Moment Map of the Action of SO.3/ on R3 R3
José Antonio Villa Morales
The aim of this extended abstract is to expose the main results of the moment map of
the action of SO.3/ on the cotangent bundle of R3 . The moment map for this action
has a strong motivation from the angular momentum studied in classical mechanics.
Suppose we have a central force and a particle which rotates around this force with
velocity v and position r, then we know that the angular momentum is given by
r v. Viewing the rotation of the particle as an action of the group SO.3/ over the
cotangent bundle of R3 we can find that the angular momentum is related to a map
between T R3 and the dual of the Lie algebra so.3/ called moment map.
1 Lie Group Actions and Symmetric Hamiltonians
Definition 1 Let M be a differential manifold and G a Lie group. An action of G on
M is a smooth map GM ! M, .g; p/ 7! gp, satisfying the following conditions:
(1) if e is the null element in G then ep D p for all p 2 M;
(2) for all g; h 2 G and p 2 M, g.hp/ D .gh/p.
A symplectic manifold is an even dimensional differentiable manifold M with
a non-degenerate and closed 2-form !. Remember that ˛ 2 k .M/ is closed if
s˛ D 0.
Definition 2 Let .M; !/ a symplectic manifold and G a Lie group acting on .M; !/.
We say that the action is symplectic if Lg ! D !, for all g 2 G.
J.A.V. Morales ()
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México and
Universidad Nacional Autónoma de México, México, México
e-mail: [email protected]
© Springer International Publishing Switzerland 2015
M. Corbera et al. (eds.), Extended Abstracts Spring 2014, Trends in Mathematics 4,
DOI 10.1007/978-3-319-22129-8_15
83
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J.A.V. Morales
Suppose we have a Lie group G acting over a differentiable manifold M. For
an arbitrary tangent vector 2 g we construct a vector field, called the infinitely
generated vector field X given by
X .p/ D
d
exp.t/p:
dt tD0
Given the symplectic manifold .M; !/, and a vector field X on M, we denote the
contraction by iX ! and it is defined by iX !.Y/ D !.X; Y/ for Y a vector field in M.
Definition 3 Let .M; !/ be a symplectic manifold, and X a given vector field on
M. If there is a function H 2 C1 .M/ such that iX ! D dH we say that H is the
Hamiltonian associate to the vector field X. Conversely, if given a differentiable
function H 2 C1 .M/ there exists a vector field X such that iX ! D dH we say that
X is the vector field associated to the function H.
We will denote by XH the vector field associated to H and, conversely, by HX the
function associated to the vector field X.
2 Symplectic Structure on the Cotangent Bundle
Given an n dimensional manifold M, it is possible to construct a 1-form on T M
which give rise to a symplectic form on T M. We only expose the construction
result on a coordinate chart; a free coordinate construction can be found in [1].
Proposition 4 Let M a differentiable manifold. Then, there exists a 1-form on
T M such that, for a chart .U; '/ on T M with coordinates .x1 ; : : : ; xn ; p1 ; : : : ; pn /,
is given by
D dx1 ^ dp1 C C dxn ^ dpn :
Definition 5 The 2-form ! WD d on T M is called the symplectic form over the
cotangent bundle.
For the next section, we will use the fact that the cotangent bundle of R3 is trivial,
so we can express the symplectic manifold .T R3 ; !/ as .R3 R3 ; !/.
3 Rotations on R3 R3
Consider the symplectic manifold .R3 R3 ; !/, and define the action of the Lie
group SO.3/ over it given by
SO.3/ .R3 R3 ; !/ ! .R3 R3 ; !/
.A; .u; v// 7! .Au; Av/:
Moment Map of the Action of SO.3/ on R3 R3
85
Proposition 6 This action of SO.3/ on .R3 R3 ; !/ is symplectic.
We give the idea of the proof: by definition ! D d, where is the
canonical 1-form of the cotangent
P bundle given in a chart .U; '/ with coordinates
.x1 ; : : : ; xn ; p1 ; : : : ; pn / by ! D ni dxi ^ dpi . With some calculations we can verify
that, for all A 2 SO.3/, LA D . And, by the commutativity of the operator d with
the pullback, it is possible to verify that LA ! D !.
Proposition 7 Let B 2 so.3/ and let XB be the corresponding infinitely generated
vector field of B. Then, the Hamiltonian associated to XB is given by
HXB .u; v/ D hAu; vi:
Proposition 8 Let B 2 so.3/ and let XB be the corresponding infinitely generated
vector field of B. Then, the Hamiltonian HXB is symmetric, i.e., for all A 2 SO.3/,
LA H D H:
3.1 Moment Map of Rotations on R3 R3
The main goal of this section is to define the moment map of a symplectic action,
and to calculate it for the specific action of SO.3/ over the cotangent bundle of R3 .
Definition 9 Let .M; !/ be a symplectic manifold, and let G be a Lie group with a
symplectic action over .M; !/. A moment map is a function W M ! g such that for
all 2 g there is a function W M ! R such that d D iX ! and .p/ D .p/./.
Proposition 10 Let .M; !/ be a symplectic manifold, let H 2 C1 .M/ be a
symmetric Hamiltonian, and consider the vector field XH associated to it. Then,
its moment map is a first integral of XH .
We have that the Lie algebra so.3/ has dimension three and then, we can identify
it with the Euclidean space R3 . We define the map so.3/ ! R3 via
0
1
0 3 2
@ 3 0 1 A ! .1 ; 2 ; 3 /:
2 1 0
In A 2 so.3/, we denote the associated vectors in R3 as A . If we take a tangent
vector B 2 so.3/ and u 2 R3 , by calculations, we can verify that
Bu D B u:
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J.A.V. Morales
Proposition 11 There exist an inner product on the Lie algebra so.3/ given by
hA; Biso.3/ D
tr.At B/
:
2
Now, consider the moment map W R3 R3 ! so.3/. By definition, dQ B D iXB !.
But, by definition of associated Hamiltonian, we also have d Q B D iXB ! D dHB . So,
we conclude Q B .u; v/ D HB .u; v/ and, using this last equality, we calculate the
moment map
..u;
Q v//.B/ D hBu; vi D hB u; vi D hu v; B i:
Hence, given a tangent vector B 2 so.3/, we have ..u; v//.B/ D hu v; B i
and, using the inner product of so.3/, we get hu v; B i D hAuv ; Biso.3/ . It is then
possible to conclude that the moment map is .u; v/ D Auv .
4 The Moment Map of the Action SO.3/ Over T SU.2/
We make the construction of the action of the group SO.3/ over the cotangent bundle
T R3 and, via the moment map, we find a first integral of the system given by this
action. Naturally, we have an action of SO.3/ over the Euclidean space R3 which is
the mathematical model of the rotations in three dimensional space. But if we make
a one point compactification of R3 , we obtain the Lie group SU.2/, so it is natural to
try to extend this action to an action over SU.2/. But following the theory developed
above, if we see the cotangent bundle of T SU.2/ as a symplectic manifold, can we
extend the action of SU.2/ to an action of SO.3/ over T SU.2/? We enumerate the
steps for constructing the possible action:
1. The action of SO.3/ over R3 is given by multiplication of a matrix and a vector
on R3 . If we make the one point compactification of R3 at the point 1, how can
we define an action SO.3/ SU.2/ ! SU.2/ such that .A; 1/ 7! A1 makes
sense?
2. SU.2/ is a Lie group and then its cotangent bundle is T SU.2/ D SU.2/ R3 .
So it is natural to define the action of SO.3/ by .A; vp / 7! .Ap; Av/. Does this
define a symplectic action?
3. How can we calculate the moment map for this action?
Reference
1. R. Abraham and J.E. Marsden, “Foundations of mechanics”. Second edition, Addison Wesley
1978.