Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces

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\rightarrow \infty $.