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Newton's Equation o...
Newton's Equation on Diffeomorphisms and Densities
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- Khesin, Boris (author)
- University of Toronto
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- Misiolek, Gerard (author)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
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- Modin, Klas, 1979 (author)
- Gothenburg University,Göteborgs universitet,Institutionen för matematiska vetenskaper,Department of Mathematical Sciences
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(creator_code:org_t)
- Cham : Springer International Publishing, 2017
- 2017
- English.
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In: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). - Cham : Springer International Publishing. - 1611-3349 .- 0302-9743. - 9783319684451 ; 10589, s. 873-873
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Abstract
Subject headings
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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 Diff(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 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 Schrodinger-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 Schrodinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and mu-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.
Subject headings
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
- NATURVETENSKAP -- Matematik -- Geometri (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Geometry (hsv//eng)
- NATURVETENSKAP -- Matematik -- Matematisk analys (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Mathematical Analysis (hsv//eng)
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Keyword
- Wasserstein distance
- Madelung transform
- Fisher-Rao metric
- Compressible Euler equations
- Newton's equation
- Newton's equation
- Wasserstein distance
- Fisher-Rao metric
- Madelung transform
- Compressible
Publication and Content Type
- kon (subject category)
- ref (subject category)
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