Conservative dynamical systems involving strong forces

Gordon, William B.

doi:10.1090/S0002-9947-1975-0377983-1

Conservative dynamical systems involving strong forces
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Trans. Amer. Math. Soc. 204 (1975), 113-135 Request permission

Abstract:

We consider conservative dynamical systems associated with potentials $V$ which have singularities at a set $S:V(x) \to - \infty$ as $x \to S$. It is shown that various “action” integrals satisfy Condition $C$ of Palais and Smale provided that the potential satisfy a certain strong force (SF) condition. Hence, e.g., we establish the existence in SF systems of periodic trajectories which wind around $S$ and have arbitrary given topological (homotopy) type and which have arbitrary given period, and also periodic trajectories which make arbitrarily tight loops around $S$. Similar results are also obtained concerning the existence of trajectories which wind around $S$ and join two given points. The SF condition is shown to be closely related to the completeness (in the riemannian sense) of certain Jacobi metrics associated with the potential $V$, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems. The SF condition excludes the gravitational case, and the action integrals do not satisfy the Palais-Smale condition in the gravitational case. The Jacobi metrics associated with gravitational potentials are not complete. For SF systems there exist trajectories which join two given points and make arbitrarily tight loops around $S$, and this is not the case in the gravitational two body problem. On the other hand, for SF systems any smooth family of $\lambda$-periodic trajectories ($\lambda$ fixed) is bounded away from $S$, and this also is not the case for gravitational systems. Thus the definition of the SF condition is “well motivated", and leads to the disclosure of certain differences between the behavior of SF systems and gravitational (and other weak force) systems.

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