Monday, January 23, 2006

Ouch


Check this out!

This is a tree we saw while we were walking aimlessly around down town Pasadina. It's a full size tree, about 16 inches in diameter with those 3/4" spikes all over it, all the way out to the smallest of branches. Amazing, not a tree you would want to tangle with!

I've dubed it: evolutions answer to tree huggers.

19 Comments:

At 1/23/2006 5:30 PM, Blogger Julianne said...

Thank goodness we don't have those trees at mt.ashland! That would really hurt! :)

 
At 1/23/2006 8:20 PM, Blogger Carson Allen said...

Like the tree the Hawks O line will be tangeling with Next Sunday. His name is "The Bus"

 
At 1/23/2006 8:54 PM, Blogger Reformed Centurion said...

We should have made you climb it Jason.

 
At 1/23/2006 9:00 PM, Blogger Jason said...

I can't imagine the pain of falling into one of these. I don't know if you could pull yourself back off.

 
At 1/24/2006 11:04 AM, Blogger Matador190 said...

Yeah well evolution can't stop me 'cuz I climbed one the other day AND hugged it. Eat it, motha nature.

 
At 1/24/2006 12:38 PM, Blogger stephen said...

hey jason remember the time in Brazil when Josh fell on that spiky branch and we had to carry him back to the boat? yep, good times.

 
At 1/24/2006 4:54 PM, Blogger Jason said...

Oh yea I remeber that he had all kinds of thorns sticking out of him. That was a long walk. What fun.

 
At 1/27/2006 10:52 AM, Blogger Gordon Cloud said...

I have met some people that remind me of this tree. You don't want to hug them either.

 
At 1/28/2006 9:13 PM, Blogger stephen said...

Hey Jason I heard about your little incident. Sounds pretty killer. Maybe it'll give you more incentive to post since you can't speak very well. Just let your fingers do the work, know what I mean? I'll be praying for you, brotha,

 
At 1/29/2006 1:22 AM, Blogger Jason said...

Thanks brother, I'll be posting something, I just need to figure out what I want to say.

 
At 1/29/2006 8:35 AM, Blogger Carson Allen said...

Jason

Our prayers are with you. I never have a hard time figuring out what to say, I just have a hard time figuring out how to spell it


Hey I will be finishing " Christ our Mediator" by C.J. today, you can borrow it if you want. It's only like 75 pages

 
At 1/29/2006 12:14 PM, Blogger stephen said...

Hey Jason, I been talking with my cousin nate who lives in santa barbara. he said we're welcome to come visit and stay at his house. We could do some surfing and diving. it would be awesome. i'll let you know if I hear more.

 
At 2/01/2006 2:13 PM, Blogger Matador190 said...

G
R
Double E
N
LEAVES!
G
R
Double E
N
LEAVES!

It's So Easy!
Happy Go Lucky...
We Are The World,
We Did It!
You! You! You! You!
Us! Us! Us! Us!

Yatta! Yatta! Yatta! Yatta!
A student just graduating!
A leader of a great business!
Do we ever get a lucky feeling
A-with our simple leafy nakedness!

Yatta! Yatta! Yatta! Yatta!
Oh-we will be winning the prize!
We will make our nation proud!
We will become a-quite-a healthy guys!
Everybody say: Yatta!

Japan, On Two!
Demo!
Tomorrow Will Be Wonderful...
Who cares if everybody is unkind? (?)
Why bother sleepin' in? (?)
Queue! Queue! Queue! Queue!
Bus! Bus! Bus! Bus!
Ohio! (or Good Morning?)

Yatta! Yatta! Yatta! Yatta!
Nine hours a night we're sleeping
An' we jump right out of bed!
Do we ever get a lucky feeling
A-when we go to work fresh and fed!

Yatta! Yatta! Yatta! Yatta!
you change-odd-weird me
world change-revise


I FEEL GREAT!!

 
At 2/01/2006 9:57 PM, Blogger Jason said...

AHHH! now he's on my blog!

Would you care to explane your self?

 
At 2/07/2006 10:52 AM, Blogger Matador190 said...

My blog has some pretty neat pictures and discussions too. Feel free to check it out at:

http://pygmychimps.blogspot.com/

 
At 2/09/2006 12:29 PM, Blogger Ana Banana said...

I don't want to kick a guy while he is down, but don't you have a little bit more time on your hands! And yet nothing new has been posted!
Just kidding, just would like to hear more from you!

 
At 2/09/2006 4:22 PM, Blogger stephen said...

Ouch! The Prodigal Son has another quote from Steve Saint, who is Resovlved: Rick Holland says Do Not Be Anxious for some Stupid Videos but look at More from the End of the Spear. MERRY CHRISTMAS! I have some Thoughts for Discussion for you!


YES! I strung all of your post titles together! I am so awesome!

 
At 2/14/2006 11:38 AM, Blogger Matador190 said...

Abstract. The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle . The structure group G = SO(3) is the vorticity group, and the bundle is the connected component of the general linear group. The base manifold is the space of positive-definite real 3 × 3 symmetric matrices, identified geometrically with the space of inertia ellipsoids. The angular momentum is not the only conserved quantity. The Kelvin circulation is also conserved as a consequence of gauge invariance. The bundle is a Riemannian manifold whose metric is determined by the kinetic energy. Nonholonomic constraints determine connexions on the bundle. In particular, the trivial connexion corresponds to rigid body motion, the natural Riemannian connexion to irrotational flow, and the invariant connexion to the falling cat.
PACS Numbers: 4550P, 9510C, 2160F, 0240



1. Introduction

This Letter formulates the classical theory of Riemann ellipsoids as a gauge theory. In the usual gauge method for isolated mechanical systems, the structure group is the rotation group, and the gauge conserved quantity is the angular momentum [1,2,3,4,5,6]. The gauge theory of Riemann ellipsoids is a fundamentally different model with the vorticity group as the structure group and the Kelvin circulation as a gauge-conserved quantity in addition to the angular momentum.

As defined by Dirichlet [7] and Riemann [8] in 1860, a Riemann ellipsoid is a self-gravitating, constant mass-density fluid with an ellipsoidal boundary and with a velocity field that is a linear function of the Cartesian position coordinates in an inertial centre-of-mass frame. The classical theory of the equilibrium and stability of rotating Riemann ellipsoids was clarified by Lebovitz [9] and Chandrasekhar [10] a century later and applied to the description of astrophysical systems [11,12,13,14,15] and gaseous plasmas [16].

With minor modifications, Riemann ellipsoid theory may be applied to fluids whose density is not uniform and to discrete systems of particles. Rapidly rotating atomic nuclei may be modelled as Riemann ellipsoids when the gravitational self-energy is replaced by the sum of the repulsive Coulomb self-energy among the protons and an attractive surface energy that approximates the strong interactions among the nucleons [17,18]. There is a quantum mechanical mean field theory of Riemann ellipsoids [19,20,21]. The equations of motion form a Hamiltonian dynamical system [22] and a finite-dimensional Lax system [23].

Because the velocity field is assumed linear, the Riemann ellipsoid configuration space is the connected component of the general linear group
(1)


The space of all inertia ellipsoids is identified with the manifold Q of all positive-definite real symmetric matrices. The relationship between and Q is given by the surjective mapping :
(2)


The group acts on itself by multiplication on the left or on the right. These are two different geometrical transformations with distinct physical interpretations. For example, left multiplication of elements by elements r of the subgroup SO(3), corresponds to physical rotations. In contrast, right multiplication by corresponds to vortex motion. The difference in interpretation originates in the distinct induced group actions on the ellipsoidal space. With respect to left multiplication by the rotation group element , an ellipsoid with inertia tensor is transformed into a rotated ellipsoid with inertia tensor . But right multiplication by the vorticity group element leaves the inertia ellipsoid invariant, , since g-1 = gt, or
(3)


The projection is right invariant with respect to the group G. Hence, the configuration space is a principal fibre bundle over the base manifold Q with structure group G [24,25].

This letter shows that the classical theory of Riemann ellipsoids is expressed naturally in terms of the differential geometry of the bundle . A connexion, or differential geometric structure, on the bundle is physically equivalent to a nonholonomic constraint on the vortex velocity field. The nonholonomic constraints to irrotational flow and the `falling cat' problem correspond to the Riemannian connexion and the invariant connexion, respectively. Littlejohn and Reinsch [26] reviewed the relationship between gauge theory and traditional physics approaches to nonholonomic constraints, especially in atomic and molecular science, while Massa and Pagani [27] and Bates and Sniatycki [28] provide mathematical overviews of the nonholonomic problem.

2. Kinematics

The kinematics of Riemann ellipsoids in the gauge formalism is obtained by certain local trivializations of the bundle , which separate the degrees of freedom into rotational, vibrational and vortex components. Every group element can be expressed as a product of three matrices, , where R, S are real orthogonal matrices and A is a diagonal matrix with real positive entries in descending order. The projection in the ellipsoidal space of a bundle point shows that the entries of the square of A are the eigenvalues of q and R is an orthogonal matrix that diagonalizes q. Physically R rotates the body into the principal axis frame, and the entries of A are the lengths of the inertia ellipsoid's principal axes. Because eigenvalues are unique, the diagonal matrix A is determined uniquely by q. The eigenspaces are also uniquely defined by q. If the eigenvalues are distinct, the eigenspaces are one dimensional and each row of R is unique up to a sign. Thus, when restricted to suitable open neighbourhoods, the matrices R and A provide a local coordinate chart for the ellipsoidal space Q. Once R and A are determined by the local chart for the base manifold Q, the orthogonal matrix S in the structure group is given uniquely. A unique decomposition , or for q = Rt A2 R and , in an open neighbourhood of is a local trivialization of the bundle . The bundle is only locally diffeomorphic to the Cartesian product of the base manifold Q and the structure group G.

With respect to left multiplication by elements r in the rotation group, the bundle point is transformed to , or a rotation r is equivalent to right multiplication of the elements R of the subgroup SO(3). With respect to right multiplication by elements g in the structure group, the bundle point is transformed to , or a gauge transformation g is equivalent to right multiplication of the elements S of the subgroup G.

2.1. Tangent space

Consider a curve in the bundle . Such a curve may be identified with the collective motion of a many-body system for which the trajectory of each particle is constrained by , where is independent of time. The reference particle distribution is chosen so that its dimensionless inertia tensor is the identity matrix. With this choice the instantaneous inertia tensor of the many-body system simplifies to .

The velocity vector for each particle is for and . Note that is a linear function of its position vector . The velocity vector may be expressed as the value of a right-invariant vector field on the group at the point ,
(4)


With respect to a local trivialization, the curve is . At each time t, define the antisymmetric matrix in the Lie algebra (3) of the rotation group and the antisymmetric matrix in the Lie algebra of the structure group. A basis for the space of 3 × 3 antisymmetric matrices is given by ei for i = 1,2,3, where . The matrix determines the angular velocity vector , and --the vortex velocity vector :
(5)


For such a local trivialization, the velocity of the curve in the bundle can be shown to be a sum of rotational, vibrational and vortex terms,
(6)


The velocity vector may be expressed alternatively as a sum of right-invariant vector fields on the bundle by using the identities,
(7)


when . Here denotes a right-invariant vector field on SO(3) and denotes a right-invariant vector field on G.

2.2. Riemannian structure

For X, Y in , define the metric at the point by
(8)


This is a positive-definite bilinear form defined on each tangent space of so that the bundle is a Riemannian manifold. The kinetic energy is proportional to the squared length of the velocity
(9)


where is a constant with the units of the moment of inertia. Expanding the velocity into the three types of motion, equation (6), the kinetic energy becomes
(10)


The last term is due to Coriolis coupling between the rotational and vortex degrees of freedom. The derivatives of the kinetic energy with respect to the angular velocity and vortex velocity are the vectors of angular momentum and circulation, respectively,
(11)
(12)


where i,j,k are cyclic.

The equations of motion are found using the Lagrangian formalism [29,18]. Suppose that the potential energy V(A) is a smooth function of the principal axes lengths. Then the potential is left- and right-invariant with respect to the rotation group and the structure group, respectively. But the metric, and hence the kinetic energy, is also left- and right-invariant with respect to the rotation and structure groups. Since the Lagrangian is the difference between the kinetic and potential energies, the two invariances, according to Noether's theorem, imply conservation laws. These are the angular momentum and Kelvin circulation. In the rotating body-fixed frame, the angular momentum and Kelvin circulation vectors precess:
(13)


The two vector conservation laws in the inertial centre-of-mass frame are
(14)


3. Connexions on the Riemann ellipsoid bundle

For many mechanical systems, there are constraint forces in addition to those described by the potential energy V(A). The simplest case is the rigid body for which the vortex velocity vanishes, . This is a holonomic constraint which reduces the configuration space to . But constraints are not typically holonomic. For example, an irrotational fluid (like a water droplet) has zero circulation, C = 0. Another example is the `falling cat' [30,31], for which the angular momentum vanishes, L = 0. In these cases the vortex velocity is proportional to the angular velocity [10]
(15)


where i, j, k are cyclic. A nonholonomic constraint for a Riemann ellipsoid is a proportionality between the vortex and angular velocity components, with a factor fk(A) that depends on the axes lengths. This proportionality is equivalent to a connexion on the bundle , as it will be shown next.

For each point in the bundle, denote the tangent space by . By definition, the vertical space is the subspace of consisting of the tangents to curves in the fibre G,
(16)


If is a Lie algebra element, then the fundamental vector field, denoted by , is the left invariant vector field on the fibre G. A basis for is the set of fundamental vector fields , where .

A connexion [25] is a smooth assignment of a horizontal subspace of the tangent space to each point such that
(17)
(18)


Because the kernel of at is the vertical subspace , its image is Tq Q, and the tangent space is a direct sum of vertical and horizontal subspaces, the linear transformation is an isomorphism from the horizontal subspace onto the tangent space of the base manifold , where . If is a tangent vector to the base manifold, then its horizontal lift is the unique horizontal vector such that . Given any basis of smooth vector fields in an open neighbourhood of the base manifold, , their unique horizontal lifts are denoted by {Fm,m = 1,...,dim Q}. The set is a basis for the horizontal subspace and
(19)


where, in a local trivialization, , and the coefficients are smooth real-valued functions on the bundle .

The second defining property of a connexion, equation (18), asserts that . In particular, when , where I is the structure group identity and , the right translation of a horizontal basis vector at the structure group identity is
(20)


The functions are the Christoffel symbols.

Consider now a basis {(fm)q, m = 1,..., 6 } for the tangent space at that consists of the three right-invariant vector fields on the rotation group SO(3) and the three vibrational vector fields . A tangent vector to a curve in the base manifold is
(21)


The curve's lift to the bundle is required to have the tangent V(t) of equation (6),
(22)


The lift is horizontal if and only if is spanned by
(23)
(24)


for i = 1, 2, 3. The Riemann ellipsoid Christoffel symbols vanish for the vibrational vectors and simplify to a diagonal form for the rotational vectors
(25)


In particular, the special rotational modes correspond to the following Christoffel symbols:
(26)


where i, j, k are cyclic. The Christoffel symbols are just functions of the axis lengths due to rotational invariance of the horizontal subspace, .

3.1. Riemannian connexion

The horizontal subspace for irrotational flow is defined as the orthogonal complement to the vertical subspace . Denote the vector space of all 3 × 3 symmetric matrices by . The orthogonal complement is given explicitly by
(27)


To prove this, suppose , , is a vertical vector and , , is a horizontal vector. These two vectors are orthogonal,
(28)


Since the sums of the dimensions of the vertical space and the horizontal space add up to the dimension of the tangent space , the tangent space is a direct sum of the horizontal and vertical subspaces. If is a horizontal vector and , then right invariance implies
(29)


or . Since the assignment of the horizontal subspace is also smooth, it defines a connexion on .

The vibrational vectors are horizontal since is a symmetric matrix. But the rotational vectors are not horizontal because
(30)


for i, j, k cyclic. Note that the inner product of two vertical vectors is also nonzero,
(31)


for a, j, k cyclic. In order for to be the horizontal lift of , it is necessary and sufficient that, for b = 1, 2, 3,
(32)


The off-diagonal Christoffels for the rotational vectors vanish, and the diagonal values are
(33)


Thus, the Riemannian connexion for which the horizontal space is perpendicular to the vertical space corresponds to irrotational flow.

3.2. Invariant connexion

The falling cat connexion is the invariant connexion on the Lie group . Since is the algebra of antisymmetric matrices and is the vector space of symmetric matrices, the Lie algebra of the group is a direct sum of vector spaces, . Moreover the vector space is invariant with respect to the adjoint group transformation, for all . These two properties of are necessary and sufficient for
(34)


to be a horizontal subspace [25]. In order to see that, note that the vertical vectors can be expressed in left-invariant form,
(35)


The tangent space to the bundle at is a direct sum of the horizontal and vertical subspaces, because every matrix is a linear combination of a symmetric matrix Y and an antisymmetric matrix . The right invariance of the horizontal subspaces is a consequence of
(36)


since for all and . The assignment of the subspaces is smooth, so is indeed a horizontal subspace.

The relation
(37)


shows that the vibrational vectors are horizontal ( is symmetric), but the rotational vectors are not horizontal since
(38)


and St A-1 ei A S is not symmetric. If the matrix StA-1 ei A S is expressed as a sum of symmetric Xs and antisymmetric Xa parts, i.e. Xs = (St A-1 ei A S -St A ei A-1 S)/2, Xa = (St A-1 ei A S + St A eiA-1 S)/2, the angular momentum may be written as a sum of horizontal and vertical vectors:
(39)


The horizontal lifts of the angular momentum vectors are
(40)


where is the horizontal lift and the two vertical vectors in the square brackets must cancel. Therefore, the invariant connexion is given by
(41)


or the Christoffel symbols are diagonal and
(42)


where i,j,k are cyclic.

3.3. Dedekind's theorem

The geometrical relationship between the Riemannian and invariant connexions is equivalent to Dedekind's theorem, which relates irrotational flow to falling cat solutions [10]. Define the Dedekind involution as the matrix transpose, . In a local trivialization, , the Dedekind map interchanges R and S. It also interchanges the angular velocity and the vortex velocity .

The differential of the Dedekind involution maps a right-invariant vectorfield into a left-invariant one, . Hence is a vector space isomorphism from the irrotational flow horizontal subspace at onto the falling cat horizontal subspace at ,
(43)


The Riemannian metric is Dedekind invariant,
(44)


because . When the potential V = V(A) is a pure function of the axis lengths, then the Lagrangian L = K -V is invariant. Therefore, the Dedekind involution transforms solutions of Lagrange's equations into other solutions, known as the adjoint solutions. If a solution is constrained to irrotational flow, then the Dedekind involution maps it into a falling cat solution, and vice versa.

4. Conclusions

The concept of a horizontal lift is physically natural. It says that a many-body system responds to rotations and vibrations (described by a curve in the base manifold) by internal vortex motions (described by a horizontally-lifted curve in the bundle). This response is determined typically by a nonholonomic constraint that depends ultimately on the nature of the forces between the particles. The constraint that the tangent to the lifted curve lies in a horizontal subspace is equivalent to a bundle connexion.

The connexions corresponding to rigid rotation, irrotational flow, and the falling cat were shown to be natural geometrical or group-theoretical concepts. Although not mathematically natural, other choices of Christoffel symbols define nonholonomic constraint forces that are not excluded by physical law. For example, the S-type Riemann ellipsoids are a sequence of special case solutions for which the angular momentum, Kelvin circulation, and the angular and vortex velocity vectors are aligned with a principal axis, say the 1-axis [9,10]. This sequence is indexed by a continuous real parameter f restricted to the interval . There is only one horizontal lift to consider and the Christoffel symbol is given by
(45)


At f = 0, the connexion yields rigid rotation, and, at f = -2, it is irrotational flow. The S-type ellipsoids are the simplest models that allow for a continuous interpolation between rigid rotation and irrotational flow. This connexion has no natural geometrical or group-theoretic significance--but it does model a variety of rotating physical systems.

An unsolved basic science problem is to determine the connexion from the interactions among the particles that form a rotating system. A complete theory of collective rotation requires equations that incorporate these interactions into the gauge theory and whose unique solution are the Christoffel symbols. They must involve a coordinate independent object and the curvature form is the obvious candidate. The Bianchi identity partially determines the Christoffel symbols, but it is not sufficient. There must be another equation that relates the bundle curvature to the microscopic physics.

The stability of equilibrium solutions is a major part of the theory of Riemann ellipsoids [32]. This classical theory might be revisited in the geometrical setting of gauge theory using modern methods [33,34].

Acknowledgments

The author would like to thank the Institute of Nuclear Physics and the organizers, J Ginocchio and F Iachello, of the programme entitled, `Algebraic Methods in Many-Body Physics', for their support. Valuable discussions with J Bryan, Ts Dankova, E Ihrig and J Troupe contributed intellectually to this work.

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At 2/15/2006 10:05 PM, Blogger Julianne said...

POST SOMETHING ELSE! (I love doing that!) :)

 

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