relationmentioning

confidence: 62%

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

et al. 2008

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The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO(n) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO(n). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

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Section: This Gives Us the Relationmentioning

confidence: 62%

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

et al. 2008

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“…Cardoso & Leite (2001) have shown that every solution of (8.1) (not necessarily orthogonal) can be written as X = J −1 (−M/2 + S), for some symmetric matrix S. The proof is immediate. Clearly, if S is symmetric, X = J −1 (−M/2+S) is a solution of (8.1).…”

Section: Reduction To a Matrix Ricatti Equationmentioning

confidence: 88%

“…The matrix ω can be computed also indirectly, via Schur real forms, as proposed by Cardoso & Leite (2001). This approach, and our explicit approach, is described in greater details in Section 8.…”

Section: The Dmv Algorithm For the Rb Equationsmentioning

confidence: 99%

“…If M 2 4 + J 2 is positive definite, it has been shown in (Cardoso & Leite 2001) that (8.2) has a unique solution S which is symmetric, positive definite, and such that the eigenvalues of…”

Section: The Dmv6 Algorithmmentioning

confidence: 99%

“…In this case, Cardoso & Leite (2001) propose the following algorithm for the computation of X, as the unique solution of (8.1) in the special orthogonal group SO(N ).…”

Section: The Dmv6 Algorithmmentioning

confidence: 99%

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The Discrete Moser?Veselov Algorithm for the Free Rigid Body, Revisited

McLachlan

Zanna²

2004

Found Comput Math

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In this paper we revisit the Moser-Veselov description for the free Rigid Body, which, in the 3×3 case, can be implemented as an explicit, second order, integrable approximation of the continuous solution. By backward error analysis, we study the modified vector field which is integrated exactly by the discrete algorithm. We deduce that the discrete Moser-Veselov (DMV) is well approximated to higher order by time-reparametrizations of the continuous equations (modified vector field). We use the modified vector field to preprocess the initial data to the DMV and show the equivalence of the DMV algorithm and the RATTLE algorithm. Numerical integration with these preprocessed initial data is several order of magnitude more accurate of the original DMV and RATTLE approach. IntroductionIn 1991 Moser and Veselov published a memorable paper (Moser & Veselov 1991) in which they described discrete versions of several classical integrable systems, among which, the Rigid Body (RB). The integrability of these discrete maps was shown with the help of a Lax-pair representation corresponding to a factorization of certain matrix polynomials.There is a large interest of the computational community towards good methods for the integration of the RB equations, since they arise naturally in a number of applications, for instance celestial mechanics and molecular dynamics. RB equations are used here to follow particles (planets, atoms, molecules, etc.) in between other body-body interactions. It is of fundamental importance that some qualitative features of the system under consideration are preserved under integration, for instance energy. However, in the recent years it has become clear that numerical preservation of energy alone is not enough to obtain 'good' qualitative descriptions of the system. Other properties like symplecticity and time-reversibility have been shown to produce favourable propagation of errors, hence better numerical methods, especially when interested over long time behaviour.Another interesting aspect of the work of Moser and Veselov is that their approach is based on the theory of discrete Lagrangians, which has become emergent in the computational mechanics community (see for instance (Marsden & West 2001) and references therein) because the methods derived by this approach naturally inherit the symplectic structure of the system under consideration, they preserve the discrete Lagrangian, preserve momentum and preserve energy. However, this approach has also some major drawbacks: i) it is difficult to obtain numerical methods of order higher than two, ii) it is usually difficult to estimate the numerical error, and iii) the derived methods (with some few exceptions) are highly implicit, hence computationally expensive. In facts, the DMV algorithm for the 3 × 3 RB is an exception, since, as we shall see in the sequel, it can be implemented as an explicit numerical method. * Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand. Email: r.mclachlan...

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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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The Moser–Veselov equation

Cited by 20 publications

References 10 publications

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

Optimal Control and Geodesics on Quadratic Matrix Lie Groups

The Discrete Moser?Veselov Algorithm for the Free Rigid Body, Revisited

Geometry of Discrete-Time Spin Systems

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