Mechanical Connections

1
Mechanical Connections

• Wayne Lawton
• Department of Mathematics National University of
  Singapore
• matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml
• (65)96314907

49th Annual Meeting of the Australian
Mathematical Society University of Western
Australia Sept 27-30, 2005
Topology and Geometry Seminar National University
of Singapore Oct 5, 2005
1
2
Contents
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
3
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?
Stand
Sunset
Sphere
Cube
Answer Not unless your brain is !!!
3
4
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.
Area
Radius 1
Holonomy rotation of tangent vectors parallel
translated around meridian area of spherical
cap.
4
5
How do Tubes Turn ?
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
6
Elroys Beanie
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
7
Rigid Body Motion
is described by
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
8
Rolling Without Turning
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
9
Material Trajectory and Holonomy
The material trajectory
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
10
Optimal Trajectory Control
Theorem Lioe2004 If
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. See
Sharpe1997.
10
11
Unconstrained Dynamics
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
12
Holonomic Constraints
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
DAlemberts 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 (xs lambdas) are computed
from m-k constraint equations and the m
equations given by .
12
13
Nonholonomic Constraints
For nonholonomic constraints DAlemberts
principle can also be applied to obtain the
existence of
such that .
where the mu-forms describe the velocity
constraints
The 2m-k variables (xs lambdas) 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
14
Level Sets and Foliations
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
15
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.
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 lt 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 lt d ? u,v lt d.
Remark. The fundamental theorem of ordinary
diff. eqn. ? evey 1 dim distribution is
integrable.
15
16
Cartans Characterization
A dim k distribution d on an m-dim manifold
arises as .
where .
are differential 1-forms.
d is integrable iff
Cartans Theorem
Proof See Chern1990 crucial link is Cartans
formula
Remark Another Cartan gem is
16
17
Ehresmann Connections
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
and image of a horizontal lift
1-form
with
We let
denote the horizontal projection.
17
18
Holonomy of a Connection
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
19
Curvature, Integrability, and Holonomy
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
20
Implicit Distribution Theorem
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
21
Distributions? Connections
Locally on M the 1-forms .
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
22
Curvature Computation
where .
where .
if and only if
if and only if
22
23
Equivalent Form for Constraints
Since the mus and omegas define the same
distribution we can obtain an equivalent system
of equations with different lambdas (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
24
Eliminating Lagrange Multipliers
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
25
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).
Constraints
25
Exercise compare with Hand-Finch solution on p 64
26
How Curved Are Your Coins ?
Let s compute the curvature for the rolling coin
system
26
27
Poisson Manifolds
Manifold
Lie algebra structure , on
such
is a derivation.
that
?
are Lie algebras and
anti homomorphism.
Symplectic Manifold
Example 1
In particular if
Example 2
Lie-Poisson bracket
Reduction Theorem
27
28
Momentum Maps
Consider a left-action of
on a Poisson manifold
by canonical maps, hence an anti homomorphism
such that commutes
and a map
is a momentum map if commutes, or
28
29
Momentum Function
is an anti homomorphism
has flow
If
has flow
then
Momentum Map for Lifted Left Action on a Manifold
Equivariance
Momentum Map for Lifted Left Action on a Lie Group
29
30
Symmetry
Noethers Theorem If
acts canonically on a Poisson
and admits a momentum map
and
is a constant
is
invariant, then
of motion for the Hamiltonian flow induced by
Proof
Corollary The Hamiltonian flow above induces a
Hamiltonian flow on each reduced space
See Marsden1990,1994.
30
31
Rigid Body Dynamics
Here
is a positive definite self-adjoint inertial
operator, and the
Hamiltonian
is LInv
The reduced dynamics on the base space of the FB
yields dynamic reconstruction using the canonical
1-form connection Marsden1990 who remarks in
Marsden1994 that reconstruction was done in 1942
Theorem Ishlinskii1952,1976 The holonomy of a
period T reduced orbit that enclosed a spherical
area A is
31
32
Further 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
PDEs KDV, Incompressible and Compressible
Fluids, Magnetohydrodynamics, Plasmas-Maxwell-Vlas
ov, Maxwell, Loop Quantum Gravity,
Representation Theory, Algebraic Geometry,...
32
33
References
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.
Sharpe1997 R. W. Sharpe, Differential Geometry-
Cartans Generalization of Kleins Erlangen
Program, Springer, New York.
Hand1998 L. Hand and J. Finch, Analytical
Mechanics, Cambridge University Press.
33
34
References
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.
34
35
References
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.
Ishlinskii1976 A. Ishlinskii, Orientation,
Gyroscopes and Inertial Navigation (in Russian).
Nauka, Moscow.
Kane1969 T. Kane and M. Scher, A dynamical
explanation of the falling phenomena, J. Solids
Structures, 5,663-670.
35
36
References
Smale1970 S. Smale, Topology and Mechanics,
Inv. Math., 10, 305-331, 11, 45-64.
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).
36