Newton's Equation on Diffeomorphisms and Densities

07/11/2017
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We develop a geometric framework for Newton-type equations on the infinite dimensional configuration space of probability densities. It can be viewed as a second order analogue of the "Otto calculus" framework for gradient  flow equations. Namely, for an n-dimensional manifold M we derive Newton's equations on the group of diffeomorphisms Di (M) and the space of smooth probability densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Poisson reduction of Newton's equation on Di (M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton-Jacobi equation of  fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrödinger-type and Newton's equations is a symplectomorphism between the corresponding phase spaces T*Dens(M) and PL2(M,C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kahler map provided that the space of densities is equipped with the (prolonged) Fisher-Rao information metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher-Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton-Jacobi, and linear and nonlinear Schrödinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and µ-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.

Newton's Equation on Diffeomorphisms and Densities

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application/pdf Newton's Equation on Diffeomorphisms and Densities Boris Khesin, Gerard Misio lek, Klas Modin
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We develop a geometric framework for Newton-type equations on the infinite dimensional configuration space of probability densities. It can be viewed as a second order analogue of the "Otto calculus" framework for gradient  flow equations. Namely, for an n-dimensional manifold M we derive Newton's equations on the group of diffeomorphisms Di (M) and the space of smooth probability densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Poisson reduction of Newton's equation on Di (M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton-Jacobi equation of  fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrödinger-type and Newton's equations is a symplectomorphism between the corresponding phase spaces T*Dens(M) and PL2(M,C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kahler map provided that the space of densities is equipped with the (prolonged) Fisher-Rao information metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher-Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton-Jacobi, and linear and nonlinear Schrödinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and µ-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.
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Newton’s Equation on Diffeomorphisms and Densities Boris Khesin1 , Gerard Misiolek2 , and Klas Modin3 1 Department of Mathematics, University of Toronto khesin@math.toronto.edu 2 Department of Mathematics, University of Notre Dame gmisiole@nd.edu 3 Mathematical Sciences, Chalmers and University of Gothenburg klas.modin@chalmers.se We develop a geometric framework for Newton-type equations on the infinite- dimensional configuration space of probability densities. It can be viewed as a second order analogue of the “Otto calculus” framework for gradient flow equa- tions. Namely, for an n-dimensional manifold M we derive Newton’s equations on the group of diffeomorphisms Diff(M) and the space of smooth probabil- ity densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Pois- son reduction of Newton’s equation on Diff(M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton–Jacobi equation of fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrödinger-type and Newton’s equations is a symplecto- morphism between the corresponding phase spaces T∗ Dens(M) and PL2 (M, C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kähler map provided that the space of densities is equipped with the (prolonged) Fisher–Rao infor- mation metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher–Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton–Jacobi, and linear and nonlinear Schrödinger equations, the framework for Newton equations encapsulates Burgers’ inviscid equation, shallow water equations, two-component and µ-Hunter–Saxton equa- tions, the Klein–Gordon equation, and infinite-dimensional Neumann problems. Keywords: Newton’s equation, Wasserstein distance, Fisher–Rao metric, Madelung transform, compressible Euler equations [1] von Renesse, M.K.: An optimal transport view of Schrödinger’s equation. Canad. Math. Bull 55, 858–869 (2012)