On the Number of Isolating Integrals in Systems with Three Degrees of Freedom (Papers appear in the Proceedings of IAU Colloquium No. 10 Gravitational N-Body Problem (ed. by Myron Lecar), R. Reidel Publ. Co. , Dordrecht-Holland.)

Details On the Number of Isolating Integrals in Systems with Three Degrees of Freedom (Papers appear in the Proceedings of IAU Colloquium No. 10 Gravitational N-Body Problem (ed. by Myron Lecar), R. Reidel Publ. Co. , Dordrecht-Holland.) On the Number of Isolating Integrals in Systems with Three Degrees of Freedom (Papers appear in the Proceedings of IAU Colloquium No. 10 Gravitational N-Body Problem (ed. by Myron Lecar), R. Reidel Publ. Co. , Dordrecht-Holland.)

Nov 1971

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1971ap%26ss..14..110f&link_type=abstract

Astrophysics and Space Science, Volume 14, Issue 1, pp.110-117

Astronomy and Astrophysics

Astrophysics

20

Scientific paper

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: left\{ begin{gathered} x_1 = x_0 + a_1 {text{ sin (}}x_0 {text{ + }}y_0 {text{)}} + b{text{ sin (}}x_0 {text{ + }}y_0 {text{ + }}z_{text{0}} {text{ + }}t_{text{0}} {text{)}} \ y_1 = x_0 {text{ + }}y_0 \ z_1 = z_0 + a_2 {text{ sin (}}z_0 {text{ + }}t_0 {text{)}} + b{text{ sin (}}x_0 {text{ + }}y_0 {text{ + }}z_{text{0}} {text{ + }}t_{text{0}} {text{) (mod 2}}π {text{)}} \ t_1 = z_0 {text{ + }}t_0 \ right. We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).

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