Global behaviors of weak KAM solutions for exact symplectic twist maps. (English) Zbl 1444.37047
Summary: We investigated several global behaviors of the weak KAM solutions \(u_c(x, t)\) parametrized by \(c \in H^1(\mathbb{T}, \mathbb{R})\). For the suspended Hamiltonian \(H(x, p, t)\) of the exact symplectic twist map, we could find a family of weak KAM solutions \(u_c(x, t)\) parametrized by \(c(\sigma) \in H^1(\mathbb{T}, \mathbb{R})\) with \(c(\sigma)\) continuous and monotonic and \[\partial_t u_c + H(x, \partial_x u_c + c, t) = \alpha(c), \quad \text{a.e.} (x, t) \in \mathbb{T}^2,\] such that sequence of weak KAM solutions \(\{ u_c \}_{c \in H^1 ( \mathbb{T} , \mathbb{R} )}\) is 1/2-Hölder continuity of parameter \(\sigma \in \mathbb{R} \). Moreover, for each generalized characteristic (no matter regular or singular) solving \[\begin{cases} \dot{x} ( s ) \in \text{co} \left[ \partial_p H \left( x ( s ) , c + D^+ u_c ( x ( s ) , s + t ) , s + t \right) \right], \\ x ( 0 ) = x_0, \quad ( x_0 , t ) \in \mathbb{T}^2, \end{cases}\] we evaluate it by a uniquely identified rotational number \(\omega(c) \in H_1(\mathbb{T}, \mathbb{R})\). This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
MSC:
| 37J11 | Symplectic and canonical mappings |
| 37E40 | Dynamical aspects of twist maps |
| 37E45 | Rotation numbers and vectors |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J46 | Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems |
| 37J51 | Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
Keywords:
exact symplectic twist map; Aubry-Mather theory; Hamilton-Jacobi equation; generalized characteristics; weak KAM solution; transition chainReferences:
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