Algebraic aspectsenvalues and the Krein type of eigenvalues on the unit circle in the complex plane. In Sections 4 to 7, we derive normal forms of symplectic matrices according to their eigenvalues. In Sections 8 and 9, we introduce the homotopy component, basic normal forms, and the ultimate type of symplectic matr
The variational functionalor Hamiltonian systems. Then in Section 2 we define the functional corresponding to the general nonlinear Hamiltonian systems on the space . of square integrable periodic functions and study its basic properties. In Section 3 we introduce the saddle point reduction method. In Section 4 we study the
Properties of index functionsndices of the reduced functional of corresponding linear Hamiltonian systems, and derive its axiom characterization. These results form the basis for our later study of the iteration theory and Morse theoretical applications to nonlinear Hamiltonian systems.
Relations with other Morse indicesems. In Section 7.1, we study the Galerkin approximation method for Hamiltonian systems. In Section 7.2, we give a simple proof for the coincidence of the Morse index of second order Hamiltonian systems with the 1-index defined in Chapter 5 of the corresponding linearized first order Hamiltonian sys
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Bott-type iteration formulaesymplectic group Sp(2.), starting from the identity matrix and studing its applications. We follow the ideas of and , and base our study upon the index function theory introduced in Chapter 5. As a special case, such a formula also works for the fundamental solutions of general linear
The common index jump theoremof nonlinear Hamiltonian systems; such a relationship is specially crucial in distinguishing Hamitonian solution orbits geometrically. For every symplectic path, we define its index jumps by certain iterated index intervals. The common index jump theorem claims that for any finite family of symplect
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Precise iteration formulael{P}_ au }left( {2n}
ight)] $$ we extend the definition of γto imes left[ {0, au }
ight] o {S_p}left( {2n}
ight)] $$ is a homotopy in the sense of Definition 5.0.3, its iteration is defined to be the map .$$[delta :left
Index Theory for Symplectic Paths with Applications