Classical irreversibility and mapping of the isosceles triangle billiard
Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle ψ=2π/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.
KeywordsField Theory Elementary Particle Quantum Field Theory Dynamical Equation Mass Point
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- Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G. (1982).Ergodic Theory, Springer-Verlag, New York.Google Scholar
- Hasegawa, H. H., and Driebe, D. J. (1992).Physics Letters A,168, 18.Google Scholar
- Hasegawa, H. H., and Driebe, D. J. (1993).Physics Letters A,176, 193.Google Scholar
- Katok, A. B., and Strelcyn, J.-M. (1986).Invariant Manifolds, Entropy and Billiards, Springer-Verlag, New York.Google Scholar
- Kerckhoff, S., Masur, H., and Smillie, J. (1986).J. Ann. Math. 124, 293.Google Scholar
- Kozlov, V. (1991).Billiards: A Generic Introduction to the Dynamics of Systems with Impacts, AMS, Providence, Rhode Island.Google Scholar
- Liboff, R. L. (1987).Journal of Physics A,20, 5607.Google Scholar
- Liboff, R. L. (1990a).Kinetic Theory: Classical, Quantum and Relativistic Descriptions, Prentice-Hall, Englewood Cliffs, New Jersey, Section 3.3.5.Google Scholar
- Liboff, R. L. (1990b).Journal of Non-Equilibrium Thermodynamics,15, 15.Google Scholar
- Reichl, L. E. (1980).A Modern Course in Statistical Physics, University of Texas Press, Austin, Texas, Chapter 8.Google Scholar
- Sinai, Ya. G. (1976).Introduction to Ergodic Theory, Princeton University Press, Princeton, New Jersey.Google Scholar