Geodesics of left invariant metrics on matrix Lie groups – Part 2 Conservation laws

In the last post, Geodesics of left invariant metrics on matrix Lie groups – Part 1,we have derived Arnold’s equation – that is a half of the problem of finding geodesics on a Lie group endowed with left-invariant metric.

Suppose is a Lie group, and is a scalar product (i.e. a nondegenerate bilinear form) on its Lie algebra . Then, using left translations defines a left invariant (Riemannian or pseudo-Riemannian) metric on the whole group If is a path in we use left translations to define the image of the tangent vector in

(1)

Usually it is written more carefully, using instead of , but I am using a simplified notation, well adapted to dealing with problems for matrix groups.

If is a geodesic for the metric , then satisfies the Arnold’s equation

(2)

where

is defined as

(3)

and are the structure constants of , that is

(4)

where form a basis for and

(5)

Equation (2) is a system of nonlinear ordinary differential equations with constant coefficients. In this form it can be found in Arnold’s 1966 paper “Sur la g\’eom\’etrie differentielle des groupes de Lie de dimension inﬁnie et ses applications \`a l’hydrodynamique des ﬂuides parfaits”, Ann. Inst.
Fourier (Grenoble) 16 (1966), 319-361.

In his blog post “The Euler-Arnold equation” Terrence Tao mentions that some people call this equation the “Euler-Arnold” equation, while some other prefer to skip Arnold’s name completely and call it “Euler-Poincare equation”. Go-figure!

Solving equations (2) is just one half of the whole problem of finding a geodesic. Once are known, we need to solve the linear differential equation with variable coefficients that results from the definition of (??):

(6)

For the case of the rotation group and free rigid body we did it in Taming the T-handle continued.

B. Kolev in his paper “Lie groups and mechanics. An introduction” shows that for the rotation group O(n) the equations of motion for the free rigid body are completely integrable. We will not need this result, but is useful to know that in general case there are always two quadratic constants of motion corresponding to “kinetic energy” and “square of the angular momentum”. We were discussing these constants of motion for the group O(3) in
“Asymmetric Spinning Top – The Hardest Concept To Grasp In Physics” – they are used in so-called Poinsot construction. We will discuss a version of it for the group O(2,1) in the future.

The first observation is that the “kinetic energy” is, in fact, a constant, independent of To see that this is the case, we differentiate:

(7)

In the previous post we wrote Eq. (2) as

(8)

Substituting into Eq. (7) we get

(9)

The result is zero because is antisymmetric in while is symmetric. That is an often used property: if and then the contraction Indeed where in the last equality we have exchanged dummy indices names . If a number is equal to its negative, it must be zero.

To get the formula for the second quadratic invariant we need to return to the Ad-invariant scalar product that we have denoted in Killing vectors, geodesics, and “Noether’s theorem”:

(10)

The fact that is Ad-invariant implies an important relation between the matrix and the structure constants
that we are going to use. Ad-invariance means that:

(11)

for all and
Differentiating at we get

(12)

Setting we get

(13)

Multiplying both sides by we obtain

(14)

We can now derive the second conservation law. The angular momentum is defined as

(15)

Notice that is a covector, a one-form on , it is in the dual of It is the metric that connects the space to its dual. While vectors in play an active role, they generate transformations, elements in the dual, one-forms from , are “passive”, they evaluate vectors to numbers. It is the metric that is the third element here, that allows the active principle to connect to the passive principle. The metric depends on the mass distribution. In application to rigid bodies the inertia tensor is encoded in the metric on the rotation group.

The second conservation law states that the square of the angular momentum evaluated with the Ad-invariant metric is constant:

(16)

To verify we differentiate and use Eq. (8) rewritten as

(17)

(18)

Now, according to Eq. (14) is antisymmetric in , while the product is symmetric, therefore we get zero:

(19)

In the following posts we will first return to the case of the rotation group in three dimensions and the rigid body, and then try to apply a similar reasoning to the case of the Lorentz group O(2,1) in 2+1 dimensions.

Geodesics of left invariant metrics on matrix Lie groups – Part 1

An elegant derivation of geodesic equations for left invariant metrics has been given by B. Kolev in his paper “Lie groups and mechanics. An introduction”.

Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at the end we will have formulas ready for applications. We will use
the conservation laws derived in the last post Killing vectors, geodesics, and Noether’s theorem. We will also use the same notation. We consider matrix Lie group with the Lie algebra The tangent space at is denoted .
Thus On we assume nondegenerate scalar product
denoted as We propagate it to the whole group using left translations as in Eqs. (8,9) of Killing vectors, geodesics, and Noether’s theorem

(1)

which implies for

(2)

The metric so constructed is automatically left-invariant, therefore for each the vector field is a Killing field.

Let be a geodesic for this metric. We denote by the tangent vector left translated to the identity:

(3)

Then, from the conservation laws derived in the last post, we know that the scalar product of with is constant. That is

(4)

The metric is left-invariant, therefore , or

(5)

We will differentiate the last equation with respect to but first let us notice that by differentiating the identity we obtain

(6)

Now, differentiating Eq. (5), and using also we obtain

(7)

We now need a certain bilinear operator on Lie(G) that is defined using the commutator and the scalar product. The commutator itself is such an operator
from But using the scalar product we can define another operator by the formula:

(8)

The right hand side is linear in and owing to the nondegeneracy of the scalar product every linear functional is represented by a scalar product with a unique vector. Therefore is well defined, and evidently is linear in both arguments.

Let be a basis in so that the structure constants are

(9)

We can also write as

(10)

Then Eq. (8) gives

(11)

or

which can be solved for using the inverse metric:

(12)

On the other hand, if we agree to lower the upper index of and with the metric, we can write Eq. (11) as

(13)

which is easy to remember.

We can now return to Eq. (7) and rewrite it as

Since , and therefore also is arbitrary, we obtain

(14)

or, using a basis and Eq. (13)

(15)

Killing vectors, geodesics, and Noether’s theorem

Consider Lie groups of matrices: SO(3) or SO(2,1). Their double covering groups are SU(2) and SU(1,1) (or, after Cayley transform, SL(2,R)). We prefer to use these covering groups as they have simpler topologies. SU(2) is topologically a three-sphere, SL(2,R) is an open solid torus. Our discussion will be quite general, and applicable to other Lie groups as well.

We denote by the Lie algebra of . It is a vector space, the set of all tangent vectors at the identity of the group. It is also an algebra with respect to the commutator.

acts on its Lie algebra by the adjoint representation. If and then

(1)

We define the scalar product on using the trace

(2)

.

In each particular case we will choose the constant so that the formulas are simple.

Due to trace properties this scalar product is invariant with respect to the adjoint representation:

(3)

We will assume that this scalar product is indeed a scalar product, that is we assume it being non-degenerate. For SO(3) and SO(2,1) it certainly is. Lie groups with this property are called semisimple.

Let be a basis in The structure constants are then defined through

(4)

We denote by the matrix of the metric tensor in the basis

(5)

The inverse matrix is denoted so that

For SU(2) the Lie algebra consists of anti-Hermitian matrices of zero trace. For the basis we can take

(6)

For the constant we chose . Then

The structure constants are

(7)

In this case, since is the identity matrix, there is no point to distinguish between lower and upper indices. But in the case of SU(1,1) it will be important.

We will now consider a general left-invariant metric on the group The discussion below is a continuation of the discussion in Riemannian metrics – left, right and bi-invariant.

That is we have now two scalar products on – the Ad-invariant scalar product with metric and another one, with metric We propagate the scalar products from the identity to other points in the group using left translations (see Eq. (1) in Riemannian metrics – left, right and bi-invariant). We have a small notational problem here, because the letter often denotes a group element, but here it also denotes the metric. Moreover, we have two scalar products and we need to distinguish between them. We will write for the scalar product with respect to the metric of two vectors tangent at Then left invariance means

(8)

which implies for tangent at

(9)

Infinitesimal formulation of left invariance is that the vector fields are “Killing vector fields for the metric” – Lie derivatives of the metric (cf. SL(2,R) Killing vector fields in coordinates, Eq.(13)) with respect to these vector fields vanish. What we need is a very important result from differential geometry: scalar products of Killing vector fields with vectors tangent to geodesics are constant along each geodesic. For the convenience of the reader we provide the definitions and a proof of the above mentioned result (a version of Noether’s theorem). Here we will assume that there are coordinates on Later on we will get rid of these coordinates, but right now we will follow the standard routine of differential geometry with coordinates.

We define the Christoffel symbols of the Levi-Civita connection

(10)

(11)

The geodesic equations are then (in Geodesics on upper half-plane factory direct we have already touched this subject)

(12)

A vector field is a Killing vector field for if the Lie derivative of with respect to vanishes, i.e.

(13)

The scalar product of the Killing vector field and the tangent vector to a geodesic is constant. That is the “conservation law”. A short proof can be found online in Sean Carroll online book “Lecture notes in General Relativity”. The discussion of the proof can be found on physics forums. But the result is a simple consequence of the definitions. What one needs is differentiating composite functions and renaming indices. Just for fun of it let us do the direct, non-elegant, brute force proof.

Suppose is a geodesic, and is a Killing field. The statement is that along geodesic the scalar product is constant. That means we have to show that

We differentiate with respect to , and we are supposed to get zero. So, let’s do it. We have derivative of a product of three terms, so we will get three terms :

Let us calculate the derivatives. After we are done, in order to simplify the notation, we will skip the arguments.

Thus

Then, from Eq. (12)

therefore

Renaming the dummy summation indices we see that the two first terms of are identical, therefore

Again, renaming the dummy summation indices we see that the first term of cancels out with therefore

For we have

Owing to the symmetry of , we can write it as

Therefore

We rename the indices to get

But the expression in parenthesis vanishes owing to Eq. (13).