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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

37G99 Local and nonlocal bifurcation theory for dynamical systems
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