Hamiltonian Symmetry Reduction via Localizations: Theory and Application to a Barbell System
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Hamiltonian Symmetry Reduction via Localizations: Theory and Application to a Barbell System Jürgen Scheurle1 · Sebastian Walcher2
Received: 1 August 2018 / Accepted: 21 November 2018 / Published online: 10 December 2018 © Springer Nature B.V. 2018
Abstract We specialize a recently introduced variant of orbit space reduction for symmetric Hamiltonian systems. This variant works with suitable localizations of the algebra of polynomial invariants of the corresponding symmetry group action, and provides reduction to a variety that is embedded in a low-dimensional affine space, which makes efficient computations possible. As an example, we discuss the mechanical system of a “barbell” in a general central force field. Keywords Linear groups · Invariant theory · Hamiltonian systems with symmetry · Relative equilibria Mathematics Subject Classification 34C20 · 13A50 · 34C14 · 37C80
1 Introduction The present paper is concerned with Hamiltonian symmetry reduction. The general theory for such reductions can be found e.g. in Arnold ([1] Appendix 5), Marsden and Weinstein [18], Kummer [13], Marsden and Ratiu [17], Ortega and Ratiu [19], and Cushman and Bates [5] (Ch. VII), and thus one may consider the theory as complete. The aspect we want to emphasize here is an efficient computational (“algebraic”) reduction procedure via invariants of the underlying symmetry group action. As is common knowledge (see for example the survey by Chossat [4] which describes strengths and weaknesses of the approach), one may employ the polynomial invariants of the symmetry group of a general dynamical system to construct a reduced system defined on an
B J. Scheurle
[email protected] S. Walcher [email protected]
1
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany
2
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany
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J. Scheurle, S. Walcher
affine algebraic variety. However, the problem with many systems is the relatively high dimension of the embedding space for the variety (as compared to the dimension of the latter), which renders any practical work with the reduced system almost impossible. We circumvent this difficulty by passing to suitable localizations of the polynomial invariant algebra. This method was recently introduced in [27], building on work by Grosshans [7]. Thus one obtains an efficient reduction, which is conceptually simple, computationally feasible, and enables a step-by-step reconstruction of the full dynamics of the original system. In the present paper, we show that this reduction procedure respects any group-invariant Hamiltonian structure of the original system, yielding an induced Poisson bracket after reduction, with the reduced system again being Hamiltonian. In particular the approach opens a path to a complete analysis of relative equilibria, including stability properties and corresponding bifurcations. We illustrate the reduction procedure with a mechanical system which we call “barbell” system (two mass points connected by a massless, but not necessaril
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