Transcript Slide 1

The Concept of Holonomy
and its Applications to
Biomechanics, Economics
and Physics
Wayne Lawton
Department of Mathematics National
University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
(65)96314907
1
Contents
Vector Fields
Differential Forms
Distributions and Connections
Holonomy and its Applications
2
Vector Fields on
V :R R
3
u:R R
3
R
3
3 is a gradient vector field iff there exists
such that
V  grad u
and is
integrable iff there exists a nowhere 0 integrating factor
a:R  R
3
such that
aV
is a gradient v. f.
Theorem 1.
V
is a gradient v. f. iff
Theorem 2.
V
is integrable iff
curl V  0
V  curl V  0
3
Vector Fields on
R
3
V :R R
U :R R
3
( U  V ) iff U  V  0 : R  R
3
3
3 is orthogonal to
3
Lemma 1. Vector fields can be represented
by first order linear differential operators


V : C (R )  C (R )

3
V f  V  grad f , f  C ( R )
3
3
that are closed under the commutator product
[U, W ]  UW  WU
Theorem 3. (Frobenius) V is integrable iff
U  V and W  V  [U, W ]  V 4
Vector Fields on
R
3
g : [a, b]  R is horizontal w.r.t. V ( g  V )
dg
iff
Lemma 2. V isolated 0’s, integrable
V

0
ds
3
3
 p  R ,   0  q  R , || p  q || 
3
such that p, q can not be connected by a horizontal path.
a : R  R be an integrating factor and
3
u : R  R such that a V  grad u
3
If g : [a, b]  R is horizontal w.r.t. V then
b
d
u(b)  u(a)   ds u( g (s)) ds  0
Proof. Let
3
a
Theorem 4. (Chow-Rashevskii-Sussmann,
Ambrose-Singer) Converse and extension of Lem 2
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
Rigid Body Motion
is described by M : R  SO(3) and its angular velocity in
T
T
space   [   ] , in the body   [    ] ,
x
y
x
z
y
z
are defined by
 0
1

MM   z
  y

 z
0
x

 x 
0 

y
,
 0
1
  
M M
z
  y

 z
0
x

 x  .
0 

y
The velocity of a material particle whose motion p ( t )  M ( t ) p ( 0 )
is
1


p (t )  M p (0)  MM p (t )    p (t )
Furthermore, the angular velocities are related by
  M ,
M
1

7
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
h  1.70 m Sunset 11.10s    0.04625

Sphere
r tan   (r  h)  r  2rh
2
r 
Cube
d
2
2h
tan2 
h
tan

Answer: Not
unless your
brain is !!!
2
2
 5220km  6370km
 2.106km
h
h

r 
r
d
d
3
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
Elroy’s Beanie
Example described on pages 3-5 in [Marsden1990]
body 2
Shape

body 1
I
inertia
2
. T (circle)

inertia I 1
Configuration
Angular 
2
Momentum
( , ) T (torus)


Conservation of Angular Momentum 
   dt
  ( I1  I 2 ) d  I 2d
Mechanical Connection
shape trajectory  configuration
   geom  dyn
 g 
 2 I 2
I1  I 2
Wind ( )
d   t
Flat Connection  Holonomy is Only Topological
6
Rigid Body Motion
is described by M : R  SO(3) and its angular velocity in
T
T
space   [   ] , in the body   [    ] ,
x
y
x
z
y
z
are defined by
 0
1

MM   z
  y

 z
0
x

 x 
0 

y
,
 0
1
  
M M
z
  y

 z
0
x

 x  .
0 

y
The velocity of a material particle whose motion p ( t )  M ( t ) p ( 0 )
is
1


p (t )  M p (0)  MM p (t )    p (t )
Furthermore, the angular velocities are related by
  M ,
M
1

7
Rolling Without Turning
on the plane z = -1 is described the by
if a ball rolls along the curve
x  0
y   1
 0  0
z  0 therefore
t  ( x(t ), y(t ),1)
 x   y 
0   y    x 
0
  0   0 
 1 0
0
0
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.
then
1
2
3
8
Material Trajectory and Holonomy
-1
2
The material trajectory u  M p : R  S ,
p  [0 0 - 1]
1
1
1

satisfies u   M MM p     u  M (  p ) hence
  p  0  || u ||  ||  || ||  ||,   u  u,   Mu  p.
Theorem [Lioe2004] If
u(0)  u(T )  p
then
M(T)  Rotz (A) where A = area bounded by u([0,T]).
Proof The no turning constraints give a connection
on the principle SO(2) fiber bundle
M u M 1 p
SO(3) 
 S  SO(3) / SO(2)
2
and the curvature of this connection, a 2-form on S 2
with values in the Lie algebra so(2) = R, coincides
with the area 2-form induced by the Riemannian metric.
9
Optimal Trajectory Control
Theorem [Lioe2004] If
M : R  SO(3) is a rotation trajectory
 M : R  Tangent(SO(3)) is a small trajectory variation
1
ˆ
and   : R  V is defined by         M M
then δ  δ    
d
d
-1
-1
-1

Proof Since

δM  δM and
M  M M M .
dt
dt
Theorem [Lioe2004] If M : [0, T]  SO(3) is the shortest
trajectory with specified M(0)  I, M(T)  SO(3)
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
Unconstrained Dynamics
The dynamics of a system with kinetic energy
T and forces F (with no constraints) is
T
F
x
For conservative
.

d  
where


 x dt x x
V we have  L
F 
0
x
x
where we define the Lagrangian L  T  V .
.
T
For local coordinates x  [ x ,..., x ]
1
m
.
we obtain m-equations and m-variables.
11
Holonomic Constraints
One method to develop the dynamics of a system with
Lagrangian L that is subject to holonomic constraints
C1  constant,...,Cmk  constant
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
.

C
mk
i
such that F 
,...,

1
mk

i 1
The 2m-k variables (x’s & lambda’s)
are computed from m-k constraint
equations and the m equations given by
i
x
L
 F.
x
12
Nonholonomic Constraints
For nonholonomic constraints D’Alemberts principle
can also be applied to obtain the existence of
1 ,...,mk
such that
F  i 1 i i
mk
where the mu-forms describe
the velocity constraints
.
i ( x)  0, i  1,...,m  k.
The 2m-k variables (x’s & lambda’s)
L
 F.
are computed from the m-k constraint
equations above and the m equations  x
On vufoils 20 and 21 we will show how to eliminate
(ie solve for) the m-k Lagrange multipliers !
13
Level Sets and Foliations
Analytic Geometry: relations & functions
synthetic geometry  algebra
Calculus: fundamental theorems local  global
Implicit Function Theorem for a smooth function F
R O
 R
n
F
m
rank( F )( p  O)  m 
Local (near p) foliation (partition into submanifolds)
consisting of level sets of F (each with dim = n-m)
F  x2  y 2  z 2
3
Example R \ {0)  O 
 R
(global) foliation of O into 2-dim spheres
14
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 < d) if v( p)  d ( p), p  E
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 Characterization
A dim k distribution d on an m-dim manifold arises as
d ( p)  v Tp (M ) : i. (v)  0, i  1,...,m  k 


where
1 ,..., m k are differential 1-forms.
.
Cartan’s Theorem d is integrable iff
di  0 mod(1 ,...,mk ), i  1,...,m  k
Proof See [Chern1990] – crucial link is Cartan’s formula
di (u, v)  u( i , v )  v( i , u )  i ([u, v])
Remark Another Cartan gem is:
di  0 mod(1 ,...,mk ) 
di  1   mk  0
16
Ehresmann Connections
Definition [Ehresmann1950] A fiber bundle is a map

E
 B between manifolds with rank = dim B,
the vertical distribution d on E is defined by
d ( p)  { v Tp (M ) |  * (v)  0 }
and a connection is a complementary distribution c
Tp ( E)  c( p)  d ( p), p  E
This defines T(E) into the bundle sum T ( E)  H ( E)  V ( E)
Theorem c is the kernel of a V(E)-valued connection

1-form T ( E) 
V ( E) and image of a horizontal lift
with v   *1 ( * (v))   (v), v Tp (E)
h

H ( E) denote the horizontal projection.
We let T ( E) 
 *1
T ( p ) ( B)  c( p ) T p( E )
17
Holonomy of a Connection

Theorem A connection on a bundle E 
 B
and points p, q in B then every path f from p to q in B
defines a diffeomorphism (holonomy) between fibers
 ( p) 
 (q)
1
h( f )
1
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
Curvature, Integrability, and Holonomy
Definition The curvature of a connection is the 2-form
~
~
(u, v)   ([h(u ), h(v )]) where
u, v T ( E) and u~, v~ are vector field extensions.
p
Theorem This definition is independent 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
Implicit Distribution Theorem
Given a dim = k distribution on a dim = m manifold M
c( p)  v Tp (M ) : i (v)  0, i  1,...,m  k 
T
we introduce local coordinates x  [ x1 ,..., xm ]
.
there exists a (m-k)
x m matrix (valued function of p) E
with rank m-k and 
 E dx hence we may re-label
the coordinate indices so that E [ B C ] where B is
.
an invertible (m-k)
x (m-k) matrix and c is defined by
  [1 ,...,mk ]
T
1
T
1
so   [1 ,...,mk ]  B   A dx
where A  [ I B C ]
i  dxi   j mk 1 Aij dx j , i  1,...,m  k
m
20
Distributions Connections
Locally on M the 1-forms
m
.
i  dxi   j  p 1 Aij dx j , i  1,..., m  k
define the distribution
.
c( x)  v Tx (M ) : i (v)  0, i  1,..., p 

Hence they also define a fiber bundle E 

B
m
k
.
where E, B is an open subset of R , R and
 ( x1,...,xmk , xmk 1,...,xm )  ( xmk 1,...,xm )
Therefore c(x) can be identified with a horizontal
~
subspace c ( x)  T ( E ) and this describes

~
an Ehresmann connection c on E 
 B
21
Curvature Computation
m
di    dx j  ij
j  p 1
p
where
.

 ij  
R
k 1
.
m
dxk  k   
Aij
xk
k  p 1
dxk
dx  dxk mod(1 ,..., p )
i
jk
j
p 1 j k m
Aij
i
where
jk
xk
R 
Aij
xk
mod (1 ,..., p )

Aik
x j

Aij
 1 x
p
Ak 
Aik
x
Aj
 0 mod(1,..., p ), i  1,..., p if and only if
i
di  1   1  0 if and only if R jk  0
22
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)
L
 F  11    mk mk
x
i ( x)  0, i  1,...,m  k.
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
Eliminating Lagrange Multipliers
We observe that we can express
m


L
L

dxk   i  dxi   Aij dx j 
x k 1  xk
i 1
j  m  k 1


m
mk
hence we solve for the Lagrange multipliers to obtain
L
i 
, i  1,..., m  k
xi
and reduced k equations
L
L
mk
 i 1 Aij
, j  m  k  1,..., m These and the
x j
xi
m-k constraint equations determine the m variables.
24
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).

d  
2 2
2 2
3
1
L  4 mR   8 mR   mgz( x, y)


 x dt x x
Constraints
x  R sin  , y  R cos  
1  dx  R sin  d  0 2  dy  R cos d  0
2
L
L


 mR   R sin  x  R cos y 
2 2
L
1
z
z
 mgR(sin  cos ),  mR   0
L

3
2
2
x
y

4
Exercise compare with Hand-Finch solution on p 64
25
How Curved Are Your Coins ?
Let ‘s compute the curvature for the rolling coin system
x1  x, x2  y, x3   , x4  
1  dx  R sin  d  dx1  R sin x3dx4
2  dy  R cos d  dx2  R cos x3dx4
A13  A23  0, A14  R sin x3 , A24  R cos x3
R  R cosx3
1
34
R  R sin x3
2
34
26
Symmetry and Momentum Maps
J
*
Definition [Marsden1990,1994] P 
 
is a momentum map if P is a Poisson manifold
with a left Hamiltonian action by a Lie group G with Lie
algebra

with linear dual

*
that satisfies
{ J( p),  , F}  v ( F ), p  P,   , F : P  R
where v  is the left-invariant vector field on P generated
by the flow ( p, t )  p exp(t), ( p, t )  P  R
The reduced space
P  J 1 () / G ,   *
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
Rigid Body Dynamics
*
*
and
P  T (SO(3))
J ( g ,  )  g
is the pullback under right translation. The Hamiltonian
 so(3)
H ( )   , I   where so(3) 
1
I
*
is a positive definite self-adjoint inertial operator, and

J ()  SO(3) 
 P  S 2
1
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
     2 H ( )T /  ,   area / 
2
28
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, …
29
References
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Applications in Mechanics, Master of Science Thesis,
National University of Singapore.
[Hand1998] L. Hand and J. Finch, Analytical
Mechanics, Cambridge University Press.
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Nonholonomic Systems, 221-273 in Mathematics
Unlimited - 2001 and Beyond, Springer, 2001.
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References
[Marsden2001] H. Cendra, J. Marsden, and T. Ratiu,
Geometric Mechanics,Lagrangian Reduction and
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Unlimited - 2001 and Beyond, Springer.
[Marsden2004] Nonholonomic Dynamics, AMS Notices
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32
References
[Berry1988] M. Berry, The geometric phase, Scientific
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Melli-Huber, Locomotion of articulated bodies in a
perfect fluid (preprint from web).
33