Reduction, symmetry, and phases in mechanics

Marsden, Jerrold E.; Montgomery, Richard; Raţiu, Tudor S.

doi:10.1090/memo/0436

Cited by 221 publications

(302 citation statements)

References 31 publications

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“…If the initial point r(0) is given, then the horizontal lift is uniquely specified, and the dynamics for its evolution are those given by (17). The horizontal lift is a useful construction for studying geometric phases [12].…”

Section: Relation Between Slices and Connectionsmentioning

confidence: 99%

Reduction and reconstruction for self-similar dynamical systems

et al. 2003

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We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied.The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.Mathematics Subject Classification: 70G65, 76M60, 60G18, 34A26, 37J15, 65P99

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Section: Relation Between Slices and Connectionsmentioning

confidence: 99%

Reduction and reconstruction for self-similar dynamical systems

et al. 2003

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“…This is called the "reconstruction problem" of the dynamics. This problem is intimately related to the concepts of geometric and dynamic phase, which play an important role in various aspects of mechanics [31] and in the study of locomotion systems (for example, in the generation of net motion by cyclic changes in shape space [18,35]). …”

Section: Reconstructionmentioning

confidence: 99%

On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

150

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Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

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“…Notably, it will become clear that the role played by solid angle in the spin 1/2 example must be filled by what we dub precession holonomy. This quantity, like solid angle, is an example of holonomy 18,19 , meaning it fits the definition of a geometric phase. It can be ascribed to any curve in the 'parameter space', which in our case is the poloidal plane.…”

Section: Introductionmentioning

confidence: 94%

“…19 related to reconstruction to guiding center motion in tokamaks. Reconstruction refers to the process of obtaining a full solution to a system of Hamiltonian equations with symmetry from a solution in the reduced phase space.…”

Section: Connections and Reconstructionmentioning

confidence: 99%

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Toroidal Precession as a Geometric Phase

Qin¹

2012

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Toroidal precession is commonly understood as the orbit-averaged toroidal drift of guiding centers in axisymmetric and quasisymmetric configurations. We give a new, more natural description of precession as a geometric phase effect. In particular, we show that the precession angle arises as the holonomy of a guiding center's poloidal trajectory relative to a principal connection. The fact that this description is physically appropriate is borne out with new, manifestly coordinate-independent expressions for the precession angle that apply to all types of orbits in tokamaks and quasisymmetric stellarators alike. We then describe how these expressions may be fruitfully employed in numerical calculations of precession.

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Reduction, symmetry, and phases in mechanics

Cited by 221 publications

References 31 publications

Reduction and reconstruction for self-similar dynamical systems

Reduction and reconstruction for self-similar dynamical systems

On the geometry of generalized Chaplygin systems

Toroidal Precession as a Geometric Phase

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