A subriemannian, or Carnot-Cartheodory, geometry is a nonintegrable distribution, or subbundle of the tangent bundle of a manifold, which is endowed with an inner product. Part I presents the basic theory and examples, focussing on the geodesies. Chapters explaining the ideas of Cartan and Gromov are included. Part II presents applications to physics. These include Berry's quantum phase and an explanation of how a falling cat rights herself to land on her feet. Library of Congress Cataloging-in-Publication Data Montgomery, R. (Richard), 1956-A tour of subriemannian geometries, their geodesies and applications / Richard Montgomery. p. cm.-(Mathematical surveys and monographs,

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the "Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every "Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computationsFigure 1 (Initial conditions computed by Carles Simó)x 1 =−x 2 =0.97000436−0.24308753i,x 3 =0; V =ẋ 3 =−2ẋ 1 =−2ẋ 2 =−0.93240737−0.86473146i T =12T =6.32591398, I(0)=2, m 1 =m 2 =m 3 =1

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Abstract.A nonholonomic system, for short "NH,'' consists of a configuration space Q n , a Lagrangian L(q,q, t), a nonintegrable constraint distribution H ⊂ T Q, with dynamics governed by Lagrange-d'Alembert's principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T −V , where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇ NH which mimics Levi-Civita's. Local geometric invariants are obtained by Cartan's method of equivalence.As an example, we analyze Engel's (2-4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study * The authors thank the Brazilian funding agencies CNPq and

Introduction 1 1. History and Background 4 2. Distribution for rolling balls 5 2.1. The distribution 5 2.2. The "obvious" symmetry 6 3. Group theoretic description of the rolling distribution 6 3.1. Shrinking the group. 8 4. A G 2 -homogeneous distribution 8 5. The maximal compact subgroup of G 2 11 5.1. Algebraic strategy of the proof. 11 5.2. Finding Maximal compacts. 12 5.3. K ≃ so 3 ⊕ so 3 12 6. Split Octonions and the projective quadric realization of Q 14 7. Summary. Lack of action on the rolling space. The theorem is done. 19 Appendix A. Covers. Two G 2 's. 19 Appendix B. The isomorphism of K and so 3 ⊕ so 3 from Proposition 3. 20 Appendix C. The rolling distribution in Cartan's thesis 22 C.1. Cartan's constructions and claims. 22 C.2. Relation with Octonions 23 C.3. Commentary and proofs of Cartan's claims. 23 References 27 * * * * * *Despite all our efforts, the "3" of the ratio 1 : 3 remains mysterious. In this article it simply arises out of the structure constants for G 2 and appears in the construction of the embedding of so 3 × so 3 into g 2 (section 5 and Appendix B). Algebraically speaking, this '3' traces back to the 3 edges in g 2 's Dynkin diagram and the consequent relative positions of the long and short roots in the root diagram (see figure 2 below) for g 2 which the Dynkin diagram is encoding.Open problem. Find a geometric or dynamical interpretation for the "3" of the 3 : 1 ratio.

We study the problem of finding the shortest loops with a given holonomy. We show that the solutions are the trajectories of particles in Yang-Mills potentials (Theorem 4), or, equivalently, the projections of Kaluza-Klein geodesies (Theorem 2). Applications to quantum mechanics (Berry's phase, Sect. 3) and the optimal control of deformable bodies (Sect. 6) are touched upon. Contents 1. The Problem and Introduction 565 2. Two Theorems and Kaluza-Klein Matrics 570 3. Pines' Motivation, Homogeneous Bundles, and Some Open Problems 574 4. Electromagnetic Analogies and Half the Proof of Theorem 1 . . . . 580 5. Sub-Riemannian Metrics and Proof of the Hard Half 585 6. The Cat's Problem 587 7. Problem of Shapere and Wilczek 589 References 590

Hamiltonian structures for 2-or 3-dimensional incompressible flows with a free boundary are determined which generalize a previous structure of Zakharov for irrotational flow. Our Poisson bracket is determined using the method of Arnold, namely reduction from canonical variables in the Lagrangian (material) description. Using this bracket, the Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated. The structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained. In the case of two-dimensional flows, the vorticity bracket is determined and the generalized enstrophy is shown to be a Casimir function, This investigation also clears up some confusion in the literature concerning the vorticity bracket, even for fixed boundary flows.

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