Euler-Poincaré reduction in principal bundles by a subgroup of the structure group

Transcription

1 Euler-Poincaré reduction in principal bundles by a subgroup of the structure group M. Castrillón López ICMAT(CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid (Spain) Joint work with Pedro L. García, University of Salamanca (Spain) César Rodrigo, Academia Millitar (Portugal). III Iberoamerican Meeting on Geometry, Mechanics and Control Salamanca, September 2012

2 . CONTENTS 1. Euler-Poincaré reduction for Field Theories 2. Reduction by a subgroup: the space 3. Reduction by a subgroup: the equations 4. Noether conservation law 5. Example: Mechanics 6. Reduction in affine principal bundles 7. Example: Molecular strands

3 REFERENCES M.C.L., P.L. García, T. Ratiu, Euler-Poincaré reduction on principal fiber bundles, Lett. Math. Phys. 58 (2001) M.C.L., T.S. Ratiu, Reduction in principal bundles: covariant Lagrange-Poincaré equations, Comm. Math. Phys. 236 (2003), no. 2, D.C.P. Ellis, F. Gay-Balmaz, D.D. Holm, V. Putkaradze, T.S. Ratiu, Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal. 197 (2010), no. 3, D.C.P. Ellis, F. Gay-Balmaz, D.D. Holm, T.S. Ratiu, Lagrange-Poincaré field equations, J. Geom. Phys. 61 (2011), no. 11, D. Holm, J.E. Marsden, T.S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math. 137, 1.81 (1998)

4 I.- EULER-POINCARÉ REDUCTION FOR FIELD THEORIES Euler-Poincaré reduction in Mechanics is paradigmatic: We have L : T G R invariant by G so that T G/G g l : g R with variations δσ = dη dt ± ad ση ( ) d δl EL(L) = 0 dt ad σ δσ = 0. In Field theories, we have sections of fiber bundles. In particular, we take π : P M and L : J 1 P R invariant with respect to the structure group G and defining a variational principle s L(j 1 s)v M

5 Theorem Given a local section s of P M and the local section σ = [j 1 s] G of (J 1 P )/G = C M, the following are equivalent: 1.-s satisfies the Euler-Lagrange equations for L, 2.-the variational principle δ M L(j 1 s)dx = 0 holds, for arbitrary variations with compact support, 3.-the Euler-Poincaré equations hold: 4.-the variational principle δ div σ δl δσ = 0, M holds, using variations of the form l(σ(x))dx = 0 δσ = σ η where η : U g is an arbitrary section of the adjoint bundle.

6 Remark: EP(l)(σ) = div σ δl δσ = 0 ( div H + ad σ H ) δl δσ = 0. Remark: If σ = [j 1 s] G, then σ is flat (Curv(σ) = 0) and the integral leaves of σ are the solutions s. EL(L)(s) = 0 loc EP(l)(σ) = 0 Curv(σ) = 0 If Θ L is the Poincaré-Cartan form, Nother theorem says d((j 1 s) i j1 BΘ L ) = 0, B g in J 1 P is equivalent to div σ δl δσ = 0.

7 II.- REDUCTION BY A SUBGROUP: THE SPACE We now consider L : J 1 P R invariant with respect to a subgroup H G. This is related with homogeneous manifolds G/H Relevant (regular) situations with symmetries fit in this context. It is connected with the notion of symmetry breaking. The reduction process contains the information of the general case. In fact, fiber bundles are generally G-bundles so that the symmetry group is usually H G. We first consider the dropped Lagrangian l : (J 1 P )/H R The geometry of the reduced space?

8 Lagrange-Poincaré approach: We fix a connection A and we have (J 1 P )/H J 1 (P/H) P/H T M h where h P/H is the adjoint bundle of the principal bundle P P/H. The reduced equations are split into two groups EL = curva EP = 0 BUT, the identification, the splitting and the gyroscopic terms are NOT canonical. Euler-Poincaré approach: Theorem Let H be a closed Lie subgroup of the structure group G of a principal bundle π : P M. Then the mapping is a fiber diffeomorphism. (J 1 P )/H C M P/H [j 1 xs] H ([j 1 xs] G, [s(x)] H )

9 Proposition For a (local) section s of P M and the induced (local) section j 1 s of J 1 P M, the induced section (σ = [j 1 s] G, s = [s] H ) of C M P/H = (J 1 P )/H, satisfies Curvσ = 0 σ s = 0. Recall that P/H M is the same as the associated bundle P G (G/H) = P (G/H) G M so that connections in P M induce covariant derivatives in P/H.

10 III.- REDUCTION BY A SUBGROUP: THE EQUATIONS Let δs be an arbitrary variation of a local section s of P M. We consider the gauge vector field X such that X Ims = δs. Recall Gauge vector fields G-invariant vertical vector fields in P sections of the adjoint bundle g M. Proposition The induced variations of the sections (σ = [j 1 s] G, s = [s] H ) of C M P/H = (J 1 P )/H are δσ = X C = σ η, δ s = (π H ) X = η P/H where π H : P P/H and X gaup η section of g.

11 We can thus define the operator P (σ, s) : gaup Γ(σ V C) Γ( s V (P/H)), P (σ, s) = (P σ, P s ) with P σ (X) = X C Im σ = σ η, P s (X) = (π H ) X Im s = ηp/h, The adjoint operator defined by P + (σ, s) : Γ(σ V C) Γ( s V (P/H)) Γ( g ) P + (σ, s) (X, Υ)(η) = X, P σ(η) + Υ, P s (η), with X Γ(σ V C) Γ(T M g ), Υ Γ( s V (P/H)), η Γ( g), and, the duality pairing, reads P + (σ, s) (X, Υ) = d X, divσ X + P + s (Υ), where P + s (Υ)(η) = Υ, η P/H.

12 Theorem Let L : J 1 P R be an H-invariant Lagrangian. Let l : C M P/H R be the reduced Lagrangian. For a (local) section s of P M and the induced section (σ, s) of C (P/H) M the following are equivalent: 1. the variational principle δ U L(j1 xs)v = 0 holds, for vertical variations δs along s with compact support, 2. the local section s satisfies the Euler-Lagrange equations for L, 3. the variational principle δ U variations of the form l(σ(x), s(x))v = 0 holds, using δσ = σ η, δ s = η P/H, for any section η of g with compact support, 4. the Euler-Poincaré equations hold: EP(l)(σ, s) := div σ δl δσ P+ s ( ) δl δ s where δl/δσ Γ(σ V C) Γ(T M g ) and δl/δ s Γ( s V (P/H)). = 0,

13 Reconstruction Theorem Let P M be a principal bundle over a simply-connected manifold and let L : J 1 P R be an H-invariant Lagrangian. Let l : C M P/H R be the reduced Lagrangian. For a (local) section s of P M and the induced section (σ, s) of C (P/H) M then EL(L)(s) = 0 EP(l)(σ, s) = 0 Curv(σ) = 0 σ s = 0 For non-simply-connected manifolds there are topological issues.

14 IV.- NOETHER CONSERVATION LAW The EP operator EP(l)(σ, s) := div σ δl δσ P+ s takes values in g with ( ) δl P + s (η) = δ s ( ) δl, δ s δl δ s, η P/H along the section s, with (π H ) η = η P/H. The elements η(x) = [s(x), B(x)] G with B : M h satisfies η P/H Im s = 0, so that for these variations the EP equations are Like a conservation law... div σ δl δσ = 0.

15 Proposition The Noether conservation law d((j 1 s) i j 1 B Θ L) = 0 for the infinitesimal symmetries B, B h, along a critical (local) section s of P M projects to the equation div σ δl = 0, δσ s h where σ = q(j 1 s), s = [s] H and div σ is the covariant divergence operator defined by the connection σ. That is, we have the Euler-Poincaré equation of l evaluated on ker P + s. The conservation law of the H-symmetry is encapsulated in the Euler-Poincaré equations.

16 For symmetries of the reduced Lagrangian l: We say that a vector field Z X(C M (P/H)) is an infinitesimal symmetry of l if it is of the form Z = (X C, (π H ) X) with X autp, and Z (lv) = 0. Proposition If (σ, s) Γ(C) Γ(P/H) is a solution of the Euler-Poincaré equations + the compatibility conditions and Z = (X C, (π H ) X), X autp, is an infinitesimal symmetry of l, the following conservation holds δl div δσ, η v + X ((l (σ, s))v) = 0, where X is the projection of X onto M and η Γ( g) = gaup is the vertical part of X with respect to the connection σ.

17 V.- EXAMPLE: MECHANICS Now M = R and P = R G. We have L : R T G R invariant with respect to H G. The Euler-Poincaré identification (time independent) (T G)/H = T R g (G/H) = g (G/H) Whereas the Lagrange-Poincaré identification is (T G)/H T (G/H) h h being the adjoint bundle of G G/H. Hint: Use the identification T (G/H) = G/H m where g h m is a reductive decomposition (through a connection).

18 For instance, we consider G = SO(3), H = SO(2) and the Lagrangian of the heavy top: L(R, Ṙ) = 1 2 K Ṙ, Ṙ mgre 3 χ In the reduced space g (G/H) = so(3) (SO(3)/SO(2)) we take coordinates Ω : R so(3) = R 3, Ω = R 1 R, Λ : R SO(3)/SO(2) = S 2, Λ = R e 3. The reduced Lagrangian is The operator P + Λ is l(ω, Λ) = 1 IΩ Ω mgλ χ 2 P + Λ : C (R, T S 2 ) C (R, so(3) ) Υ Λ Υ.

19 The Euler-Poincaré equation is EP(l)(Ω, Λ) = div Ω δl δω P+ Λ ( ) δl δλ = d (IΩ) + Ω IΩ + mgλ χ = 0. dt The compatibility condition CurvΩ = 0 is trivial. The condition Ω Λ = 0 reads dλ dt + Ω Λ = 0. These are the classical equations of the reduced heavy top through semidirect product reduction. Consider T G (G/H) and the Lagrangian Then ˆL is G invariant. L( ) = ˆL(, [e] H ) ˆL(v, [g] H ) = L(g 1 v) (T G (G/H))/G = g (G/H).

20 Noether of the H = SO(2)-symmetry The (encapsulated) equation applied to kerp + Λ = {λλ} gives equivalent to IΩ Λ d (IΩ) Λ + Ω IΩ Λ = 0 dt (the vertical angular momentum) constant The field X = t is an infinitesimal symmetry of the reduced Lagrangian. The conservation law for l is d dt IΩ Ω+ d ( ) 1 IΩ Ω mgλ χ = d dt 2 dt conservation of the total energy. ( ) 1 IΩ Ω + mgλ χ 2 = 0

21 VI.- REDUCTION IN AFFINE PRINCIPAL BUNDLES Given a principal bundle P M and a linear action of G on a vector space V we consider and the associated bundle G aff = GSV E = P G V = (P V )/G. The so-called induced affine bundle is P aff = P M E Think of P = LF (M), G = Gl(n; R) and V = R n. Then G aff = Aff(R n ), P aff = AffF (M)

22 Let L : J 1 P aff R be a Lagrangian invariant with respect to G G aff. The reduced Lagrangian is defined in the reduced space Due to the decomposition (J 1 P aff )/G = C aff P aff /G g aff = g V connections in P aff split in two factor. Then In addition P aff /G = E. C aff = C (T M E) Then with variables l : C (T M E) E R (σ, h; s).

23 The Euler-Poincaré equation (in ( g aff ) = g E ) are div σ δl δσ + h δl δh δl δ s s = 0 div σ δl δh δl δ s = 0 The compatibility conditions are σ aff s aff = σ s + h = 0, Curvσ aff = (Curvσ, σ h) = (0, 0) which, taking into account that σ σ = Curvσ are equivalent to σ s = h, Curvσ = 0.

24 VII.- EXAMPLE: MOLECULAR STRANDS We consider P = R 2 SO(3) R 2 and V = R 3. Then E = R 2 R 3 R 2, P aff = R 2 SE(3) R 2, with SE(3) = SO(3)SR 3. Let L : J 1 P aff R invariant with respect to SO(3).

25 The Euler-Poincaré identification gives a reduced Lagrangian defined in C (T M E) E = (T R 2 so(3)) (T R 2 R 3 ) (R 2 R 3 ) with variables σ = Ωdx + ωdt h = Γdx + γdt s(x, t) = ((x, t), ρ(x, t)) with the coordinates in R 2 are (x, t) and the functions Ω, ω, Γ, γ, ρ : R 2 R 3.

26 Euler-Poincaré equation takes the form ( x δl δω + Ω δl ) δω + ( t ( x δl δω + ω δl ) + Γ δl δω δγ + γ δl δγ ρ δl δρ = 0, δl δγ + Ω δl ) ( δl + δγ t δγ + ω δl ) δl δγ δρ = 0. The compatibility equations are ρ x + Ω ρ + Γ = 0, ρ t + ω ρ + γ = 0, Ω t ω x + ω Ω = 0.

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