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Effective Hamiltonians and Averaging for Hamiltonian Dynamics II
"... Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study ..."
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Cited by 48 (27 self)
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Abstract. This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE methods to understand the structure of certain Hamiltonian flows. The main point is that the “cell”or “corrector”PDE, introduced and solved in a weak sense by Lions, Papanicolaou 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. We investigate to what extent this observation can be made rigorous in the case that the Hamiltonian is strictly convex in the momenta, given that the relevant PDE does not usually in fact admit a smooth solution. 1. Introduction. This is the first of a projected series of papers that develop PDE techniques to understand certain aspects of Hamiltonian dynamics with many degrees of freedom. 1.1. Changing variables.
Two Approximations for Effective Hamiltonians Arising from Homogenization of HamiltonJacobi Equations
, 2003
"... Effective Hamiltonian appears in homogenization of HamiltonJacobi equations, semiclassical limit of Schrödinger equation, MajdaSouganidis combustion model, Hamiltonian dynamics and many other applications. We propose two new numerical methods for computing effective Hamiltonians by solving Hamilt ..."
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Cited by 11 (0 self)
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Effective Hamiltonian appears in homogenization of HamiltonJacobi equations, semiclassical limit of Schrödinger equation, MajdaSouganidis combustion model, Hamiltonian dynamics and many other applications. We propose two new numerical methods for computing effective Hamiltonians by solving HamiltonJacobi equations numerically: Small&delta; Method and LargeT Method. Small&delta; method is based on solving an approximate cell problem directly by using a monotone numerical scheme. LargeT method is based on solving a nonlinear eigenvalue problem, which in turn reduces to evolving a pseudotime dependent HamiltonJacobi equation to a large time. Numerical examples show the accuracy and efficiency of the algorithms.
VISCOUS STABILITY OF QUASIPERIODIC LAGRANGIAN TORI
"... Abstract. We consider a smooth Tonelli Lagrangian L: T T n → R and its viscosity solutions u(x, P) characterized by the cell equation H(x, P + Dxu(x, P)) = H(P), where H: T ∗ T n → R is the Hamiltonian associated with L. We will show that if P0 corresponds to a quasiperiodic Lagrangian invariant t ..."
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Abstract. We consider a smooth Tonelli Lagrangian L: T T n → R and its viscosity solutions u(x, P) characterized by the cell equation H(x, P + Dxu(x, P)) = H(P), where H: T ∗ T n → R is the Hamiltonian associated with L. We will show that if P0 corresponds to a quasiperiodic Lagrangian invariant torus, then Dxu(x, P) is Hölder continuous in P at P0 with Hölder exponent arbitrarily close to 1, and if both H and the torus are real analytic and the frequency vector of the torus is Diophantine, then Dxu(x, P) is Lipschiz continuous in P at P0, i.e., there is a constant C> 0 such that ‖Du(x, P) − Du(x, P0)‖ ∞ ≤ C‖P − P0 ‖ as ‖P − P0 ‖ ≪ 1. Similar Pregularity of the Peierls barrier for L will be obtained, and applications to viscosity solutions near KAM tori in configuration space in a nearly integrable Hamiltonian system will also be considered.
EFFECTIVE HAMILTONIANS AND QUANTUM STATES
"... Abstract. We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M12, MF], augmented by the PDE theory from Fathi [F1,2]and from [EG1]. This earlier work provides us with a Lipschitz continuous function u solving the eik ..."
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Abstract. We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M12, MF], augmented by the PDE theory from Fathi [F1,2]and from [EG1]. This earlier work provides us with a Lipschitz continuous function u solving the eikonal equation a.e. and a probability measure σ solving a related transport equation. We present some elementary formal identities relating certain quantum states ψ and u, σ. We show also how to build out of u, σ an approximate solution of the stationary Schrödinger eigenvalue problem, although the error estimates for this construction are not very good. 1. Introduction. This paper records a few observations and comments concerning the possible implications for quantum mechanics of Fathi’s “weak KAM”theory from [F12] and the recent paper [EG1], which discusses connections between the “effective Hamiltonian”introduced by Lions–Papanicolaou–Varadhan [LPV], Mather’s theory of action minimizing measures