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### Mechanical Connections

### Contents

### Is the Earth Flat ?

### How Are Tangent Vectors Connected ?

### How do Tubes Turn ?

### Elroy’s Beanie

### Rigid Body Motion

### Rolling Without Turning

### Material Trajectory and Holonomy

### Optimal Trajectory Control

### Unconstrained Dynamics

### Holonomic Constraints

### Nonholonomic Constraints

### Level Sets and Foliations

### Frobenius Distributions

### Cartan’s Characterization

### Ehresmann Connections

### Holonomy of a Connection

### Curvature, Integrability, and Holonomy

### Implicit Distribution Theorem

### Distributions Connections

### Equivalent Form for Constraints

### Eliminating Lagrange Multipliers

### Rolling Coin

### Symmetry and Momentum Maps

### Rigid Body Dynamics

### Boundless Applications

### References

### References

### References

### References

Wayne Lawton

Department of Mathematics National University of Singapore

matwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

(65)96314907

1

3-6 Earth, Tangents, Tubes, Beanies

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

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?”

Stand

Sunset

Sphere

Cube

Answer: Not

unless your

brain is !!!

3

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.

Area =

Radius = 1

Holonomy: rotation of tangent vectors parallel translated around meridian = area of spherical cap.

4

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.

5

Example described on pages 3-5 in [Marsden1990]

body 2

inertia

Shape

.

body 1

inertia

Configuration

Angular

Momentum

Conservation of Angular Momentum

Mechanical Connection

shape trajectory configuration

Flat Connection Holonomy is Only Topological

6

and its angular velocity in

space

in the body

are defined by

The velocity of a material particle whose motion

is

Furthermore, the angular velocities are related by

7

on the plane z = -1 is described the by

therefore

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

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

is a rotation trajectory

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.

10

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.

.

11

One method to develop the dynamics of a system with Lagrangian L that is subject to 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

For nonholonomic constraints D’Alemberts principle can also be applied to obtain the existence of

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

Analytic Geometry: relations & functions

synthetic geometry algebra

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

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.

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

A dim k distribution d on an m-dim manifold arises as

.

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

Definition [Ehresmann1950] A fiber bundle is a map

between manifolds with rank = dim B,

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

Theorem A connection on a bundle

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

Definition The curvature of a connection is the 2-form

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

Given a dim = k distribution on a dim = m manifold M

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

.

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

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].

23

We observe that we can express

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

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

Exercise compare with Hand-Finch solution on p 64

25

is a momentum map if

is a Poisson manifold

with a left Hamiltonian action by a Lie group G with Lie

algebra

with linear dual

that satisfies

where

is the left-invariant vector field on

generated

by the flow

The reduced space

is a PM.

Theorem [Marsden1990,1994] If H : P R is G-inv. then it induces a Hamiltonian flow on the red. space.

27

Here

is the pullback under right translation. The Hamiltonian

where

is a positive definite self-adjoint inertial operator, and

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

28

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, …

29

[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.

30

[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.

[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.

31

[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

[Ishlinskii1952] A. Ishlinskii, Mechanics of special gyroscopic systems (in Russian). National Academy Ukrainian SSR, Kiev.

[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.

32

[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).

33

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