• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文会议>International conference on geometric science of information >Newton's Equation on Diffeomorphisms and Densities
【24h】

Newton's Equation on Diffeomorphisms and Densities

机译：牛顿的微分同构和密度方程

获取原文
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

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 PL~2(M,C). This improves on the previous symplec-tic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kaehler 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 μ-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.
机译：我们在概率密度的无穷维配置空间上为牛顿型方程开发了一个几何框架。可以将其视为梯度流方程的“奥托演算”框架的二阶类似物。即，对于n维流形M，我们在微分群Diff（M）和光滑概率密度Dens（M）的空间上推导牛顿方程，并描述与它们相关的汉密尔顿约简。例如，可压缩的Euler方程是通过Diff（M）上具有对称组的体积保持微分对称性的牛顿方程的Poisson约简而获得的，而流体力学的Hamilton-Jacobi方程则对应于潜在的解。我们还证明了Schrodinger型方程和牛顿方程之间的Madelung变换是对应相空间T〜* Dens（M）和PL〜2（M，C）之间的辛同态。这比以前的冯·雷尼塞斯（Ven Renesse）[1]的辛式潜水结果要好。此外，我们证明只要密度空间配备了（延长的）Fisher-Rao信息度量，并且描述了它的动态应用，Madelung变换就是一个Kaehler映射。 Madelung变换的这种几何设置阐明了经典Fisher-Rao度量与其量子对等物Bures度量之间的关系。除了可压缩的Euler，Hamilton-Jacobi以及线性和非线性Schrodinger方程外，牛顿方程的框架还封装了Burgers的无粘性方程，浅水方程，二元和μ-Hunter-Saxton方程，Klein-Gordon方程以及无限维Neumann问题。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号