DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 261–265

’FALLING CAT’ CONNECTIONS CONSTRUCTED

FROM THE MOMENTUM MAP

Marián FECKO

We consider a standard symplectic dynamics on T M generated by a natural Lagrangian L. The Lagrangian is assumed to be invariant with respect to the action T R_{g} of a Lie group G lifted from the free and proper action R_{g} of G on M. It is shown that under these conditions a connection on principal bundle π : M → M/G can be constructed based on the momentum map corresponding to the action T R_{g}. A simple explicit formula for the connection form is given. For the special case of the standard action of G = SO(3) on M = ℝ^{3} × ⋯ × ℝ^{3} corresponding to a rigid rotation of a N-particle system the formula obtained earlier by Guichardet and Shapere and Wilczek is reproduced.

1. Introduction

A.Guichardet [1] and A.Shapere and F.Wilczek [2] pointed out that the phenomenon of reorientation of deformable bodies (molecules in [1] or cats, astronauts etc. in [2]) in space, based physically on the angular momentum conservation, lends itself to a simple and powerful description within the framework of the theory of connections. They showed, in particular, that in the center-of-mass system (P⃗ = 0) the condition of vanishing of the total angular momentum (L⃗ = 0) can be rephrased in terms of the SO(3)-connection in the principle bundle π : M → M/SO(3), where M is the configuration space of the deformable body, where SO(3) acts by rigid rotations (without deformation). In more detail the trajectories fulfilling L⃗ = 0 represent the horizontal curves in the sense of the connection (”vibrational curves” in [1] as opposed to purely rotational ones given by rigid rotations). In what follows we try to understand the origin of the connection within the standard framework [3] of lagrangian mechanics on T M.

The central object providing the link between the symmetries and conserved quantities in symplectic dynamics is the momentum map [4,5]. Now both P⃗ and L⃗ result (being linear in velocities) from the symmetries of rather special type, namely those lifted to T M from M. That is why the situation under consideration is the following : we study a lagrangian system (T M, L) with appropriate action of a Lie group G lifted from the configuration space M. Then we show how one can construct (under some restrictions on the Lagrangian L) a connection in the principal bundle π : M → M/G. This connection happens to coincide with the one in [1,2] in the particular case treated there, i.e. for G = SO(3), M being the configuration space of N-particle system.

1991 Mathematics Subject Classification. 70D10, 58F05, 53B15.

Key words and phrases. Lagrangian system with symmetry on T M, connection on principal bundle, momentum map.

This paper is in final form and no version of it will be submitted for publication elsewhere.

2. Construction of the connection form

Let

R_{g} : M → M

be a right action of a Lie group G on a manifold M. Then the lifted action of G on T M is

T R_{g} : T M → T M

Let L : T M → ℝ be a G-invariant Lagrangian, i.e.

L ∘ T R_{g} ≡ (T R_{g})^{∗}L = L

for all g ∈ G. The (exact) symplectic form [3] on T M is given by

ω_{L} = dθ_{L} = dS(dL)

where (1,1)-type tensor field S on T M (almost tangent structure ≡ vertical endomorphism) is a lift of the identity tensor on M (S = I^{↑}; in canonical local coordinates x^{i}, v^{i} on T M, S = dx^{i} ⊗ \frac{∂}{∂v^{i}} or S = dx^{i} ⊗ \frac{∂}{∂ẋ^{i}} if the notation v^{i} ≡ ẋ^{i} is used). Since ω_{L} is to be maximum rank 2-form, the condition

det(\frac{∂^{2}L}{∂v^{i}∂v^{j}}) ≠ 0

must be fulfilled (nondegenerate Lagrangian). The momentum map associated with the (exact symplectic) action T R_{g} on T M

P : T M → G^{∗}

is given by

< P (v), a >:= P_{a}(v) v ∈ T M

where [7]

P_{a} = X_{a}^{↑}L

X_{a}^{↑} being the vertical lift of the fundamental field X_{a} of the action R_{g} corresponding to the element a ∈ G, where G is the Lie algebra of G. Fixing a basis E_{α}, α = 1, …, dimG in G and the dual one E^{α} in G^{∗} one can write

P = P_{α}E^{α}

where

P_{α} ≡ P_{E_{α}} : T M → ℝ

are the components of P with respect to E^{α}.

It holds

(T R_{g})^{∗}P = Ad_{g}^{∗}P

where Ad_{g}^{∗} : G^{∗} → G^{∗} is the coadjoint action of G on G^{∗}. If Ω^{k}(M, ρ) denotes the space of k-forms of type ρ on G-space M (i.e. V-valued k-forms on M obeying R_{g}^{∗}σ = ρ(g^{−1})σ, ρ being a representation of G in V), we see that

P ∈ Ω^{0}(T M, Ad^{∗})

- it is G^{∗} - valued 0-form of type Ad^{∗} on T M. Thus P ∈ Ω^{0}(T M, Ad^{∗}) is available. What we need is a connection on

π : M → M/G

i.e. a connection form

A ∈ Ω^{1}(M, Ad)

such that

< A, X_{a} >= a

There is a way to construct some A from P. It is done in 3 steps :

1. if P_{α} are linear in velocities, then

P_{α}(x, v) = P_{αi}(x)v^{i} ↦ P̃_{α} ≡ P_{αi}(x)dx^{i} ∈ Ω^{1}(M, Ad^{∗})

This restrics L to ”natural” lagrangian

L(x, v) = \frac{1}{2}g_{ij}(x)v^{i}v^{j} − U(x)

2. Ad^{∗} changes to Ad when some ĥ : G → G^{∗} is used :

Â := ĥ^{−1} ∘ P̃ ∈ Ω^{1}(M, Ad)

3. It holds

< Â, X_{a} >= C(x)(a) C(x) : G → G
C_{α}^{γ} := g_{αβ}(x)h^{βγ} g_{αβ}(x) := g(X_{α}, X_{β})

Then

A := C^{−1} ∘ Â = C^{−1} ∘ ĥ^{−1} ∘ P̃

is already a connection form . Explicitly [7]

A = A^{α}E_{α} = (g^{αβ}P̃_{β})E_{α} = g^{αβ}(♭_{g}X_{β})E_{α}

(ĥ dropped from the resulting formula - it is not necessary).

3. Properties of A

1. The physical meaning of horizontality : Let γ : ℝ → M be a curve on M representing some motion of the system under consideration. Then

< A, γ̇ >= 0 ⇒ P_{α}(γ̂(t)) = 0

(γ̂ is the lift of γ to T M). Thus a horizontal curve is such motion of the system that all conserved quantities P_{α} have all the time zero value (remember P⃗ = 0, L⃗ = 0 in the Introduction).

2. if W horizontal, then

0 =< A, W >= g^{αβ}g(X_{β}, W)E_{α}

or

V er_{x}M ⊥ Hor_{x}M

(can serve [1] as a definition of connection).

4. Special cases

1. A man walking on a gramophone disc. Let us have a gramophone disc (its moment of inertia with respect of the axis being I) and a ”man” (point mass m) on it. If the angle α measures the orientation of the disc with respect to the outer space and r, φ are the polar coordinates of the man with respect to the disc, the Lagrangian of the system is

L(r, φ, α, ṙ, φ̇, α̇) = \frac{1}{2}Iα̇^{2} + \frac{1}{2}m(ṙ^{2} + r^{2}(α̇ + φ̇)^{2}) − U(r, φ)

(interaction of the point mass m with the disc depends only on their relative position). Then the computation gives (G = SO(2) here)

A = A^{1}E_{1} = g^{11}(♭_{g}X_{1})E_{1} = \frac{1}{I + mr^{2}} ((I + mr^{2})dα + mr^{2}dφ) = dα + \frac{mr^{2}}{I + mr^{2}} dφ

(one can take E_{1} = 1 since G = ℝ). A curve γ ↔ (r(t), φ(t), α(t)) is horizontal if

Iα̇(t) + mr^{2}(t)(α̇(t) + φ̇(t)) = 0

- just vanishing of the total angular momentum of the system.

2. N-particle system (deformable body) Let r⃗_{a} , a = 1, … N denote the radius vector of a-th particle , x_{a}^{i} its i-th component (i = 1,2,3), m_{a} its mass. There is a natural action of the Euclidean group G = E(3) on the configuration space of the N-particle system, consisting in rigid rotations and translations

r⃗_{a} ↦ r⃗_{a}B + b⃗ B ∈ SO(3)

Connection form computed in this case is

A = A_{tr} + A_{rot} = \frac{m_{a}dx_{a}^{i}}{m} E_{i}+

+(−I^{ij}(r⃗_{1}, …, r⃗_{N})(∑_{a}m_{a}r⃗_{a} × dr⃗_{a})_{j})E_{i} ≡

≡ \frac{p̃^{i}}{m}E_{i} − I^{ij}L̃_{j}E_{i}

where I_{ij}(r⃗_{1}, … r⃗_{N}) is the inertia tensor of the configuration (the part A_{rot} coincides with the results obtained in [1,2]). Horizontal motion :

m_{a} ˙r⃗_{a}(t) ≡ P⃗ (t) = 0
∑_{a}m_{a}r⃗_{a} × ˙ r⃗_{a} ≡ L⃗ = 0

i.e. such that the total (linear) momentum P⃗ as well as the total angular momentum L⃗ of the system vanish.

5. References

[1] A.Guichardet: ”On rotation and vibration motions of molecules”, Ann. Inst. Henri Poincaré, Vol.40, n.3, 1984, p.329-342

[2] A.Shapere, F.Wilczek: ”Gauge kinematics of deformable bodies”, Am.J.Phys. 57 (6),514-518, June 1989

[3] M.Crampin, F.A.E.Pirani : ”Applicable Differential Geometry”, Cambridge Univ. Press, Cambridge, 1987

[4] J.M.Souriau : Structure des systèmes dynamiques, Dunod, Paris, 1970

[5] V.I.Arnold : Mathematical Methods of Classical Mechanics, Benjamin/Cummings Reading MA, 1978, Appendix 5

[6] R.Montgomery : ”Isoholonomic Problems and Some Applications”, Commun. Math. Phys. 128, 565-592 (1990)

[7] M.Fecko : ”’Falling cat’ connections and the momentum map”, to appear in J.Math.Phys. (dec.’95)

Department of Theoretical Physics, Comenius University Mlynská dolina F2, 842 15 Bratislava, SLOVAKIA

E–mail : fecko@fmph.uniba.sk