The geometry of Euler s equations. lecture 2

Transcription

1 The geometry of Euler s equations lecture 2

2 A Lie group G as a configuration space subgroup of N N matrices for some N G finite dimensional Lie group we can think of it as a matrix group Notation: a,b, elements of the group Basic point: we can move objects around (vector fields, forms, etc.) by left (or right) multiplication. ξ=d/dt a(t) t=0, with a(0)=a tangent vector at a a left mult. by b ba tangent vector at ba b ξ = d/dt (b a(t)) t=0 for the right multiplication we can define ξ ξ b in a similar way co vectors ( forms): the dual of (ξ b ξ ) moves forms from T* ba G to T* a G α T* ba G b* α T* a G (abusing the notation: b* is not the adjoint matrix)

3 The tangent space at the unit element: g (the Lie algebra of G) The adjoint action of G on g: ξ a ξ a 1 =Ad(a) ξ Lie bracket on g: Let η g and ḃ(t) t=0 = η. Then [η, ξ] = d/dt t=0 Ad (b(t)) ( ξ Structural constants of g: e 1,...,e n basis of g, [e i,e j ]=c k ij e k the group operation induces the Lie algebra structure Jacobi identity [ξ, [η, ζ]] + [η, [ζ, ξ]] + [ζ, [ξ, η]] = 0

4 Coordinates on TG and T*G: move the basis e 1,, e n by the left multiplication to each point of the group get a frame of left invariant vector fields, still denoted e 1,, e n Identify TG with G g or with G R n by using the frame. (a, ξ) G g a ξ T a G or ( a ; ξ 1,...,ξ n ) a (ξ i e i ) T a G

5 Similarly, T G is identified with G g or with G R n (a, α) G g (a ) 1 α T a G or ( a ; y 1,...,y n ) G R n (a ) 1 (y j e j ) T a G dual basis to e 1,,e n The coordinates y 1,...,y n can also be thought of as functions on T G. Prolongation of the action of G on itself by the left multiplication to T G: b ( a ; y 1,...,y n ) ( ba, y 1,...,y n ) Notethat that y 1,, y n considered asfunctions on T*M areinvariantunder (the prolongation of) the left multiplication. This is more or less by definition.

6 C (G \ T*G) are exactly the functions of y 1,, y n. In fact, in this case Y = G \ T*G can be identified with R n or g Interpretation: T*G we follow a(t) and d/dt y(t) (momentum) Y= G \ T*G reduced phase space: we forget a(t) and follow only d/dt y(t) The functions y 1,, y n provide coordinates on Y (which are global) C (Y) inherits a Poisson bracket from T*M. What is the bracket? equations for y: d/dt y(t) = {H,y} enough to calculate l {y Thi d li l k h i i i, y j } Calculation: {y = k I, y j } c ij y k, This needs a little work, the main point is to use the formula de* j (e k, e l )= e* j ([e k, e l ]) for the invariant forms/fields see next hamiltonian HH() H=H(y) where c k ij are the structure constants of the Lie algebra g

7 Canonical form of T*G e_1*,, e_n* frame of the left invariant forms, dual to the frame e_1,,e_n Recall the natural coordinates on T*G: ( a ; y 1,, y n ) G R n y j e* j The canonical 1 form on T*G: α = y j e* j The symplectic form: ω = dα = dy j Æ e* j + y j de* j = = dy Æe* j j j y j c kle* k Æ e* l where we used the Cartan formula dα(ξ,η) = ξ α (η) η α (ξ) α([ξ,η]) with α = e* p, ξ=e q, η=e r = 0 in our case [e p, e q ] = c r pq e r

8 In the local frame in T*G given by y 1,,y n, e 1,,e n the form omega is given by the matrix µ 0 I I C(y) with notation y k c k ij C(y) The inverse ω rs of this (anti symmetric) matrix is given by µ C(y) I I 0 The Poisson bracket is (in our conventions) {f,g} = ω rs f s g r In particular, {y i, y j } = c k ij y k,

9 Remark: Note that the group G was any Lie group. We have shown: If G is a Lie group and g is its Lie algebra with structural constants c k ij, then on the dual g of g the formula defines a Poisson bracket on g. {y = k i, y j } cijy k This was already known to S. Lie, but he did explore the implications. In the 1960s this structure was used by A. A. Kirillov to obtain important results in representations of nilpotent groups, and to develop his method of orbits.

10 b G the left shifts a ba extend to a symplectic diffeom. of T*G: represent the form y i e* i at a in the coordinates ( a, y 1,,y n ): b (a, y 1,, y n ) = (ba, y 1,, y n ) the infinitesimal i i version of these df deformations: ξ g generates an infinitesimal symplectic deformation of T G ba=(1+² ξ) a y does not change, since the coordinate functions y i are invariant under the left shifts ( a, y ) ( a+² ξ a, y ) Recall: Infinitesimal sympl. defs. functions (perhaps modulo corrections)

11 What is the generating function f for ( a, y ) ( a+² ξ a, y )? In the coordinates of lecture 1 it would be f = p i ξ i We need to express this in the presently used coordinate frame y 1,, y n, e 1,, e n coordinates of the infinitesimal deformation in this frame: first coordinate no shift in y ( 0,, 0, [Ad (a 1 ) ξ ] 1,, [Ad (a 1 ) ξ ] n ) arises from ξ a = a a 1 ξ a = a Ad (a 1 )ξ need to use this form because our coordinate frame is left invariant (not right invariant). invariant).

12 Recall Noether s Theorem: recall f = p i ξ i of lecture 1 in coordinates The generating function of the infinitesimal symplectic transformation above will be f (a,y) = < y, Ad(a 1 ) ξ > = < Ad(a 1 )* y, ξ > Moreover, f(a,y) will be conserved for any Hamiltonian depending only on y Exercise: check by hand that { f, y i } = 0 for each i Definition: M(a,y) = Ad(a 1 )* y g Example: rigid body rotation y momentum in the coordinate frame moving with the body Ad(a 1 )y momentum in the coordinate frame fixed in space is called the moment map. For any Hamiltonian depending only on y the evolution preserves M, or dm/dt = 0. This is because the ξ above can be taken as any element of g

13 determined by C i (y)=c i, C i the Casimir functions, {C i, y j }=0 for each j=1,,n Symplectic leaves of the Poisson manifold g (a connected tdg) y A subspace through y generated by all possible vectors dy/dt={h,y}, as H runs through all H=H(y) is contained in the tangent space to the orbit O y = {Ad*(a)y, a G} (because of the conservation of M for any such H, for example). orbit O y Vice versa, any vector in T y O y can be obtained in this way. Hence: Symplectic leaves orbits (See A.A.KirillovA Kirillov s book for more and implications to representations.) Hence the evolution on g given by H(y) and {, } really describes a family of hamiltonian systems (parametrized by the orbits)

14 Example: non degenerate stationary points are (locally) parametrized by the orbits: manifold of stationary points (transversal to the orbit foliation) Non degenerate critical point ofh ononeone orbits implies critical points in neighboring orbits orbits

15 Example: the simplest non commutative group G (Commutative G leads to {y i, y j }=0) µ a b affine transformation of R G = { A = ; a > 0,, b R} } ( p ) 0 1 µ x y g = { X = 0 0 ; x, y R} µ p q g = { P = 0 0 ; p, q R} the group of orientation preserving (maps of the form x ax+b ) with a>0 adjoint orbits y x <P,X>= px + qy co adjoint orbits q Ad(A) µ µ b Ad(A) b a 0 a p

16 Left invariant geodesics G b e 2 =a / b e 2 * = db/a given by Hamiltonian H=H(p,q)=(p 2 +q 2 )/2 {p,q} = q note the change of notation both p and q are components ofthe momentum e 1 =a / a e 1 *=da/a e 1 dp/dt = {H,p} = q 2 dq/dt = {H,q} = pq 1 a Momentum conservation: geodesics da/dt = pa, db/dt = qa Example of solutions: a(t) = a 0 tanh(t) b(t) = a 0 /cosh(t) G Poincare model of the hyperbolic plane

17 Remark: Direct Lagrangian approach (see also V.I.Arnold s book) L left invariantlagrangian on TG a(t) curve in G v(t) = a 1 1 da/dt velocity in the coordinates of the left invariant frame t1 t2 L(v(t)) dt action a ² (t) = a(t)b ² (t) with b ² 1+²ξ nearby trajectory v ² = a ² 1 da ² /dt = Ad (b ² 1 ) v + b ² 1 db ² /dt velocity of the nearby trajectory v ² = v + ² ( [ξ,v] + dξ/dt ) + O(² 2 ) L(v ² ) = L(v) + ²L v( [ξ,v] ξ + dξ/dt) ξ + O(² 2 ) t1 t2 (L(v ² ) L(v)) dt= ² t1 t2( <dl v /dt, ξ>+<l v, [ξ,v]>) dt + O(² 2 ) F(v) defined by <F(v),ξ> = <L v, [ξ,v]>, or (abusing notation), F(v) = Ad*(v) L v Equation for v: dl v /dt = F(v) (does not involve a(t); recovering a: da/dt= a(t) v(t))

18 Other calculable geodesics for left invariant metrics: Sl(2,R) motion of a rigid body in the hyperbolic plane H 3 tree dimensional Heisenberg group, Caratheodory metrics S 3 or SO(3) three dimensional rigid body (Euler s equations for rotating bodies) and more. We expect: 3d group with a left invariant hamiltonian co adjoint orbits have dimension at most 2 equations for geodesics solvable by quadratures On the other hand: 6d groups (such as Sl(2,C) or SO(4)) will often have 4d orbits potential for chaos in the reduced equations Special Hamiltonians can still give integrability: Kovalevskaya s top, n dim rigid body in Euclidean space (Manakov)

19 n unit normal Ideal Incompressible Fluids u Configuration space: G=Diff GDiff( ) 0 volume preserving diffeomorphisms of (group under map composition) Motion the fluid: curves in G, t φ t φ t (x) t φ t (x) trajectory of a particle x Symmetries: φ t is a solution, ψ G φ t ψ is a solution particle re labeling

20 Hamilton s principle: the actual fluid motions solutions n are extremals of Z t2 Z t 1 φt (x) 2 dx dt among curves φ t in G with φ t 1 and φ t 2 fixed Ω u φ t (x) Thisleads to Euler s equations for u(x, t) = φ( φ 1 (x, t),t) we are now using right shifts, rather then the left shifts. x It s not hard to get the equation by direct calculation but we will follow the Hamiltonian approach, which also immediatelly gives the conservation laws coming from the invariance by G ( particle re labelings )

21 The Lie algebra of GDiff( ) G=Diff 0 and its dual U = T identity G the Lie algebra of G div free vector fields in tangent to the boundary u,v U [u,v] the usual Lie bracket of u,v Adjoint action: Ad(φ) u = φ # u (φ # u)(x) = Dφ (φ 1 (x)) u(φ 1 (x)) (is exactly the bracket induced from the group) u φ 1 (x) φ x φ # u The dual U* linear functionals on U can be obtained most naturally by where a is viewed as a 1 form. u a i (x) u i (x) dx = <a,u>

22 The co adjoint action (of G=Diff 0 ( ) on U*) <a, φ # u> = <φ*a, u> φ*a the usual pull back Ad*(φ) a = φ*a [ φ*a (φ(x)) ] i = a j (x) φ j,i(x) So it seems natural to identify U* with one forms, but the problem is that the correspondence is not one to one: a = df <a,u> = 0 for each u U: fu = f (div u) = 0 To get a one to one correspondence replace 1 forms by (1 forms)/(exact differentials df)

23 1 forms / differentials df d via the volume element 2 forms vector fields a dx i a ij dx j Ædx i ω i / x i with ω i =² ijk i i,j /, a kj k,j Or, in the vector calculus notation: a (a 1, a 2, a 3 ) ω = curl a Action of Diff 0 ( ) in the a coordinates In the ω coordinates Ad*(φ) a = φ*a Ad*(φ)ω = φ # ω

24 Duality in between U and U* in the ω coordinates on U* a a i dx i da curl a ω ω i / x i vector potential of u, analogue of the 2d stream function u U, u=curl Ψ, div Ψ = 0 in, ΨÆn= 0 on when is topologically trivial) for simplicity it you can also think of = R 3 a u dx = ω Ψ dx (Check that the boundary term vanishes due to the boundary conditions) In these coordinates the basic variable is ω = ω(x,t) ( vorticity ) The vector fields u U are identified with their vector potentials Ψ The Hamiltonian = kinetic energy : ω U*, curl u = ω, div u = 0, un =0, Ψ= ω, div Ψ = 0, ΨÆn = 0 H = H(ω) = ½ u 2 dx = ½ωΨ dx

25 Poisson bracket Lie bracket in U in terms of the vector potentials Ψ u = curl Ψ, v = curl Φ U, w=[u,v], w=curl Θ w = [u,v] = u v v u= curl (uæv) (we used div u = div v = 0) Θ = curl Ψ Æcurl Φ Ψ and Φ can be considered aslinearfunction on U* l Ψ : ω ωψ and l Φ : ω ωφ Poisson bracket: {l Ψ, l Φ } = l Θ = l curlψ Æcurl Φ variational derivative, expressing F and G using the duality with U (need some smothness) general functionals: { F, G } (ω) = ω ( curl (δf/δω) Æ curl(δg/δω) ) dx

26 Example: H(ω) = ½ ω Ψ ( the Euler Lagrangian introduced earlier) δh/δω = Ψ (with Ψ = ω + boundary cond.) ω = curl u, div u = 0 + boundary cond., u = curl Ψ d/dt l Φ (ω) = ω t Φ = { H, l Φ } (ω) = ω( curl Ψ Æ curl Φ) = curl(ωæu) Φ evolution of the coordinate given by Φ Φ is arbitrary div free with ΦÆn= 0 at the boundary Analogous to the finite dimensional equation dy i /dt={h,y i } ω t + curl (uæω) = 0 or ω t + [ u, ω ] = 0 t vorticity formulation of Euler s equations Noether s theorem (conservation of the moment function) Ad*(φ t ) 1 ω(t) = ω(0) or ω(t) = φ # ω(0) Helmholtz s law vorticity moves with the flow

27 Consequences of Helmholtz s law Vortex filament at initial time curve γ velocity u ω γ u i dx i is conserved (γ moving with the flow) section S S ωn is conserved (S moving with the flow)

28 folding necessary not to increase energy volume ofthe filament must be preserved velocity near the filament increases eventually it will presumably become very complicated ω vorticity is stretched must become very thin at many places (preservation of volume) high velocities, a lot of folding necessary so that energy is not increased

29 A benefit of the co adjoint orbit approach: some reduced systems (such as point vortices, vortex fillaments, ) come with natural hamiltonian structure. 1) 2d.. sets of a given number of points can be moved around by Diff 0 ( ) finite dimensional orbits finite dimensional Hamiltonian system point vortices (Hamiltonian can be taken from the original Euler s eq. if we remove the infinite self energy of each vortex. 2) 3d.. curves can be moved around by Diff 0 ( ) get a natural symplectic structure on the manifold of curves for certain weak filaments energy length curves with the sympl. structure, Hamiltonian = length, flow of curves by binormal curvature

30 Geometric picture of the steady states for 2d Euler: manifold of stationary points (transversal to the orbit foliation) Non degenerate critical point of H on one orbits implies critical points in neighboring orbits orbits Can be established rigorously in 2d under some (reasonable) assumptions (A. Choffrut, V.S.)

31 References: V.I.Arnold: Mathematical Methods of Classical Mechanics V.I.Arnold, B. A. Khesin: Topological Fluid Mechanics A.A.Kirillov: Lectures on the orbit method J.Marsden, A.Weinstein: Coadjoint orbits and Clebsch variables for incompressible fluids, Physica D 7 (1983), no. 1 3,