The Concept of Holonomy and its Applications to Biomechanics, Economics and Physics Wayne Lawton...
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Transcript of The Concept of Holonomy and its Applications to Biomechanics, Economics and Physics Wayne Lawton...
The Concept of Holonomy and its Applications to
Biomechanics, Economics and Physics
Wayne Lawton
Department of Mathematics National University of Singapore
http://www.math.nus.edu.sg/~matwml
(65)963149071
Contents
Vector Fields
2
Differential Forms
Distributions and Connections
Holonomy and its Applications
Vector Fields on
is a gradient vector field iff there exists
such that
3R33: RRV
integrable iff there exists a nowhere 0 integrating factor
3
RRu 3: uV grad
RRa 3: such that Va is a gradient v. f.
Theorem 1. V is a gradient v. f. iff 0 curl V
Theorem 2. V is integrable iff 0 curlV V
and is
Vector Fields on
is orthogonal to
(
3R33: RRU
iff
4
VU Lemma 1. Vector fields can be represented by first order linear differential operators
)()(: 33 RCRCV
that are closed under the commutator product
WUUWU, W ][
33: RRV ) RRVU 3:0
)(, grad 3RCffVfV
Theorem 3. (Frobenius) V is integrable iff
V][and U, WV WV U
Vector Fields on is horizontal w.r.t. (
3R3],[: Rbag
iff
5
Vg Lemma 2. isolated 0’s, integrable
||||,0, 33 qpRqRp
V )
0Vdsdg
Proof. Let
V
be an integrating factor and
V
such that p, q can not be connected by a horizontal path.
RRu 3:RRa 3:
such that uVa gradIf
3],[: Rbag is horizontal w.r.t. then
0))(()()( dssguaubub
a dsd
Theorem 4. (Chow-Rashevskii-Sussmann, Ambrose-Singer) Converse and extension of Lem 2
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.
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
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 =
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 definition is independent 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
ij dxdxk
ij
k
ij
11
),...,( mod 11
p
m
pjijji 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).