Geodesic Deviation Equation Approach to Chaos

Szczęsny, Jerzy; Dobrowolski, Tomasz

doi:10.1006/aphy.1999.5957

Cited by 13 publications

(17 citation statements)

References 6 publications

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“…Eisenhart's metric [18], which is consistent with Hamilton's least-action principle, is probably the most convenient choice. It offers easy calculation of the Riemann tensor, and it avoids spurious results traceable to the singular boundary of the perhaps better-known Jacobi metric that is derivable from Maupertius's leastaction principle [19] [16,20] derive an expression for in terms of the curvature and its standard deviation averaged over the microcanonical ensemble. The idea is that, as t !…”

Section: Estimated Time Scale For Chaotic Mixingmentioning

confidence: 99%

Chaotic orbits in thermal-equilibrium beams: Existence and dynamical implications

Bohn

Sideris

2003

Phys. Rev. ST Accel. Beams

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Phase mixing of chaotic orbits exponentially distributes these orbits through their accessible phase space. This phenomenon, commonly called ''chaotic mixing,'' stands in marked contrast to phase mixing of regular orbits which proceeds as a power law in time. It is operationally irreversible; hence, its associated e-folding time scale sets a condition on any process envisioned for emittance compensation. A key question is whether beams can support chaotic orbits, and if so, under what conditions? We numerically investigate the parameter space of three-dimensional thermal-equilibrium beams with space charge, confined by linear external focusing forces, to determine whether the associated potentials support chaotic orbits. We find that a large subset of the parameter space does support chaos and, in turn, chaotic mixing. Details and implications are enumerated.

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Section: Estimated Time Scale For Chaotic Mixingmentioning

confidence: 99%

Chaotic orbits in thermal-equilibrium beams: Existence and dynamical implications

Bohn

Sideris

2003

Phys. Rev. ST Accel. Beams

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“…where R is the Riemann curvature tensor (see, for example, [27] or [25]). This generalizes [29, Lemma 1] and reminds us of the intimate relationship between the curvature of a manifold and the 'spreading' of its geodesics.…”

Section: ) T Combot and T Watersmentioning

confidence: 99%

Integrability conditions of geodesic flow on homogeneous Monge manifolds

Combot

Waters²

2013

Ergod. Th. Dynam. Sys.

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“…6 Eisenhart's metric operates over a combined spacetime manifold in which the geodesics are parameterized by the real time t, i .e., ds 2 = dt 2 with…”

Section: Preliminariesmentioning

confidence: 99%

Nonperturbative Geometrodynamic Calculation of Chaotic Mixing Time in Charged-Particle Beams

Bohn¹

2000

The Physics of High Brightness Beams

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The time scale for irreversible mixing in a charged-particle bunch as a consequence of time-independent, nonlinear space-charge forces is estimated analytically to be a few plasma periods, much shorter than the two-body relaxation time. The basis for the estimate is a metric tensor inferred from Hamilton's least-action principle. Geodesics derived from the metric tensor correspond to particle trajectories. Their behavior reflects the properties of the curvilinear manifold in which they are embedded, among which irregularities associated with parametric resonances are of foremost importance. Exponential separation of nearby chaotic trajectories is thereby accessible to the geometrodynamic approach. The e-folding time associated with dispersing an initially localized perturbation throughout the bunch characterizes the process of irreversible mixing. It thereby constrains both the placement and size of hardware for emittance compensation that may be needed, for example, to undo phase-space degradation arising from coherent synchrotron radiation in magnetic bends. These constraints are estimated for linacs powering modern infrared and x-ray free-electron lasers.

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Geodesic Deviation Equation Approach to Chaos

Cited by 13 publications

References 6 publications

Chaotic orbits in thermal-equilibrium beams: Existence and dynamical implications

Chaotic orbits in thermal-equilibrium beams: Existence and dynamical implications

Integrability conditions of geodesic flow on homogeneous Monge manifolds

Nonperturbative Geometrodynamic Calculation of Chaotic Mixing Time in Charged-Particle Beams

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