A groupoid approach to geometric mechanics Daniel Fusca* Contents 1 Introduction 3 1.1 Main results . .4 1.2 Methods and applications . .7 1.2.1 Euler-Lagrange-Arnold equations and reduction . .8 2 Lie groupoids and Lie algebroids 10 2.1 Definitions and examples . 10 2.1.1 Lie groupoids . 10 2.1.2 Lie algebroids . 12 2.2 Connections on Lie algebroids . 15 2.2.1 Vector bundle connections . 15 2.2.2 A-connections, Levi-Civita connections . 18 2.3 Algebroid morphisms . 20 2.3.1 Algebroid morphisms from T I2 to A ................... 23 3 Mechanics on algebroids 24 3.1 Hamilton's principle and the Euler-Lagrange-Arnold equation . 25 3.2 Examples of Euler-Lagrange-Arnold equations . 26 3.2.1 Euler-Lagrange equations, Newton's equations, and geodesic equations 27 3.2.2 Euler-Poincar´eequations . 28 3.2.3 Dynamics on action algebroids . 29 3.3 Relation to Hamiltonian mechanics . 30 3.3.1 Natural Poisson bracket on the dual of an algebroid . 30 3.3.2 Hamilton's equations and the Legendre transform . 32 3.4 Lagrangian reduction . 34 3.5 Recovery of some known results on reduction . 37 *Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; e-mail: [email protected] 1 3.5.1 Euler-Poincar´ereduction . 38 3.5.2 Semidirect product reduction . 38 3.6 Riemannian submersions of algebroids . 40 4 Fluid-body kinematics 42 4.1 Rigid body kinematics . 42 4.1.1 Velocities of the body . 43 4.1.2 Metric on the body configuration space . 45 4.2 The fluid-body configuration space . 47 4.3 Fluid densities on exterior domains . 48 4.4 Velocities of the fluid-body system and the Hodge decomposition . 49 4.4.1 The Hodge decomposition . 50 4.4.2 Splitting the fluid-body velocity bundle . 53 5 Fluid-body dynamics 54 5.1 Unreduced dynamics . 54 5.1.1 Incompressible dynamics as geodesic flow . 55 5.1.2 Compressible dynamics as Newton's equation . 55 5.2 Reduced incompressible dynamics . 56 5.2.1 Metric and connection for the incompressible system . 60 5.2.2 Levi-Civita connection . 61 5.2.3 Incompressible fluid-body equations . 63 5.3 Reduced compressible dynamics . 66 5.3.1 Compressible fluid-body algebroid . 67 5.3.2 Metric and connection for the compressible system . 70 5.3.3 Compressible fluid-body equations . 75 5.4 Kirchhoff dynamics . 80 6 Reduction of fluid-body systems 81 6.1 Incompressible fluid-body dynamics as geodesic flow . 82 6.2 Compressible fluid-body dynamics as Newton's equation . 82 6.3 Projection of potential solutions to the Kirchhoff system . 83 7 The Madelung transform 86 7.1 Geometric preliminaries . 88 7.1.1 Hamiltonian structures of non-linear Schr¨odingerand the quantum hydrodynamical system . 88 7.1.2 Lie group and Lie algebra actions on the space of wave functions . 89 7.2 The Madelung transform and a geometric interpretation . 90 7.2.1 The Madelung transform . 90 7.2.2 The Madelung transform as a momentum map . 92 2 1 Introduction In 1966 V. Arnold proved that the Euler equation for an incompressible fluid describes the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms of the fluid’s domain [1]. This remarkable observation led to numerous advances in the study of the Hamiltonian properties, instabilities, and topological features of fluid flows. However, Arnold's approach does not apply to systems whose configuration spaces do not have a group structure. A particular example of such a system is that of a fluid with moving boundary. More generally, one can consider a system describing a rigid body moving in a fluid. Here the configurations of the fluid are identified with diffeomorphisms mapping a fixed reference domain to the exterior of the (moving) body. In general such diffeomorphisms cannot be composed, since the domain of one will not match the range of the other. The systems we consider are 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 diffeomor- phism 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. This extends the recent work [11], where a groupoid approach was developed to study incompressible fluid flows with vortex sheets. 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 incompress- ible 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 sen. In order to prove these results, we further develop the theory of Lagrangian mechanics on algebroids. This approach to mechanics was initiated by Weinstein [27]. Significant advances were made by Martinez [20] et al. [5], [21]. 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. There are two limiting cases of interest: the case of a fluid of zero density describes the motion of a body alone, while the case where the body is the empty set describes the motion of an ideal compressible or incompressible fluid. 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 [15], and more recently studied in [12] and [26]. 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. Namely, the non-linear Schr¨odingerequation is Hamiltonian with respect to the constant Poisson structure on the space of wave functions, which are complex valued (fast 3 decaying) smooth functions on Rn. On the other hand, the compressible Euler equation is Hamiltonian with respect to the natural Lie-Poisson structure on the space of pairs consisting of (fast decaying at infinity) fluid momenta µ and fluid densities ρ. This space is the dual of the Lie algebra of the semidirect product of the diffeomorphism group of Rn times the space of real-valued fast decaying functions, which is the configuration space of a compressible fluid. The thesis is divided into three parts. The first consists of Sections 2 and 3, where the general theory of Lagrangian mechanics on algebroids is presented. The second part of the thesis comprises Sections 4 through 6, where the general theory is applied to various systems of a rigid body moving in a fluid. The last part, Section 7, studies geometric and group properties of the Madelung transform. 1.1 Main results In this thesis we consider the following dynamical equations governing the motion of a rigid body in a fluid. In the simplest case, the fluid is incompressible and irrotational and there is no circulation around the body.1 In this case, there are so many constraints on the fluid that its motion is completely determined by the motion of the body. The effect of the fluid is to add to the body's effective inertia. The governing equations for the body's motion are the Kirchhoff equations: ¢ ¨ d ¨ ! !; r¥ λ l l ¦ dt ¨ ¨ d ¨ λ rλ : ¤ dt Here ! and λ are the effective angular and linear momenta of the fluid-body system, r and l are the angular and linear velocities, and the diamond product l ¢ Rn Rn so n is defined by `λ l l; re ¢ `λ, rle If the fluid is no longer constrained to be irrotational, but still assumed to be incom- pressible, then the system is governed by the incompressible fluid-body equations: ¨¢ d ¨ u u ©u ©P ¨dt ¨ d ¨ n ¨m l S P n ind q ¦ dt @Fx ¨ d ¨ T n ¨ rIx S P n q qx ind q ¨ @F ¨dt x ¨ d ¨ x ξ : ¤dt 1The condition that there be no circulation around the body follows from irrotationality of the fluid if the exterior of the body is simply connected. We will always assume this. 4 Here Fx is the domain of the fluid around the body located at position x and n is the outward pointing normal of the surface of the body @Fx. A superscript T denotes the transpose. The first equation is the incompressible Euler equation for the fluid with velocity u. The second and third equations are Newton's law for the body's linear momentum ml and angular > momentum rIx respectively. The last equation relates the body's position x SE n to its velocity ξ > TxSEn. The function P is the pressure, which in addition to ensuring that the fluid motion remains incompressible throughout the motion, also ensures that the fluid’s normal velocity at the boundary is equal to the body's normal velocity. Finally, if there are no restrictions on the fluid, then the motion of system is governed by the compressible fluid-body equations: ¢ © ¨ d © P1 © ¨ dt u u u ρ~ P2 ¨ ¨ d ρ~ © ρu~ 0 ¨ dt ¨ d « 1 ¦m dt l P2 ρ~ P1 n inρ @F ¨ x ¨ d « 1 T ¨ dt rIx P2 ρ~ P1 n q qx inρ ¨ @F ¨ x ¨ d ¤ dt x ξ : Compared to the incompressible case, the fluid density ρ ρd~ nq is an additional dynamical quantity.