SIGMA 9 (2013), 059, 18 pages arXiv:1310.2335
Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
Oksana Bihun a and Francesco Calogero b
a) Department of Mathematics, Concordia College at Moorhead, MN, USA
b) Physics Department, University of Rome ''La Sapienza'', Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
Received June 07, 2013, in final form October 02, 2013; Published online October 09, 2013
The class of solvable many-body problems ''of goldfish type'' is extended by including
(the additional presence of) three-body forces.
The solvable N-body problems thereby identified are characterized by Newtonian equations of motion
featuring 19 arbitrary ''coupling constants''.
Restrictions on these constants are identified which cause these systems – or appropriate variants of
them – to be isochronous or asymptotically isochronous, i.e. all their
solutions to be periodic with a fixed period (independent of the initial data) or to have
this property up to contributions vanishing exponentially as t→ ∞.
many-body problems; N-body problems; partial differential equations; isochronous systems.
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- Bihun O., Calogero F., Solvable and/or integrable many-body models on a circle,
J. Geom. Symmetry Phys. 30 (2013), 1-18.
- Bihun O., Calogero F., Yi G., Diophantine properties associated to the
equilibrium configurations of an isochronous N-body problem,
J. Nonlinear Math. Phys. 20 (2013), 158-178.
- Calogero F., Motion of poles and zeros of special solutions of nonlinear and
linear partial differential equations and related "solvable" many-body
problems, Nuovo Cimento B 43 (1978), 177-241.
- Calogero F., Classical many-body problems amenable to exact treatments,
Lecture Notes in Physics. New Series m: Monographs, Vol. 66,
Springer-Verlag, Berlin, 2001.
- Calogero F., The neatest many-body problem amenable to exact treatments (a
"goldfish"?), Phys. D 152-153 (2001), 78-84.
- Calogero F., Isochronous systems, Oxford University Press, Oxford, 2008.
- Calogero F., An integrable many-body problem, J. Math. Phys.
52 (2011), 102702, 5 pages.
- Calogero F., Another new goldfish model, Theoret. and Math. Phys.
171 (2012), 629-640.
- Calogero F., New solvable many-body model of goldfish type,
J. Nonlinear Math. Phys. 19 (2012), 1250006, 19 pages.
- Calogero F., Two quite similar matrix ODEs and the many-body problems related
to them, Int. J. Geom. Methods Mod. Phys. 9 (2012),
1260002, 6 pages.
- Calogero F., A linear second-order ODE with only polynomial solutions,
J. Differential Equations 255 (2013), 2130-2135.
- Calogero F., On the zeros of polynomials satisfying certain linear second-order
ODEs featuring many free parameters, J. Nonlinear Math. Phys.
20 (2013), 191-198.
- Calogero F., A solvable many-body problem, its equilibria, and a
second-order ordinary differential equation whose general solution is
polynomial, J. Math. Phys. 54 (2013), 012703, 13 pages.
- Calogero F., Yi G., A new class of solvable many-body problems, SIGMA
8 (2012), 066, 29 pages, arXiv:1210.0651.
- Calogero F., Yi G., Can the general solution of the second-order ODE
characterizing Jacobi polynomials be polynomial?, J. Phys. A:
Math. Theor. 45 (2012), 095206, 4 pages.
- Calogero F., Yi G., Diophantine properties of the zeros of certain Laguerre
and para-Jacobi polynomials, J. Phys. A: Math. Theor. 45
(2012), 095207, 9 pages.
- Calogero F., Yi G., Polynomials satisfying functional and differential
equations and Diophantine properties of their zeros, Lett. Math.
Phys. 103 (2013), 629-651.
- Gomez-Ullate D., Sommacal M., Periods of the goldfish many-body problem,
J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 351-362.