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Multiplicité des trajectoires fermées de systèmes hamiltoniens connexes. (Multiplicity of closed trajectories of convex Hamiltonian systems). (French) Zbl 0633.58034
Let $$S\subset R^{2n}$$ be a compact hypersurface of class $$C^ 2$$, being the boundary of an open convex set containing the origin, for $$x\in S$$ let n(x) be the unit outward normal vector of S and $$J=\left[ \begin{matrix} 0\\ -I_ n\end{matrix} \begin{matrix} I_ n\\ 0\end{matrix} \right]$$. The following theorem is proved: if $$n\geq 3$$, and S has a strictly positive Gaussian curvature then the flow $$\dot x=Jn(x)$$ over S has two closed trajectories, at least. This theorem has the following corollary: if $$H: R^{2n}\mapsto R$$ is of class $$C^ 2$$, the level surface $$S=H^{- 1}(h)$$ satisfies the conditions above, and $$H'(x)=0$$, $$x\in S$$ then the problem $$\dot x=JH'(x)$$, $$x(0)=x(T)$$, $$H(x)=h$$ has two solutions $$(x_ 1,T_ 1)$$, $$(x_ 2,T_ 2)$$, at least $$(x_ 1\neq x_ 2)$$.
Reviewer: M.Farkas

MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:
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