A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants |
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Authors: | Francesco Calogero |
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Institution: | 1. Physics Department, University of Rome “La Sapienza”, 00185 Rome, Italy;2. and Istituto Nazionale di Fisica Nucleare, Sezione di Roma |
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Abstract: | 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 N2 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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Keywords: | Solvable N-body problems integrable N-body problems integrable dynamical systems |
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