Mechanical Connections

1. Mechanical Connections Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml (65)96314907 49th Annual Meeting of the Australian Mathematical Society University of Western Australia Sept 27-30, 2005 Topology and Geometry Seminar National University of Singapore Oct 5, 2005 1

2. 3-6 Earth, Tangents, Tubes, Beanies Contents 7-10 Rolling Ball Kinematics 11-13 Nonholonomic Dynamics – Formulation 14-22 Distributions and Connections 23-24 Nonholonomic Dynamics - Solution 25-26 Rolling Coin Dynamics 27 Symmetry and Momentum Maps 28 Rigid Body Dynamics 29 Boundless Applications 30-33 References 2

3. 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?” Is the Earth Flat ? Stand Sunset Sphere Cube Answer: Not unless your brain is !!! 3

4. 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. How Are Tangent Vectors Connected ? Area = Radius = 1 Holonomy: rotation of tangent vectors parallel translated around meridian = area of spherical cap. 4

5. Tubes used for anatomical probing (imaging, surgery) can bend but they can not twist. So how do they turn? How do Tubes 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. 5

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

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

8. on the plane z = -1 is described the by therefore Rolling Without Turning if a ball rolls along the curve 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

9. The material trajectory Material Trajectory and Holonomy satisfies hence Theorem [Lioe2004] If then 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

10. Theorem [Lioe2004] If is a rotation trajectory Optimal Trajectory Control is a small trajectory variation and is defined by then Proof Since and Theorem [Lioe2004] If is the shortest trajectory with specified 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. Potential Application: Rotate (real or virtual) rigid body by moving a computer mouse. See [Sharpe1997]. 10

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

12. One method to develop the dynamics of a system with Lagrangian L that is subject to holonomic constraints Holonomic Constraints 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 . such that The 2m-k variables (x’s & lambda’s) are computed from m-k constraint equations and the m equations given by . 12

13. For nonholonomic constraints D’Alemberts principle can also be applied to obtain the existence of Nonholonomic Constraints such that . 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 ! 13

14. Analytic Geometry: relations & functions synthetic geometry  algebra Level Sets and Foliations Calculus: fundamental theorems local  global 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) Example (global) foliation of O into 2-dim spheres 14

15. 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. Frobenius Distributions 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. Definition A vector field v : E  T(E) is subordinate to a distribution d (v < d) if 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

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

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

18. Theorem A connection on a bundle Holonomy of a Connection and points p, q in B then every path f from p to q in B defines a diffeomorphism (holonomy) between fibers 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. 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

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

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

21. Locally on M the 1-forms . Distributions Connections define the distribution . Hence they also define a fiber bundle . where is an open subset of and Therefore can be identified with a horizontal and this describes subspace an Ehresmann connection . on . 21

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

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) Equivalent Form for Constraints 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]. 23

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

25. 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). Rolling Coin Constraints Exercise compare with Hand-Finch solution on p 64 25

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

27. Manifold & Lie algebra structure { , } on such Poisson Manifolds is a derivation. that  are Lie algebras and anti homomorphism. Example 1 Symplectic Manifold In particular if Example 2 Lie-Poisson bracket Reduction Theorem 27

28. Consider a left-action of on a Poisson manifold Momentum Maps by canonical maps, hence an anti homomorphism and a map such that commutes is a momentum map if commutes, or 28

29. Momentum Function is an anti homomorphism If has flow then has flow Momentum Map for Lifted Left Action on a Manifold Equivariance Momentum Map for Lifted Left Action on a Lie Group 29

30. Noether’s Theorem If acts canonically on a Poisson Symmetry and admits a momentum map and is invariant, then is a constant of motion for the Hamiltonian flow induced by Proof Corollary The Hamiltonian flow above induces a Hamiltonian flow on each reduced space See [Marsden1990,1994]. 30

31. Here Rigid Body Dynamics is a positive definite self-adjoint inertial operator, and the Hamiltonian is LInv The reduced dynamics on the base space of the FB yields dynamic reconstruction using the canonical 1-form connection [Marsden1990] who remarks in [Marsden1994] that reconstruction was done in 1942 Theorem [Ishlinskii1952,1976] The holonomy of a period T reduced orbit that enclosed a spherical area A is 31

32. 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 Further Applications PDE’s – KDV, Incompressible and Compressible Fluids, Magnetohydrodynamics, Plasmas-Maxwell-Vlasov, Maxwell, Loop Quantum Gravity, … Representation Theory, Algebraic Geometry,... 32

33. [Halliday2001] D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, Ext. Sixth Ed. John Wiley. References [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. [Sharpe1997] R. W. Sharpe, Differential Geometry- Cartan’s Generalization of Klein’s Erlangen Program, Springer, New York. [Hand1998] L. Hand and J. Finch, Analytical Mechanics, Cambridge University Press. 33

34. [Frobenius1877] G. Frobenius, Uber das Pfaffsche Probleme, J. Reine Angew. Math., 82,230-315. References [Chern1990] S. Chern, W. Chen and K. Lam, Lectures on Differential Geometry, World Scientific, Singapore. [Ehresmann1950] C. Ehresmann, Les connexions infinitesimales dans ud espace fibre differentiable, Coll. de Topologie, Bruxelles, CBRM, 29-55. [Hermann1993] R. Hermann, Lie, Cartan, Ehresmann Theory,Math Sci Press, Brookline, Massachusetts. [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. 34

35. [Marsden2001] H. Cendra, J. Marsden, and T. Ratiu, Geometric Mechanics,Lagrangian Reduction and Nonholonomic Systems, 221-273 in Mathematics Unlimited - 2001 and Beyond, Springer. References [Marsden2004] Nonholonomic Dynamics, AMS Notices [Ishlinskii1952] A. Ishlinskii, Mechanics of special gyroscopic systems (in Russian). National Academy Ukrainian SSR, Kiev. [Ishlinskii1976] A. Ishlinskii, Orientation, Gyroscopes and Inertial Navigation (in Russian). Nauka, Moscow. [Kane1969] T. Kane and M. Scher, A dynamical explanation of the falling phenomena, J. Solids Structures, 5,663-670. 35

36. [Smale1970] S. Smale, Topology and Mechanics, Inv. Math., 10, 305-331, 11, 45-64. References [Montgomery1990] R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128,565-592. [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). 36