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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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##### References:
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