Investigations in Phase Space

  • Horst Leipholz


Let us assume that a mechanical process, similar to that which we have already discussed, is described by the system of differential equations q i = F i (q, α 0).We will restrict ourselves to autonomous systems. In order to investigate the stability of a particular solution q i 0 = F i (q 0, α 0) we will attempt to obtain our answer as easily as possible, as described in previous chapters. In other words, we will attempt to reach conclusions regarding stability without the complete integration of the system of differential equations whenever possible. In addition to the possibilities discussed in Section 1.3, we may use the intermediate integrals for this purpose, which geometrically represent phase surfaces and which have already been mentioned. The course of the solutions lying along the phase surfaces can be obtained from the properties of the phase surfaces. We may at least determine whether orbital stability exists or not.


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  1. 1.
    k is the number of the intermediate integral.Google Scholar
  2. 1.
    If the second derivatives of Fi are also zero, third-order terms with respect to k must be considered, etc.Google Scholar
  3. 1.
    LaSalle, J., Quart. App!. Math. 7, pp. 1–19, 1949.Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1987

Authors and Affiliations

  • Horst Leipholz
    • 1
  1. 1.University of WaterlooWaterlooCanada

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