On the number of isolating integrals in systems with three degrees of freedom |
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| Authors: | Claude Froeschle |
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| Institution: | 1. Observatoire de Nice, Le Mont-Gros, 06, Nice, France
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| Abstract: | 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: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ 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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