References
45
(1974). It should be noted, however, that (Benettin and Strelcyn 1978) have found
a transition region from regular to chaotic flow for a r. We will return to the
stadium billiard when we discuss quantum systems.
2.8 Conclusions
In this chapter, we have introduced concepts and model systems that will recur
repeatedly throughout the remainder of the book. For example, the stadium and the
bakers map will reappear in Chap. 7, where their quantum analogs will be studied.
It is interesting to note that (Ramani et al. 1989) have described a method,
different from that discussed in this chapter, to determine if a system is integrable.
They study the singularities of the differential equations and categorize them in
terms of those singularities. They conjecture that systems of equations with the
Painlevi property (the only moving singularities are poles) are integrable.
In this book, we will not discuss ergodic theory, which is a theory that attempts to
lay the dynamical foundations of statistical mechanics. Suffice it to say that systems,
such as the Sinai billiard, that are globally K-flows are also ergodic and mixing.
Excellent discussions about the relation between ergodic theory and dynamics may
be found in Farquhar (1964), Arnol’d and Avez (1968), and Ornstein (1974). Shorter
discussions may be found in Farquhar (1972) and Lebowitz and Penrose (1973).
References
Arnol’d VI (1963) Russ Math Surv 18:9; 18:85
Arnol’d VI, Avez A (1968) Ergodic problems of classical mechanics. W.A. Benjamin, New York
Barr AD, Barr AR, Porter MD, Reichl LE (2017) Chaos 27:104604
Barrar R (1970) Celestial Mech 2:494
Benettin G, Strelcyn JM (1978) Phys Rev A 17:773
Benettin G, Galgani L, Strelcyn JM (1976) Phys Rev A 14:2338
Benettin G, Froeshle C, Scheidecker JP (1979) Phys Rev A 19:2454
Berry MV (1978) AIP conference proceedings, vol. 46. American Institute of Physics, New York,
p. 16. Reprinted in [MacKay and Meiss 1987]
Birkhoff GD (1927) Acta Math 50:359
Bunimovich LA (1974) Funct Anal Appl 8:254
Casartelli M, Diana E, Galgani L, Scotti A (1976) Phys Rev A 13:1921
Chirikov B (1979) Phys Rep 52:263
Date E, Tanaka S (1976) Prog Theor Phys 55:457; Prog Theor Phys Suppl 59:107
Farquhar IE (1964) Ergodic theory in statistical mechanics. Wiley-Interscience, New York
Farquhar IE (1972) In: Beil J, Rae J (eds) Irreversibility in the Many-Body problem. Plenum Press,
New York
Flaschka H (1974) Phys Rev B 9:1924
Ford J, Stoddard DS, Turner JS (1973) Prog Theor Phys 50:1547
Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading
Henon M (1974) Phys Rev B 9:1921
45
(1974). It should be noted, however, that (Benettin and Strelcyn 1978) have found
a transition region from regular to chaotic flow for a r. We will return to the
stadium billiard when we discuss quantum systems.
2.8 Conclusions
In this chapter, we have introduced concepts and model systems that will recur
repeatedly throughout the remainder of the book. For example, the stadium and the
bakers map will reappear in Chap. 7, where their quantum analogs will be studied.
It is interesting to note that (Ramani et al. 1989) have described a method,
different from that discussed in this chapter, to determine if a system is integrable.
They study the singularities of the differential equations and categorize them in
terms of those singularities. They conjecture that systems of equations with the
Painlevi property (the only moving singularities are poles) are integrable.
In this book, we will not discuss ergodic theory, which is a theory that attempts to
lay the dynamical foundations of statistical mechanics. Suffice it to say that systems,
such as the Sinai billiard, that are globally K-flows are also ergodic and mixing.
Excellent discussions about the relation between ergodic theory and dynamics may
be found in Farquhar (1964), Arnol’d and Avez (1968), and Ornstein (1974). Shorter
discussions may be found in Farquhar (1972) and Lebowitz and Penrose (1973).
References
Arnol’d VI (1963) Russ Math Surv 18:9; 18:85
Arnol’d VI, Avez A (1968) Ergodic problems of classical mechanics. W.A. Benjamin, New York
Barr AD, Barr AR, Porter MD, Reichl LE (2017) Chaos 27:104604
Barrar R (1970) Celestial Mech 2:494
Benettin G, Strelcyn JM (1978) Phys Rev A 17:773
Benettin G, Galgani L, Strelcyn JM (1976) Phys Rev A 14:2338
Benettin G, Froeshle C, Scheidecker JP (1979) Phys Rev A 19:2454
Berry MV (1978) AIP conference proceedings, vol. 46. American Institute of Physics, New York,
p. 16. Reprinted in [MacKay and Meiss 1987]
Birkhoff GD (1927) Acta Math 50:359
Bunimovich LA (1974) Funct Anal Appl 8:254
Casartelli M, Diana E, Galgani L, Scotti A (1976) Phys Rev A 13:1921
Chirikov B (1979) Phys Rep 52:263
Date E, Tanaka S (1976) Prog Theor Phys 55:457; Prog Theor Phys Suppl 59:107
Farquhar IE (1964) Ergodic theory in statistical mechanics. Wiley-Interscience, New York
Farquhar IE (1972) In: Beil J, Rae J (eds) Irreversibility in the Many-Body problem. Plenum Press,
New York
Flaschka H (1974) Phys Rev B 9:1924
Ford J, Stoddard DS, Turner JS (1973) Prog Theor Phys 50:1547
Goldstein H (1980) Classical mechanics. Addison-Wesley, Reading
Henon M (1974) Phys Rev B 9:1921