S ep 2 01 6 Maslov-type indices and linear stability of elliptic Eulersolutions of the three-body problem
In this paper, we use the central configuration coordinate de composition to study the linearized Hamiltonian system near the 3-body elliptic Euler solution s. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute theω-indices and obtain certain properties of linear stability of the Euler elliptic soluti ons of the classical three-body problem.
Index Theory for Symplectic Paths with Applications
I The Symplectic Group Sp(2n).- 1 Algebraic Aspects.- 1.1 Symplectic matrices.- 1.2 Symplectic spaces.- 1.3 Eigenvalues of symplectic matrices.- 1.4 Normal forms for the eigenvalue 1.- 1.5 Normal forms for the eigenvalue ?1.- 1.6 Normal forms for eigenvalues in U?R.- 1.7 Normal forms for eigenvalues outside U.- 1.8 Basic normal forms.- 1.9 Perturbations basic normal forms.- 2 Topological Aspects.- 2.1 Structures of Sp(2) and its subsets.- 2.2 The global structure of Sp(2n,R).- 2.3 Hyperbolic symplectic matrix set.- 2.4 Structure of regular sets.- 2.5 Structures of singular sets.- 2.6 Transversality of rotation paths.- 2.7 Orientability of M?,(2n) in Sp(2n).- II The Variational Method.- 3 Hamiltonian Systems and Canonical Transformations.- 3.1 Canonical transformations.- 3.2 Generating functions.- 4 The Variational Functional.- 4.1 The Galerkin approximation.- 4.2 The L2-Variational Structure.- 4.3 The saddle point reduction.- 4.4 The dimension theorem on kernels.- 4.5 Certain estimates.- III Index Theory.- 5 Index Functions for Symplectic Paths.- 5.1 Paths in Sp(2).- 5.2 Non-degenerate paths in Sp(2n).- 5.3 Index properties of non-degenerate paths.- 5.4 Perturbations of degenerate paths.- 6 Properties of Index Functions.- 6.1 Index functions and Morse indices.- 6.2 An axiom approach and further properties.- 7 Relations with other Morse Indices.- 7.1 The Galerkin approximation.- 7.2 Second order Hamiltonian systems.- 7.3 Lagrangian systems.- IV Iteration Theory.- 8 Precise Iteration Formulae.- 8.1 Paths in Sp(2).- 8.2 Hyperbolic and elliptic paths.- 8.3 General symplectic paths.- 9 Bott-type Iteration Formulae.- 9.1 Splitting numbers.- 9.2 Bott-type formulae.- 9.3 Abstract precise iteration formulae.- 10 Iteration Inequalities.- 10.1 Estimates via mean index and initial index.- 10.2 Successive estimates.- 10.3 Controlling iteration numbers via indices.- 11 The Common Index Jump Theorem.- 11.1 A common selection theorem.- 11.2 The common index jump theorem.- 12 Index Iteration Theory for Closed Geodesics.- 12.1 Morse index theory.- 12.2 Splitting numbers.- V Applications.- 13 The Rabinowitz Conjecture.- 13.1 Minimax principle preparations.- 13.2 Controlling the minimal period via indices.- 13.3 Asymptotically linear Hamiltonian systems.- 13.4 Superquadratic Hamiltonian systems.- 13.5 Second order systems.- 13.6 Subharmonics.- 13.7 Notes and comments.- 14 Periodic Lagrangian Orbits on Tori.- 14.1 Critical module preparations.- 14.2 The finite energy homology theory.- 14.3 Critical modules and isomorphisms.- 14.4 Global homological injectivity.- 14.5 Global homological vanishing.- 14.6 Notes and comments.- 15 Closed Characteristics on Convex Hypersurfaces.- 15.1 Index theorem for dual action principle.- 15.2 Variational properties.- 15.3 Critical orbits and index jumps.- 15.4 Existence and multiplicity.- 15.5 Stability results.- 15.6 Symmetric hypersurfaces.- 15.7 Notes and comments.
Central configurations of the N-body problem via equivariant Morse theory
In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4.The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.