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Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries. (English) Zbl 1152.58007
Consider \(J=\left( \begin{smallmatrix} 0 & -I\\ I & 0 \end{smallmatrix}\right)\) and \(N=\left( \begin{smallmatrix} -I & 0\\ 0 & I \end{smallmatrix}\right)\) where \(I\) is the identity matrix of \({\mathbb R}^n\) and let \(Sp(2n)\), \({\mathcal L}_s({\mathbb R}^{2n})\) denote the symplectic group and the space of symmetric matrices \({2n}\times {2n}\), respectively. According to the paper by Y. Long, D. Zhang and C. Zhu [Adv. Math. 203, No. 2, 568–635 (2006; Zbl 1118.58006)], a path in \({\mathcal L}_s({\mathbb R}^{2n})\), \(B(t)=\left( \begin{smallmatrix} B_1(t) & B_2(t)\\ B_3(t) & B_4(t) \end{smallmatrix}\right)\) satisfies condition \((B_1)\) if \(B(t)\) is 1-periodic, \(B_1\) and \(B_4\) are even, \(B_2\) is odd and \(B_3\) is odd about the midpoint \(1/2\).
The paper concerns brake orbits of the Hamiltonian system
\[ \dot x=JH_x'(t,x),\tag{1} \]
\[ x(t+1)=x(t), x(t+1/2)=Nx(-t+1/2)\tag{2} \]
where \(H\) is \(C^1\) with linearizations \(B_0(t)x\), \(B_\infty(t)x\) at zero and infinity, respectively. Moreover
\[ H(t,x)=H(t,-x)=H(t,Nx), \quad H(t+1,x)=H(t,x)=H(-t,x)\;\forall(t,x). \]
(A periodic solution \(x=(p,q): {\mathbb R}\to{\mathbb R}^{2n}\) of the Hamiltonian system is called a brake orbit if it satisfies \(p(-t)=-p(t)\) and \(q(-t)=q(t)\).)
For paths in \({\mathcal L}_s({\mathbb R}^{2n})\) that satisfy \((B_1)\) there exists a Maslov-type index, defined as a pair of integers, in terms of a Maslov type index for symplectic paths (see also loc. cit.). Now, given two paths \(B_1\), \(B_2\) in \({\mathcal L}_s({\mathbb R}^{2n})\) that satisfy \((B_1)\) the author introduces a notion of relative Morse index of the pair. A theorem is given relating the relative Morse index of \((B_1, B_2)\) and the Maslov-type indices of \(B_1\), \(B_2\).
A deep result is the main theorem of the paper: it gives a lower bound for the number of pairs of nontrivial brake orbits of (1)–(2) in terms of the difference of the Maslov-type indices of \(B_0\) and \(B_1\). As an application, a new proof is given of the main theorem in article of V. Benci [Trans. Am. Math. Soc. 274, 533–572 (1982; Zbl 0504.58014)].

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
70H05 Hamilton’s equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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