The algebraic entropy of classical mechanics

McLachlan, Robert I.; Ryland, Brett N.

doi:10.1063/1.1576904

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…This allows methods, usually known as 'Runge-Kutta-Nyström' methods, with smaller errors and often with fewer force evaluations as well. A full analysis of the Lie algebra generated by f 1 and f 2 for simple mechanical systems is provided in [96].…”

Section: Simple Mechanical Systemsmentioning

confidence: 99%

Geometric integrators for ODEs

McLachlan

Quispel

2006

J. Phys. A: Math. Gen.

156

161

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Geometric integration is the numerical integration of a differential equation, while preserving one or more of its 'geometric' properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature.

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Section: Simple Mechanical Systemsmentioning

confidence: 99%

Geometric integrators for ODEs

McLachlan

Quispel

2006

J. Phys. A: Math. Gen.

156

161

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“…It is, however, distinct from this established definition in that there is no underlying set of recurrence inequalities to provide additional structure (see, for example, [37,Lemma 4.1.7]). The proposed entropy concept also coincides in certain special cases (modulo a logarithm) with the notion of entropy defined for graded algebras [40,41]. But that connection does not appear to be so important for the present work.…”

Section: Introductionmentioning

confidence: 72%

Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

Gray¹

2022

SIGMA

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This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions.

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“…The main point here is that in the case (69) (we will refer to that as the RKN case), [[[F [b] , F [a] ], F [b] ], F [b] ] = 0 identically. This is equivalent to F [babb] = 0 in (53), which introduces some linear dependencies among higher order terms in the expansion of log(Ψ(h)) (see [59] for a detailed study). This means that the characterization given in Theorem 1 for a splitting integrator (36) to be of order r (for r ≥ 4) is no longer applicable if one restricts to the case (69).…”

Section: Runge-kutta-nyström Methodsmentioning

confidence: 99%

Splitting and composition methods in the numerical integration of differential equations

Blanes,

Casas,

Murua

2008

Preprint

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We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.

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The algebraic entropy of classical mechanics

Cited by 7 publications

References 21 publications

Geometric integrators for ODEs

Geometric integrators for ODEs

Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

Splitting and composition methods in the numerical integration of differential equations

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