Abstract
Three planar classical particles interact via a potential proportional to the area of the triangle they form. This system is equivalent to two oscillators attached to the origin, the nearest being repelled by and the other being attracted to it (piecewise integrable Hamiltonian). Numerical simulations show two types of trajectories: those apparently escaping to infinity, and those in confined quasiperiodic orbits. Adiabatic theories lead to discrete recurrence relations and allow for the second type only. A general method allowing prediction of first return time of the slow motion as well as a short/long-period relation is presented. The issue of the possibly metastable nature of escaping trajectories is raised.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Bavaud, Statistical mechanics of convex bodies,J. Stat. Phys. 57:1059–1068 (1989); F. Bavaud, Isoperimetric phase transitions of two-dimensional droplets,Commun. Math. Phys. 132:549–554 (1990).
A. Carnegie and I. C. Percival, Regular and chaotic motion in some quartic potentials,J. Phys. A 17:801–813 (1984).
B. Simon, The classical limit of quantum partition function,Commun. Math. Phys. 71:247–276 (1980); B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum,Ann. Phys. (N.Y.) 146:209–220 (1983).
T. Dagaeff and C. Rouvinez, On the discontinuities of the boundary in billiards,Physica D 67:166–187 (1993).
D. G. Kendall, Exact distributions for shapes of random triangles in convex sets,Adv. Appl. Prob. 17:308–329 (1985); C. R. Goodall and K. V. Mardia, Multivariate aspects of shape theory,Ann. Stat. 21:848–866 (1993).
L. Landau and E. Lifchitz,Mécanique (Mir, Moscow, 1969).
Author information
Authors and Affiliations
Additional information
Dedicated to Prof. Philippe Choquard.
Rights and permissions
About this article
Cite this article
Bavaud, F. Triangular dynamics under pressure. J Stat Phys 76, 645–660 (1994). https://doi.org/10.1007/BF02188679
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02188679