1. KINEMATICS of a ROLLING BALL Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg Lecture based on my student’s MSc Thesis Symmetry and Its Applications in Mechanics by Lioe Luis Tirtasanjaya (B.Sc., ITB)

2. KINEMATICS versus MECHANICS Kinematics is the geometry of motion Example: If the height h of an object is a function h = t^2 of time t, then the graph of the function h is a parabola, one of three conic sections. What are the other two ? Why are they called conics ? Example: (Kepler) The planets move around the sun on elliptical orbits with the sun at one focus, their radius vectors sweep out equal areas in equal times, and their period is proportional to the 3/2 power of the semi-major axis of their orbits. Mechanics is the physics of motion Example: (Newton) Acceleration = Force / Mass, this gives the height if an object falling near Earth’s surface as a function of t Example: (Newton) Force = G M m / R^2, this gives the trajectories of the planets

3. AFFINE SPACE Is a triple is the set of points in the space of real numbersis a vector space over the field is a translation map that satisfies where we define and we define

4. EUCLIDEAN SPACE Is an quadruplet is an affine space positive definite : bilinear : a linear function of each argument is a mapping that is symmetric : Definitionare orthogonal if Definition have distance

5. BASES Theorem 1. Ifis three dimensional, then a basis of by a column vectora representation of by a matrix and gives a representation of a linear map whose columns are the vectors where composition of mapsmatrix representation is the standard basis for The following diagram commutes, this means that [MN] = [M][N] gives

6. ROTATION LIE GROUP Theorem 2. Proof The adjoint of M, defined by is the matrix representation of a linear mapIf with respect to an orthonormal basis then satisfies, the second statement then is equivalent to M preserving orientation

7. RIGID TRANSFORMATIONS Definition Theorem 3is a rigid transformation iff Proof Exercise, note that M is determined Remark M is determined, c is arbitrary, v is determined by M and c Definition This is rotation by M about c followed by translation by u

8. RIGID TRAJECTORIES Definition Theorem 4is a rigid trajectory iff Proof Follows from Theorem 3

9. ROTATION LIE ALGEBRA Proof The first assertion follows since Theorem 5.so(3) is a Lie Algebra under the commutator product For an orthonormal basis [so(3)] are the skew-symmetric matrices and the second assertion follows since

10. VECTOR ALGEBRA Theorem 6.so(3) is isomorphic tounder the vector product Proof Definition Remark

11. ANGULAR VELOCITY Theorem 7. Ifis differentiable then Proof Definition are the and where and angular velocity in space angular velocity in the body and

12. ADJOINT REPRESENTATIONS Proof Exercise Theorem 8 The adjoint representations defined above define (1) a homomorphism of the group SO(3) into the group of linear isomorphisms of so(3), and (2) a homomorphism of the Lie algebra of linear maps of so(3) into so(3) with the commutator product. Furthermore, if M and B are differentiable functions of t

13. VELOCITY Theorem 9 The velocity of a point trajectory p(t) is given by Proof Differentiate the following formula obtained from Theorem 4

14. SURFACE MOTION OF A BALL Introduce orthonormal bases with x, y, z coordinates for V, consider a spherical ball with radius 1 moving so as to be tangent to and above a plane P that is parallel to the x-y subspace in V Theorem 10. The rigid trajectory of the center of the ball is and for every Proof The first assertion follows since the center of the ball must move parallel to P and hence to the x-y subspace of V, the second assertion follows from Theorem 4 since lies on the ball at time 0 and at time t = T it lies on the ball and P

15. NO SLIDING CONDITION Definition We say that a ball moves without sliding if the point on the ball in the plane P has zero velocity Theorem 11. A ball moves without sliding iff Proof Follows from Theorems 9 and 10 since at time t = T

16. NO TURNING CONDITION Definition A ball moves no turning if Theorem 12. A ball rolls iff Definition A ball rolls if it moves with no sliding and no turning Proof Follows from Theorem 10 and the definitions above

17. EQUATIONS OF A ROLLING BALL Theorem 13. The rotation equations of a moving ball are and these together with the condition M(0) = I determine M, and approximate trajectory in SO(3) is given by Proof The first assertion follows from the definition the second assertion follows since exp maps so(3) onto SO(3)

18. ROLLING ON A LINE Theorem 14. Rolling on a straight line with unit velocity u results in a rotation around a unit vector v, obtained by rotating u counter clockwise by 90 degrees (so v is parallel to the x-y plane), by an amount t in a counterclockwise direction Proof Consider the action of both sides of the expression above on vectors parallel and orthogonal to v

19. TURNING BY ROLLING Theorem 15 If a ball rolls along the straight line from [0 0 0] to [pi/2 0 0], then along the straight line to [pi/2 -d 0], then along the straight line to [0 d 0], the net effect is a rotation by angle d in the counterclockwise direction around v = [0 0 1]. Proof The net rotation where

20. MATERIAL COORDINATES Theorem 16 The unit vector-valued function such thatis the unit vector defined by coordinate of the point of contact of ball and the plane P Definition The material coordinate of a point q on the ball at time t Proof Ifthen so Definition Let d in V be defined so that is the trajectory of the material

21. MATERIAL TRAJECTORY Theorem 17 The material trajectorysatisfies and u determines M by Proof and is an orthornormal basis of V and the last since andhence the third equation follows since assertion follows from the definition of

22. HOLONOMY and CURVATURE Theorem 18 If material trajectorysatisfies thenwhere area and that the curvature of this connection, a 2-form with values in the Lie algebra so(2) = R, coincides with the area 2-form induced by the Riemannian metric. The detailed proof is developed in Luis’s MSc Thesis. Proof This is an extension of Theorem 15. It is based on the fact that the rolling constraints are described by a connection on the principle SO(2) fiber bundle is the directed area bounded by the curve

23. VARIATIONAL EQUATION Theorem 19 If and is a rotation trajectory and is a small trajectory variation Proof Follows from the fact that is defined by then the fact that and the definition

24. OPTIMAL TRAJECTORIES Theorem 20 The shortest trajectory with specifiedsatisfies Proof Follows from Theorem 19 using the calculus of variations, details are in the MSc Thesis of Luis c ( [0,T] ) is an arc of a circle in the plane P 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