Differentially Constrained Dynamics

1. with monetary applications to rolling coins* Differentially Constrained Dynamics Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg (65)96314907 *the methods described herein are intended for quantum gravitational speculation, the author disclaims any financial responsibility for fools’ attempts to apply these methods in Sentosa Casinos 1

2. Our objective is to explain the physics behind some of the results in the paper Nonholonomic Dynamics by Anthony M. Bloch, Jerrold E. Marsden and Dmitry V. Zenkov,, Notices AMS, 52(3), 2005. Introduction Recall from the vufoils titled Connections.ppt that a distribution (in the sense of Frobenius) on an m-dim connected manifold M is defined by a smoothly varying subspace c(x) of the tangent space T_x(M) at x to M at every point x in M. We note that dim(c(x)) is constant and define p = m – dim(c). Definition the Grassman manifold G(m,k) consists of all k-dim subpaces of R^m, it is the homogeneous space O(m)/(O(p)xO(m-p)) so has dimension p(m-p). Remark c can be described a section of the bundle over M whose fibers is homeomorphic to G(m,p) 2

3. c can be defined by a collection of p linearly independent . differential 1-forms . Distribution Forms If we introduce local coordinates . then there exists a p x m matrix (valued function of x) E . that has rank p and . hence we may re-label the coordinate indices so that . where B is . an invertible p x p matrix and c is defined by the forms . where . hence . 3

4. The dynamics of a system with kinetic energy T and forces F (with no constraints) is . Unconstrained Dynamics where . we have . For conservative . where we define the Lagrangian . For local coordinates . we obtain m-equations and m-variables. . 4

5. 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 (m-p) 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 (m+p) variables (x’s & lambda’s) are computed from p constraint equations and the m-equations given by . 5

6. Here m = 2, p = 1 and for suitable coordinates . Example: Particle on Inclined Plane and . where . is the (fixed) angle of the inclined plane. Therefore . and . and . and . and . 6

7. 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 (m+p) variables (x’s & lambda’s) are computed from the p constraint equations above and the m-equations 7

8. Since the mu’s and omega’s give the same connection we can obtain an equivalent system of equations with different lambda’s (Lagrange multipliers) Equivalent Form for Constraints On the following page we will show how to eliminate the Lagrange multipliers so as to reduce these equations to the form given in Equation (3) on page 326 of the Nonholomorphic Dynamics paper mentioned on page 2 of this paper. 8

9. We observe that we can express . Eliminating Lagrange Multipliers hence we solve for the Lagrange multipliers to obtain and reduced (m-p) equations . which with . the p constraint equations determine the m variables. . 9

10. We consider a more general form of the rolling coin problem described on p 62-64 of Analytical Mechanics by Louis N. Hand and Janet D. Finch. Here theta is the angle the radius R, mass m coin makes with the y-axis, phi is the rotation angle as it rolls along a surface described by the graph of the height function z(x,y). So for this problem m = 4 (variables phi, theta, x, y), p=2. Let the Coins Roll and constraints Exercise compare with Hand-Finch solution on p 64 10

11. Locally on M the 1-forms . Ehresmann Connection define the distribution . Hence they also define a fiber bundle . where . is an open subset of . and . Therefore . can be identified with a horizontal . subspace . and this describes . an Ehresmann connection . on . 11

12. Our objective is to explain the relationship of this Ehresmann connection to the differentially constrained dynamics, in particular to prove the assertion, made on the top right of page 326 in paper by Bloch, Marsden and Zenkov, that its curvature tensor vanishes if and only if the distribution c is involutive or integrable. This means that the differential constraints are equivalent to holonomic constraints. Theorem 2.4 on page 82 of Lectures on Differential Geometry by S. S. Chern, W. H. Chen and K. S. Lam, this is equivalent to the condition When are the constraints holonomic ? 12

13. Calculation where . where . if and only if if and only if 13

14. Let ‘s compute the curvature for the rolling coin system Curved Coins 14

15. We observe that each fiber of is homeomorphic to and that the 2-form on E Curvature has values in the tangent spaces to the fibers. Exercise Show that the expression above equals the curvature, as defined on page 8 of Connections.ppt, of the Ehresmann connection . Reference: Geometric Mechanics,Lagrangian Reduction and Nonholonomic Systems by H. Cendra, J. E. Marsden, and T. S. Ratiu, p. 221-273 in Mathematics Unlimited - 2001 and Beyond, Springer, 2001. 15