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Mechanical Connections. Wayne Lawton Department of Mathematics National University of Singapore (65)96314907. 1. 3-6 Earth, Tangents, Tubes, Beanies. Contents. 7-10 Rolling Ball Kinematics. 11-13 Nonholonomic Dynamics – Formulation.

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mechanical connections

Mechanical Connections

Wayne Lawton

Department of Mathematics National University of Singapore




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


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?”

Is the Earth Flat ?





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.

How Are Tangent Vectors Connected ?

Area =

Radius = 1

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


how do tubes turn

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.


elroy s beanie

Example described on pages 3-5 in [Marsden1990]

body 2


Elroy’s Beanie



body 1





Conservation of Angular Momentum 

Mechanical Connection

shape trajectory  configuration

Flat Connection  Holonomy is Only Topological


rigid body motion

is described by

and its angular velocity in

Rigid Body Motion


in the body

are defined by

The velocity of a material particle whose motion


Furthermore, the angular velocities are related by


rolling without turning

on the plane z = -1 is described the by


Rolling Without Turning

if a ball rolls along the curve


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.


material trajectory and holonomy

The material trajectory

Material Trajectory and Holonomy



Theorem [Lioe2004] If


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.


optimal trajectory control

Theorem [Lioe2004] If

is a rotation trajectory

Optimal Trajectory Control

is a small trajectory variation


is defined by


Proof Since


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.


unconstrained dynamics

The dynamics of a system with kinetic energy

T and forces F (with no constraints) is

Unconstrained Dynamics


For conservative


we have

where we define the Lagrangian


For local coordinates


we obtain m-equations and m-variables.



holonomic constraints

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



nonholonomic constraints

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 !


level sets and foliations

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)


(global) foliation of O into 2-dim spheres


frobenius distributions

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.


cartan s characterization

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


Cartan’s Characterization



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:


ehresmann connections

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


and image of a horizontal lift


We let

denote the horizontal projection.


holonomy of a connection

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.


curvature integrability and holonomy

Definition The curvature of a connection is the 2-form

Curvature, Integrability, and Holonomy



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.


implicit distribution theorem

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




distributions connections

Locally on M the 1-forms


Distributions Connections

define the distribution


Hence they also define a fiber bundle



is an open subset of



can be identified with a horizontal

and this describes


an Ehresmann connection





curvature computation

Curvature Computation





if and only if

if and only if


equivalent form for constraints

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].


eliminating lagrange multipliers

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.


rolling coin

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


Exercise compare with Hand-Finch solution on p 64


symmetry and momentum maps

Definition [Marsden1990,1994]

Symmetry and Momentum Maps

is a momentum map if

is a Poisson manifold

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


with linear dual

that satisfies


is the left-invariant vector field on


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.


rigid body dynamics



Rigid Body Dynamics

is the pullback under right translation. The Hamiltonian


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


boundless applications

Boundless Applications

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, …



[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.



[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.



[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.



[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).