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Effective Hamiltonians and averaging for Hamiltonian dynamics. I. (English) Zbl 0986.37056
This paper is one of the first attempts to develop PDE techniques to understand certain aspects of Hamiltonian dynamics with many degrees of freedom. More precisely: given a smooth Hamiltonian \(H:\mathbb{R}^n \times\mathbb{R}^n \to\mathbb{R}\) with (1) \(\dot x=D_p H(p,x)\), \(\dot p=-D_xH(p,x)\).
The authors examine the Hamiltonian flow (1) under a canonical change of variables \((p,x)\mapsto (P,X)\) (2) \(p=D_xu(P,x)\), \(X=D_pu(P,x)\).
The key point is that the “corrector” PDE, introduced and solved in a weak sense by P. Lions, Papanicolau and Varadhan in their study of periodic homogenization for Hamilton-Jacobi equations, formally induces a canonical change of variables, in terms of which the dynamics are trivial. The authors study to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in momenta.

37J50 Action-minimizing orbits and measures (MSC2010)
35F20 Nonlinear first-order PDEs
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