Existence of a capacity
This chapter is devoted to the existence proof of a distinguished capacity function, denoted by c0, and introduced by H. Hofer and E. Zehnder in . It is intimately related to periodic orbits of Hamiltonian systems. The capacity c0(M,ω) measures the minimal C0-oscillation of special Hamiltonian functions H : M → ℝ, needed in order to conclude the existence of a distinguished periodic orbit having small period and solving the associated Hamiltonian system X H on M. For 2-dimensional manifolds, c0 agrees with the total area, and in the special case of convex, open and smooth domains U ⊂ (ℝ2n,ω0) it is represented by a distinguished closed characteristic of the boundary ∂U having minimal (reduced) action equal to c0(U,ω0). This can be used, for example, to exhibit a class of compact hypersurfaces in ℝ2n, which are not symplectically diffeomorphic to convex ones. The proof will be based on the action principle. The variational approach will be explained in detail. In the analytical framework of the Sobolev space H1/2, we shall introduce a special minimax argument which originated in the work of P. Rabinowitz. The construction of the capacity c0 completes the proof of the symplectic rigidity phenomena described in Chapter 2. The special features of c0, however, will also be useful later on for the dynamics of Hamiltonian systems.
KeywordsPeriodic Solution Hamiltonian System Symplectic Manifold Hamiltonian Equation Capacity Function
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