Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures

G.B. Byrnes, F.A. Haggar and G.R.W. Quispel

Physica A: Statistical Mechanics and its Applications, 1999, vol. 272, issue 1, 99-129

Abstract: We present a number of alternative sufficient conditions for the existence of pre-symplectic or pre-implectic (Poisson) structures, for both ordinary differential (ODE) and ordinary difference (OΔE) equations. Four alternative sets of conditions are obtained for ODEs and OΔEs in n dependent variables: (1) A vector field in involution with the ODE and an integral (or two symmetries for an OΔE) imply a pre-implectic structure; (2) volume preservation and n−2 symmetries imply a pre-symplectic structure; (3) volume preservation and n−2 integrals imply a pre-implectic structure; (4) complex implectic structure implies infinitely many real implectic structures. In all but the first case the methods can give a set of distinct structures.

Keywords: Ordinary differential equations; Mappings; Symmetries; Integrals; Poisson structures; Implectic structures; Symplectic structures (search for similar items in EconPapers)
Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:272:y:1999:i:1:p:99-129

DOI: 10.1016/S0378-4371(99)00094-1

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