A groupoid approach to geometric mechanics

Fusca, Daniel (2022) A groupoid approach to geometric mechanics. Doctoral thesis, UIN SAIZU Purwokerto.

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Abstract

We consider numerous variations of a rigid body in an inviscid fluid. The different cases are specified by the properties of the fluid; the fluid may be compressible or incompressible, irrotational or not. By using groupoids we generalize Arnold’s diffeomorphism group framework for fluid flows to show that the well-known equations governing the motion of these various systems can be viewed as geodesic equations (or more generally, Newton’s equations) written on an appropriate configuration space. We also show how constrained dynamical systems on larger algebroids are in many cases equivalent to dynamical systems on smaller algebroids, with the two systems being related by a generalized notion of Riemannian submersion. As an application, we show that incompressible fluid-body motion with the constraint that the fluid velocity is curl- and circulation-free is equivalent to solutions of Kirchhoff’s equations on the finite-dimensional algebroid se�n�. In order to prove these results, we further develop the theory of Lagrangian mechanics on algebroids. Our approach is based on the use of vector bundle connections, which leads to new expressions for the canonical equations and structures on Lie algebroids and their duals. The case of a compressible fluid is of particular interest by itself. It turns out that for a large class of potential functions U, the gradient solutions of the compressible fluid equations can be related to solutions of Schr¨odinger-type equations via the Madelung transform, which was first introduced in 1927. We prove that the Madelung transform not only maps one class of equations to the other, but it also preserves the Hamiltonian properties of both equations. i

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