zbMATH — the first resource for mathematics

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)].

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)
PDF BibTeX Cite
Full Text: DOI
[1] Benci, V., On critical point theory for indefinite functionals in the presences of symmetries, Trans. amer. math. soc., 274, 533-572, (1982) · Zbl 0504.58014
[2] Benci, V.; Rabinowitz, P., Critical point theorems for indefinite functionals, Invent. math., 52, 241-273, (1979) · Zbl 0465.49006
[3] Booss, B.; Furutani, K., The Maslov-type index - A functional analytical definition and the spectral flow formula, Tokyo J. math., 21, 1-34, (1998) · Zbl 0932.37063
[4] Chang, K.C.; Liu, J.Q.; Liu, M.J., Nontrivial periodic resonance Hamiltonian systems, Ann. inst. H. Poincaré anal. non linéaire, 14, 1, 103-117, (1997) · Zbl 0881.34061
[5] Cappell, S.E.; Lee, R.; Miller, E.Y., On the Maslov-type index, Comm. pure appl. math., 47, 121-186, (1994) · Zbl 0805.58022
[6] Dong, Y., Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems, Pacific J. math., 221, 2, 253-280, (2005) · Zbl 1124.37033
[7] Fei, G., Relative Morse index and its application to Hamiltonian systems in the presence of symmetries, J. differential equations, 122, 302-315, (1995) · Zbl 0840.34032
[8] Long, Y.; Zhang, D.; Zhu, C., Multiple brake orbits in bounded convex symmetric domains, Adv. math., 203, 568-635, (2006) · Zbl 1118.58006
[9] Long, Y.; Zhu, C., Maslov-type index theory for symplectic paths and spectral flow (II), Chinese ann. math. ser. B, 21, 1, 89-108, (2000) · Zbl 0959.58017
[10] Rabinowitz, P., On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear anal., 11, 5, 599-611, (1987)
[11] Robbin, J.; Salamon, D., The Maslov indices for paths, Topology, 32, 827-844, (1993) · Zbl 0798.58018
[12] Seifert, H., Periodische bewegungen mechanischer systeme, Math. Z., 51, 197-216, (1948) · Zbl 0030.22103
[13] D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems, preprint · Zbl 1126.37042
[14] Zhu, C.; Long, Y., Maslov-type index theory for symplectic paths and spectral flow, I, Chinese ann. math. ser. B, 20, 4, 413-424, (1999) · Zbl 0959.58016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.