1. The Concept of Holonomy and its Applications to Biomechanics, Economics and Physics Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml (65)96314907 1

2. Contents Vector Fields 2 Differential Forms Distributions and Connections Holonomy and its Applications

3. Vector Fields on is a gradient vector field iff there exists such that integrable iff there exists a nowhere 0 integrating factor 3 such thatis a gradient v. f. Theorem 1.is a gradient v. f. iff Theorem 2.is integrable iff and is

4. Vector Fields on is orthogonal to (iff 4 Lemma 1. Vector fields can be represented by first order linear differential operators that are closed under the commutator product ) Theorem 3. (Frobenius)is integrable iff

5. Vector Fields on is horizontal w.r.t.( iff 5 Lemma 2. isolated 0’s, integrable ) Proof. Letbe an integrating factor and such that p, q can not be connected by a horizontal path. such that Ifis horizontal w.r.t.then Theorem 4. (Chow-Rashevskii-Sussmann, Ambrose-Singer) Converse and extension of Lem 2

6. How do Tubes Turn ? 5 Tubes used for anatomical probing (imaging, surgery) can bend but they can not twist. So how do they turn? Unit tangent vector of tube  curve on sphere, Normal vector of tube  tangent vector to curve. Tube in plane  geodesic curve on sphere No twist  tangent vector parallel translated (angle with geodesic does not change)  holonomy = area enclosed by closed curve.

7. Rigid Body Motion is described by are defined by The velocity of a material particle whose motion space and its angular velocity in in the body is Furthermore, the angular velocities are related by 7

8. Is the Earth Flat ? Page 1 of my favorite textbook [Halliday2001] grabs the reader with a enchanting sunset photo and the question: “How can such a simple observation be used to measure Earth?” 3 Sphere StandSunset Cube Answer: Not unless your brain is !!!

9. How Are Tangent Vectors Connected ? The figure on page 44 in [Marsden1994] illustrates the parallel translation of a Foucault pendulum, we observe that the cone is a flat surface that has the same tangent spaces as the sphere ALONG THE MERIDIAN. 4 Holonomy: rotation of tangent vectors parallel translated around meridian = area of spherical cap. Radius = 1 Area =

10. Elroy’s Beanie Example described on pages 3-5 in [Marsden1990] 6 body 1 inertia Shape. shape trajectory  configuration Conservation of Angular Momentum  body 2 inertia Configuration Mechanical Connection Angular Momentum Flat Connection  Holonomy is Only Topological

11. Rigid Body Motion is described by are defined by The velocity of a material particle whose motion space and its angular velocity in in the body is Furthermore, the angular velocities are related by 7

12. Rolling Without Turning on the plane z = -1 is described the by if a ball rolls along the curve therefore then Astonishingly, a unit ball can rotate about the z-axis by rolling without turning ! Here are the steps: 1. [0 0 -1]  [pi/2 0 -1] 2. [pi/2 0 -1]  [pi/2 -d -1] 3. [pi/2 -d -1]  [0 d -1] The result is a translation and rotation by d about the z-axis. 8

13. Material Trajectory and Holonomy The material trajectory satisfieshence Theorem [Lioe2004] Ifthen where A = area bounded by u([0,T]). Proof The no turning constraints give a connection on the principle SO(2) fiber bundle and the curvature of this connection, a 2-form on with values in the Lie algebra so(2) = R, coincides with the area 2-form induced by the Riemannian metric. 9

14. Optimal Trajectory Control and is a rotation trajectory is a small trajectory variation Proof Since is defined by then and 10 is the shortest the ball rolls along an arc of a circle in the plane P and u([0,T]) is an arc of a circle in the sphere. Furthermore, M(T) can be computed explicitly from the parameters of either of these arcs. Theorem [Lioe2004] If trajectory with specified Potential Application: Rotate (real or virtual) rigid body by moving a computer mouse.

15. Unconstrained Dynamics 11 The dynamics of a system with kinetic energy T and forces F (with no constraints) is where For conservative. we have where we define the Lagrangian. For local coordinates. we obtain m-equations and m-variables..

16. Holonomic Constraints 12 such that is to assume that the constraints are imposed by a constraint force F that is a differential 1-form that kills every vector that is tangent to the k dimensional submanifold of the tangent space of M at each point. This is equivalent to D’Alembert’s principle (forces of constraint can do no work to ‘virtual displacements’) and is equivalent to the existence of p variables. One method to develop the dynamics of a system with Lagrangian L that is subject to holonomic constraints The 2m-k variables (x’s & lambda’s) are computed from m-k constraint equations and the m equations given by.

17. Nonholonomic Constraints 13 such that. For nonholonomic constraints D’Alemberts principle can also be applied to obtain the existence of where the mu-forms describe the velocity constraints The 2m-k variables (x’s & lambda’s) are computed from the m-k constraint equations above and the m equations On vufoils 20 and 21 we will show how to eliminate (ie solve for) the m-k Lagrange multipliers !

18. Level Sets and Foliations Analytic Geometry: relations & functions synthetic geometry  algebra Implicit Function Theorem for a smooth function F Local (near p) foliation (partition into submanifolds) consisting of level sets of F (each with dim = n-m) Calculus: fundamental theorems local  global Example (global) foliation of O into 2-dim spheres 14

19. Frobenius Distributions Definition A dim = k (Frobenius) distribution d on a manifold E is a map that smoothly assigns each p in E A dim = k subspace d(p) of the tangent space to E at p. Definition A vector field v : E  T(E) is subordinate to a distribution d (v < d) if Example A foliation generates a distribution d such that point p, d(p) is the tangent space to the submanifold containing p, such a distribution is called integrable. The commutator [u,v] of vector fields is the vector field uv-vu where u and v are interpreted as first order partial differential operators. Theorem [Frobenius1877] (B. Lawson  by Clebsch & Deahna) d is integrable iff u, v < d  [u,v] < d. Remark. The fundamental theorem of ordinary diff. eqn.  evey 1 dim distribution is integrable. 15

20. Cartan’s Characterization A dim k distribution d on an m-dim manifold arises as. 16 d is integrable iffCartan’s Theorem where. are differential 1-forms. Proof See [Chern1990] – crucial link is Cartan’s formula Remark Another Cartan gem is:

21. Ehresmann Connections the vertical distribution d on E is defined by Definition [Ehresmann1950] A fiber bundle is a map between manifolds with rank = dim B, Theorem c is the kernel of a V(E)-valued connection and a connection is a complementary distribution c This defines T(E) into the bundle sum and image of a horizontal lift 1-form with 17 We letdenote the horizontal projection.

22. Holonomy of a Connection Theorem A connection on a bundle Proof Step 1. Show that a connection allows vectors in T(B) be lifted to tangent vectors in T(E) Step 2. Use the induced bundle construction to create a vector field on the total space of the bundle induced by a map from [0,1] into B. Step 3. Use the flow on this total space to lift the map. Use the lifted map to construct the holonomy. and points p, q in B then every path f from p to q in B defines a diffeomorphism (holonomy) between fibers Remark. If p = q then we obtain holonomy groups. Connections can be restricted to satisfy additional (symmetry) properties for special types (vector, principle) of bundles. 18

23. Curvature, Integrability, and Holonomy Definition The curvature of a connection is the 2-form 19 where andare vector field extensions. Theorem A connection is integrable (as a distribution) iff its curvature = 0. Theorem This definition is independent of extensions. Theorem A connection has holonomy = 0 iff its curvature = 0.

24. Implicit Distribution Theorem 20 there exists a (m-k) x m matrix (valued function of p) E we introduce local coordinates. with rank m-k and an invertible (m-k) x (m-k) matrix and c is defined by the coordinate indices so that. hence we may re-label where B is where so Given a dim = k distribution on a dim = m manifold M

25. Distributions  Connections Locally on M the 1-forms. 21 define the distribution. Hence they also define a fiber bundle. whereis an open subset ofand Thereforecan be identified with a horizontal subspace and this describes an Ehresmann connection. on.

26. Curvature Computation 22 where. where. if and only if

27. Equivalent Form for Constraints 23 Since the mu’s and omega’s define the same distribution we can obtain an equivalent system of equations with different lambda’s (Lagrange multipliers) On the next page we will show how to eliminate the Lagrange multipliers so as to reduce these equations to the form given in Eqn. (3) on p. 326 in [Marsden2004].

28. Eliminating Lagrange Multipliers 24 We observe that we can express and reduced k equations hence we solve for the Lagrange multipliers to obtain These and the m-k constraint equations determine the m variables.

29. Rolling Coin General rolling coin problem p 62-64 [Hand1998]. Theta = angle of radius R, mass m coin with y-axis phi = rotation angle rolling on surface of height z(x,y). Constraints 25 Exercise compare with Hand-Finch solution on p 64

30. How Curved Are Your Coins ? Let ‘s compute the curvature for the rolling coin system 26

31. Symmetry and Momentum Maps Definition [Marsden1990,1994] 27 is a momentum map if is a Poisson manifold with a left Hamiltonian action by a Lie group G with Lie algebra that satisfies where is the left-invariant vector field on with linear dual by the flow generated Theorem [Marsden1990,1994] If H : P  R is G-inv. then it induces a Hamiltonian flow on the red. space. The reduced space is a PM.

32. Rigid Body Dynamics Here 28 and is the pullback under right translation. The Hamiltonian is a positive definite self-adjoint inertial operator, and where is a fiber bundle whose connection (canonical 1-form on the symplectic manifold P) gives dynamic reconstruction from reduced dynamics. Theorem [Ishlinskii1952] (discovered 1942) The holonomy of a period T reduced orbit that enclosed a spherical area A is

33. Boundless Applications 29 Falling Cats, Heavy Tops, Planar Rigid Bodies, Hannay-Berry Phases with applications to adiabatics and quantum physics, molecular vibrations, propulsion of microorganisms at low Reynolds number, vorticity free movement of objects in water, PDE’s – KDV, Maxwell-Vlasov, …

34. References 30 [Halliday2001] D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, Ext. Sixth Ed. John Wiley. [Marsden1994] J. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag. [Marsden1990] J. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics, Memoirs of the AMS, Vol 88, No 436. [Lioe2004] Luis Tirtasanjaya Lioe, Symmetry and its Applications in Mechanics, Master of Science Thesis, National University of Singapore. [Hand1998] L. Hand and J. Finch, Analytical Mechanics, Cambridge University Press.

35. References 31 [Hermann1993] R. Hermann, Lie, Cartan, Ehresmann Theory,Math Sci Press, Brookline, Massachusetts. [Ehresmann1950] C. Ehresmann, Les connexions infinitesimales dans ud espace fibre differentiable, Coll. de Topologie, Bruxelles, CBRM, 29-55. [Frobenius1877] G. Frobenius, Uber das Pfaffsche Probleme, J. Reine Angew. Math., 82,230-315. [Chern1990] S. Chern, W. Chen and K. Lam, Lectures on Differential Geometry, World Scientific, Singapore. [Marsden2001] H. Cendra, J. Marsden, and T. Ratiu, Geometric Mechanics,Lagrangian Reduction and Nonholonomic Systems, 221-273 in Mathematics Unlimited - 2001 and Beyond, Springer, 2001.

36. References 32 [Ishlinskii1952] A. Ishlinskii, Mechanics of special gyroscopic systems (in Russian). National Academy Ukrainian SSR, Kiev. [Marsden2001] H. Cendra, J. Marsden, and T. Ratiu, Geometric Mechanics,Lagrangian Reduction and Nonholonomic Systems, 221-273 in Mathematics Unlimited - 2001 and Beyond, Springer. [Marsden2004] Nonholonomic Dynamics, AMS Notices [Kane1969] T. Kane and M. Scher, A dynamical explanation of the falling phenomena, J. Solids Structures, 5,663-670. [Montgomery1990] R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128,565-592.

37. References 33 [Berry1988] M. Berry, The geometric phase, Scientific American, Dec,26-32. [Guichardet1984] On the rotation and vibration of molecules, Ann. Inst. Henri Poincare, 40(3)329-342. [Shapere1987] A. Shapere and F. Wilczek, Self propulsion at low Reynolds number, Phys. Rev. Lett., 58(20)2051-2054. [Kanso2005] E. Kanso, J. Marsden, C. Rowley and J. Melli-Huber, Locomotion of articulated bodies in a perfect fluid (preprint from web).