Indices and Stability of the Lagrangian System on Riemannian Manifold
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Indices and Stability of the Lagrangian System on Riemannian Manifold Gao Sheng ZHU School of Mathematics, Tianjin University, Tianjin 300072, P. R. China E-mail : [email protected] Abstract In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincar´e map of the Lagrangian system at critical point x d Lq (t, x, x) ˙ − Lp (t, x, x) ˙ =0 dt is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices. Keywords Nonorientable, spectral flow, Morse index, Maslov-type indices, Lagrangian system, Hamiltonian system MR(2010) Subject Classification
1
58E05, 37J45, 34C25
Introduction and Main Results
The aim of this paper is to discuss the relations of stability and indices of Lagrangian systems with Legendre convexity condition. Morse developed his index theory to treat closed geodesics on manifolds in the 1930s, Bott established Morse index iteration theory to further study closed geodesics in his celebrated work [5] in 1956. Since then, many interesting and significant results have been obtained via Morse index theory. Maslov index theory is a natural and difficult generalization of Morse index theory, which is successful to deal with the related problems of period solutions of Hamiltonian systems by many mathematicians. A systematic work on Maslov index and its iteration theory are in [6, 16] and the references there. In some cases, one has to investigate the orbits with some symmetric property, such as n-body problem in celestial mechanics [10] etc.; Meanwhile, the periodic orbits of Lagrangian systems with Legendre convexity condition on Riemannian manifolds can be transformed into some symmetric orbits in Euclidean space, hence Maslov index theory has to be extended to some symmetric cases [7, 8, 13–15, 25]. Received July 26, 2019, accepted July 1, 2020 Supported by NSFC (Grant Nos. 11871356, 11871368)
Zhu G. S.
2
In this paper, we suppose that M be a compact Riemannian manifold of dimension m with metric ·, ·, the functional 1 L(t, x(t), x(t)), ˙ ∀x ∈ W 1,2 (S1 , M), (1.1) A(x) = 0
where Lagrangian function L ∈ C 2 (S1 × T M, R) with S1 = R/Z satisfies the Legendre convexity condition Lqq (t, p, q) > 0, ∀(t, p, q) ∈ S1 × T M. (1.2) Suppose x is a critical point of A, then we obtain the following Lagrange system d ˙ − Lp (t, x, x) ˙ = 0. Lq (t, x, x) dt
(1.3)
To simplify notations, we write P (t) = Lqq (t, x, x), ˙
Q(t) = Lqp (t, x, x), ˙
R(t) = Lpp (t, x, x) ˙ + R0 ,
where R0 is defined by (3.8). The Hessian of A at x is given by 1 D2 A(x)(v, w) = {P ∇v + Qv, ∇w + QT ∇v, w + Rv,
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