Mechanical Connections

Wayne Lawton

Department of Mathematics National University of Singapore

matwml@nus.edu.sg

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

(65)96314907

1

Contents3-6 Earth, Tangents, Tubes, Beanies

2

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

29 Boundless Applications

27 Symmetry and Momentum Maps

28 Rigid Body Dynamics

30-33 References

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

h

rr

d

rhrhrr 2)(tan 2222 Sphere

km6370km52202tan2

hr

Stand mh 70.1 Sunset04625.010.11 s

Cube

h

d

km106.2tan hd

Answer: Not unless your brain is !!!

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 =

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 sphereNo twist tangent vector parallel translated (angle with geodesic does not change) holonomy = area enclosed by closed curve.

Elroy’s BeanieExample described on pages 3-5 in [Marsden1990]

6

body 1inertia

1I

Shape.

shape trajectory configuration

Conservation of Angular Momentum

body 2inertia

Configuration

)(circleT

)(),( 2 torusT

Mechanical Connection

dt dIdII 221 )(

2I

dyngeom

AngularMomentum

)(21

22 WindIII

g td

Flat Connection Holonomy is Only Topological

Rigid Body Motion

is described by

,

0

0

0

MM1

xy

xz

yz

are defined by

)3(: SORM

The velocity of a material particle whose motion

space ,][T

zyx and its angular velocity in

)()(MM)0()(1

tptppMtp

,][T

zyx in the body

.

0

0

0

MM1

xy

xz

yz

is

)0()()( ptMtp

Furthermore, the angular velocities are related by

1

,

MM

7

Rolling Without Turningon the plane z = -1 is described the by

if a ball rolls along the curve

0z)1),(),(( tytxt

00000

001

010

0

y

x

x

y

y

x

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 androtation by d about the z-axis. 8

1

2

3

Material Trajectory and HolonomyThe material trajectory 1]- 0 [0p, SR:pMu

2-1

satisfies hence

,uu

)(111

pMupMMMu

||,|||||||||| u

pTuu )()0(Theorem [Lioe2004] If then

(A)RotM(T)z

where A = area bounded by u([0,T]).

Proof The no turning constraints give a connection on the principle SO(2) fiber bundle

)2(/)3()3( 21

SOSOSSO pMuM

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

0p .puM

2S

Optimal Trajectory Control

and

is a rotation trajectory

is a small trajectory variation

Proof Since

SO(3)R:M SO(3))(TangentR:M

VR: is defined by

δδthen

1MMˆ

MδδMdt

d .111

MMMMdt

d --- and

10

is the shortest

SO(3)M(T)I,M(0) 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.

SO(3)T][0,:M Theorem [Lioe2004] If

Theorem [Lioe2004] If

trajectory with specified

Potential Application: Rotate (real or virtual) rigid body by moving a computer mouse.

Unconstrained Dynamics

11

The dynamics of a system with kinetic energy T and forces F (with no constraints) is

Fx

T

wherexxdt

d

x

For conservative. x

VF

0x

L

we have

where we define the Lagrangian .

.VTL For local coordinates.

Tmxxx ],...,[ 1

we obtain m-equations and m-variables..

Holonomic Constraints

12

such that

.Fx

L

constant,...,constant1 kmCCis 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 .

km ,...,1

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

x

CF ikm

i i

1

The 2m-k variables (x’s & lambda’s) are computed from m-k constraint equations and the m equations given by

.

Nonholonomic Constraints

13

such that

.

.Fx

L

km ,...,1

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

i

km

i iF

1

where the mu-forms describe the velocity constraints

.,...,1,0)( kmixi 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 !

Level Sets and FoliationsAnalytic Geometry: relations & functionssynthetic geometry algebra

Implicit Function Theorem for a smooth function F

mFn ROR

Local (near p) foliation (partition into submanifolds) consisting of level sets of F (each with dim = n-m)

Calculus: fundamental theorems local global

mOpFrank ))((

Example ROR zyxF 222

)0{\3

(global) foliation of O into 2-dim spheres 14

Frobenius DistributionsDefinition 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 subordinateto 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.

Eppdpv ),()(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

Cartan’s CharacterizationA dim k distribution d on an m-dim manifold arises as

.

16

kmivMTvpd ip ,...,1,0)(:)()(

d is integrable iffCartan’s Theorem

where .

km ,...,1 are differential 1-forms.

Proof See [Chern1990] – crucial link is Cartan’s formula

kmid kmi ,...,1),,...,( mod 0 1

]),([),(),(),( vuuvvuvud iiii Remark Another Cartan gem is:

),...,( mod 0 1 kmid 01 kmid

Ehresmann Connections

BE

the vertical distribution d on E is defined by

Definition [Ehresmann1950] A fiber bundle is a map

}0)(|)({)( * vMTvpd p

between manifolds with rank = dim B,

EppdpcETp ),()()(

Theorem c is the kernel of a V(E)-valued connection

and a connection is a complementary distribution c

)()( EVET

This defines T(E) into the bundle sum )()()( EVEHET

and image of a horizontal lift

1-form )()()(

1*

)( ETpcBT pp

with )(),())(( *1

* ETvvvv p

17We let )()( EHET h denote the horizontal projection.

Holonomy of a ConnectionTheorem A connection on a bundle BE

Proof Step 1. Show that a connection allows vectors in T(B) be lifted to tangent vectors inT(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

)()( 1)(1 qp fh

Curvature, Integrability, and HolonomyDefinition The curvature of a connection is the 2-form

)])~(),~(([),( vhuhvu

19

where

)(, ETvu p and

vu ~,~ are vector field extensions.

Theorem A connection is integrable (as a distribution) iff its curvature = 0.

Theorem This defintion is independed of extensions.

Theorem A connection has holonomy = 0 iff its curvature = 0.

Implicit Distribution Theorem

20

kmivMTvpc ip ,...,1,0)(:)()(

there exists a (m-k) x m matrix (valued function of p) E

we introduce local coordinates.

with rank m-k and

Tmxxx ],...,[ 1

dxE

an invertible (m-k) x (m-k) matrix and c is defined by

the coordinate indices so that.

][ CBEhence we may re-label

where B is

dxABT

km

1

],...,1[

where ][1CBIA

so

m

kmj jijii kmidxAdx1

,...,1,

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

Tkm ],...,[ 1

Distributions ConnectionsLocally on M the 1-forms.

21

pivMTvxc ix ,...,1,0)(:)()( define the distribution.

Hence they also define a fiber bundle .where

m

pj jijii kmidxAdx1

,...,1,

BE km RR ,is an open subset of

),...,(),...,,,...,( 111 mkmmkmkm xxxxxx BE, and

Therefore

)()(~ ETxc )(xc can be identified with a horizontal

subspace and this describes

an Ehresmann connection.

BE c~ on.

Curvature Computation

22

where.

mkjp

pkjijk dxdxR

11 ),...,( mod

jxAp

kx

A

xA

x

Aijk AAR ikij

j

ik

k

ij

1

m

pkkx

Ap

kkkx

A

j dxdxk

ij

k

ij

11

),...,( mod 11

p

m

pjjji dxd

where.

pip ,...,1),,...,( mod 0 1 if and only if

0ijkR011 id if and only if

Equivalent Form for Constraints

23

kmkmFx

L

11

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)

.,...,1,0)( kmixi 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].

Eliminating Lagrange Multipliers

24

km

i

m

kmjjijii

m

kk

k

dxAdxdxx

L

x

L

1 11

We observe that we can express

kmii

i x

L ,...,1,

and reduced k equations

hence we solve for the Lagrange multipliers to obtain

mkmjx

LA

x

L km

ii

ijj

,...,1,1

These and the

m-k constraint equations determine the m variables.

Rolling CoinGeneral rolling coin problem p 62-64 [Hand1998]. Theta = angle of radius R, mass m coin with y-axisphi = rotation angle rolling on surface of height z(x,y). ),(22

8122

43 yxmgzmRmRL

0sin1 dRdx

Constraints

25

0cos2 dRdy

02241

mRL

yL

xLL RRmR

cossin22

23

),cos(sin yz

xzmgR

cos,sin RyRx

Exercise compare with Hand-Finch solution on p 64

xxdt

d

x

How Curved Are Your Coins ?Let ‘s compute the curvature for the rolling coin system 4321 ,,, xxyxxx

4311 sinsin dxxRdxdRdx

26

3243142313 cos,sin,0 xRAxRAAA

3134 cos xRR

4322 coscos dxxRdxdRdy

3234 sin xRR

Symmetry and Momentum MapsDefinition [Marsden1990,1994]

*J P

27

is a momentum map if

P is a Poisson manifold with a left Hamiltonian action by a Lie group G with Lie

algebra

that satisfies RPFPpFvFp :,,),(},),(J{

where

*

v is the left-invariant vector field on

with linear dual

Pby the flow

RPtptptp ),(),exp(),( generated

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

The reduced space

*1 ,/)(J GP is a PM.

Rigid Body DynamicsHere

))3((* SOTP

28

and *),( ggJ is the pullback under right translation. The Hamiltonian

is a positive definite self-adjoint inertial operator, and

where

*)3()3( soso I

is a fiber bundle whose connection (canonical 1-form on the symplectic manifold P) gives dynamic reconstruction from reduced dynamics.

1,)( IH

21 )3()(J SPSO

2/,/)(2 areaTH

Theorem [Ishlinskii1952] (discovered 1942) The holonomy of a period T reduced orbit that enclosed a spherical area A is

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, …

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.

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.

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.

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).