Goldfishing by gauge theory

Calogero, F.; Langmann, Edwin

doi:10.1063/1.2235035

Cited by 20 publications

(7 citation statements)

References 16 publications

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“…This last step requires that such an ansatz exist, and that someone discover it. Several successful examples have been recently reported [8][9][10][11][12][13][14] (for a review see Ref. [4]), and they also led, as already mentioned above, to Diophantine findings and conjectures.…”

Section: Introductionmentioning

confidence: 73%

Isochronous Dynamical System and Diophantine Relations I

Calogero¹,

Iona²

2021

JNMP

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We identify a solvable dynamical system-interpretable to some extent as a many-body problem-and point out that-for an appropriate assignment of its parameters-it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data). We then identify its equilibrium configurations and investigate its behavior in their neighborhood. We thereby identify certain matrices-of arbitrary order-whose eigenvalues are all rational numbers: a Diophantine finding.

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Section: Introductionmentioning

confidence: 73%

Isochronous Dynamical System and Diophantine Relations I

Calogero¹,

Iona²

2021

JNMP

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“…We do not display explicitly the equations of motion of this second model in the simplest N " 2 case because they can be immediately obtained by setting r " 0 in those of the first model (see (16)). …”

Section: Resultsmentioning

confidence: 99%

“…(The original goldfish model is the special case of these equations of motion with r " 0 and f nˆ z,¨ z˙" 0; after its first identification as a solvable model [7], and its tentative recognition as a "goldfish" [8], this N-body problem and some of its extensions have been investigated in several publications (see, for instance, [9][10][11][12][13][14][15][16][17][18][19][20]). Notation 1.1.…”

Section: Introductionmentioning

confidence: 99%

Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

Calogero

2016

Symmetry

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Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer (N ě 2). These models are characterized by nonlinear Newtonian ("accelerations equal forces") equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods T{p with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future.Keywords: solvable many-body problems; integrable many-body problems; integrable dynamical systems; solvable many-body problems of goldfish type

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“…Recently a technique has been introduced [1] which allows a rather straightforward identification of solvable N-body problems "of goldfish type" (for the identification of this class of solvable N-body problems, and a selection of previous developments concerning this class, see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). The usefulness of this technique to identify new many-body problems characterized by certain Newtonian ("acceleration equal force") equations of motion describing the motion of N nonlinearly interacting pointlike unit-mass particles moving in the complex z-plane has been demonstrated by several examples [1,[19][20][21].…”

Section: Introductionmentioning

confidence: 99%

A Solvable N-body Problem of Goldfish Type Featuring N² Arbitrary Coupling Constants

Calogero¹

2021

JNMP

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A N-body problem "of goldfish type" is introduced, the Newtonian ("acceleration equal force") equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model-for arbitrary N-is solvable, namely its configuration (positions and velocities of the N "particles") at any later time t can be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N 2 arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous-so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.

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Goldfishing by gauge theory

Cited by 20 publications

References 16 publications

Isochronous Dynamical System and Diophantine Relations I

Isochronous Dynamical System and Diophantine Relations I

Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

A Solvable N-body Problem of Goldfish Type Featuring N² Arbitrary Coupling Constants

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Goldfishing by gauge theory

Cited by 20 publications

References 16 publications

Isochronous Dynamical System and Diophantine Relations I

Isochronous Dynamical System and Diophantine Relations I

Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

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A Solvable N-body Problem of Goldfish Type Featuring N² Arbitrary Coupling Constants