《微分几何：辛几何引论》课程教学资源（英文讲义）Lec06 GEOMETRY OF HAMILTONIAN SYSTEMS

LECTURE6:GEOMETRYOFHAMILTONIANSYSTEMSCONTENTS11.Geometry of Hamiltonian vector fields42.ThePoisson structure73.Completely integrableHamiltonian systems1.GEOMETRYOFHAMILTONIANVECTORFIELDS Symplectic vectorfield v.s.Hamiltonian vector field.Let (M,w)be a symplectic manifold. Then the non-degeneracy of w gives us alinear isomorphism between vector fields and 1-forms on M:W:Vect(M)→2(M),三→=w.Recall that a vector field E on M is called symplectic if t=w is a closed 1-form onM, and it is called Hamiltonian if tgw is an exact 1-form. So if we denote the set ofsymplectic vector fields by Vect(M,w) and the set of Hamiltonian vector fields byVectHam(M,w), then the restriction of gives us linear isomorphismsw : Vect(M,w) → Z'(M)andw : VectHam(M,w) → B'(M),where z(M) is the space of closed 1-forms on M, and B'(M) the space of exact1-forms.As a consequence, the quotient Vect(M,w)/VectHam(M,w) is just the firstdeRham cohomology group H'(M), and we have an exact sequence of vector spaces(1)0 → Vect(M,w) → VectHam(M,w) →H'(M) →0.In particular, we seeProposition 1.1. If H'(M) = [0), then every symplectic vector field on M isHamiltonian1

LECTURE 6: GEOMETRY OF HAMILTONIAN SYSTEMS Contents 1. Geometry of Hamiltonian vector fields 1 2. The Poisson structure 4 3. Completely integrable Hamiltonian systems 7 1. Geometry of Hamiltonian vector fields ¶ Symplectic vector field v.s. Hamiltonian vector field. Let (M, ω) be a symplectic manifold. Then the non-degeneracy of ω gives us a linear isomorphism between vector fields and 1-forms on M: ω˜ : Vect(M) → Ω 1 (M), Ξ 7→ ιΞω. Recall that a vector field Ξ on M is called symplectic if ιΞω is a closed 1-form on M, and it is called Hamiltonian if ιΞω is an exact 1-form. So if we denote the set of symplectic vector fields by Vect(M, ω) and the set of Hamiltonian vector fields by VectHam(M, ω), then the restriction of ˜ω gives us linear isomorphisms ω˜ : Vect(M, ω) → Z 1 (M) and ω˜ : VectHam(M, ω) → B 1 (M), where Z 1 (M) is the space of closed 1-forms on M, and B1 (M) the space of exact 1-forms. As a consequence, the quotient Vect(M, ω)/VectHam(M, ω) is just the first deRham cohomology group H1 (M), and we have an exact sequence of vector spaces (1) 0 → Vect(M, ω) → VectHam(M, ω) → H 1 (M) → 0. In particular, we see Proposition 1.1. If H1 (M) = {0}, then every symplectic vector field on M is Hamiltonian. 1

2LECTURE6:GEOMETRYOFHAMILTONIANSYSTEMS Smooth function v.s. Hamiltonian vector field.Recallweshowedinlecture4thatifE,.三aretwosvmplecticvectorfieldsonM, then [三i, 三2] is a Hamiltonian vector field on M. In fact, one has(2)(31,=2)W = -d(w(三1,三2),This implies[Vect(M,w), Vect(M,w)] C VectHam(M,w),In other words, as a Lie algebra, Vect Ham(M,w) is an ideal of Vect(M,w). (This isof course related to the fact that Ham(M,w) is an ideal of the group Symp(M,w).)So the short exact sequence (1) is in fact a short exact sequence of Lie algebras, ifwe endowed with H'(M) the trivial Lie bracket.Modulo locally constant functions,Hamiltonian vector fields are in one-to-onecorrespondence with smooth functions on M,in the following sense:So if E ishamiltonian, then there exists a smooth function f e Co(M) so that tw =df:Conversely, since w is non-degenerate, for any f e Co(M), there is a unique vectorfield 三f on M so thatl=rw=df.The space of locally constant functions on M is the Oth de Rham cohomology groupH'(M).So we get another short exact sequence of vector spaces(3)0 -→ H°(M) →C(M) → VectHam(M,w) → 0.We shall see soon that this is again a short exact sequence of Lie algebras, providedwedefineasuitableLiealgebrastructureonCoo(M)(whichis,however,differentfrom the Poisson bracket we will define below by a negative sign).Recall: The vector field Ef is called the Hamiltonian vector field associated totheHamiltonian function f.The flow generated by Eyis called the Hamiltonianflowassociated to f.Erample. Consider M =R2n with canonical symplectic form w =de;Adei. Thenafde(ofdridf =>OriDEItfollowsofaaf a(4)Ef=os;OciOriosSo the integral curve of 三, is a curve (r(t),s(t)) satisfyingaf(t) =f(t) =DOr,This set of equations is known as Hamiltonian equations

2 LECTURE 6: GEOMETRY OF HAMILTONIAN SYSTEMS ¶ Smooth function v.s. Hamiltonian vector field. Recall we showed in lecture 4 that if Ξ1, Ξ2 are two symplectic vector fields on M, then [Ξ1, Ξ2] is a Hamiltonian vector field on M. In fact, one has (2) ι[Ξ1,Ξ2]ω = −d(ω(Ξ1, Ξ2)). This implies [Vect(M, ω), Vect(M, ω)] ⊂ VectHam(M, ω). In other words, as a Lie algebra, VectHam(M, ω) is an ideal of Vect(M, ω). (This is of course related to the fact that Ham(M, ω) is an ideal of the group Symp(M, ω).) So the short exact sequence (1) is in fact a short exact sequence of Lie algebras, if we endowed with H1 (M) the trivial Lie bracket. Modulo locally constant functions, Hamiltonian vector fields are in one-to-one correspondence with smooth functions on M, in the following sense: So if Ξ is hamiltonian, then there exists a smooth function f ∈ C ∞(M) so that ιΞω = df; Conversely, since ω is non-degenerate, for any f ∈ C ∞(M), there is a unique vector field Ξf on M so that ιΞfω = df. The space of locally constant functions on M is the 0th de Rham cohomology group H0 (M). So we get another short exact sequence of vector spaces (3) 0 → H 0 (M) → C ∞(M) → VectHam(M, ω) → 0. We shall see soon that this is again a short exact sequence of Lie algebras, provided we define a suitable Lie algebra structure on C ∞(M) (which is, however, different from the Poisson bracket we will define below by a negative sign). Recall: The vector field Ξf is called the Hamiltonian vector field associated to the Hamiltonian function f. The flow generated by Ξf is called the Hamiltonian flow associated to f. Example. Consider M = R 2n with canonical symplectic form ω = Pdxi∧dξi . Then df = X ∂f ∂xi dxi + ∂f ∂ξi dξi  . It follows (4) Ξf = X ∂f ∂ξi ∂ ∂xi − ∂f ∂xi ∂ ∂ξi  So the integral curve of Ξf is a curve (x(t), ξ(t)) satisfying x˙(t) = ∂f ∂ξi , ˙ξ(t) = − ∂f ∂xi . This set of equations is known as Hamiltonian equations

3LECTURE6:GEOMETRYOFHAMILTONIANSYSTEMSRemark. Obviously the formula (4) also gives the local expression of Ef on an arbitrary symplectic manifold if one uses theDarboux coordinates.Thevector field 三fis also called the symplectic gradient of f. Gradient vector field v.s. Hamiltonian vector field.Back to the R2n example. Note that the usual gradient vector field of f is(af aof a)vf =>orOrosa)So the symplectic gradient and the usual gradient of f are related byJ(E) = Vf.where J is theusual complex structure on R2n, i.e.aga0ariThis observation is easily extended to Kahler manifolds, or more generally anysymplectic manifold M with compatible triple (w,J,g).In this theusual gradientvectorfield of f is thevectorfieldf onM sothatg(Vf, ) = df().Using the fact g(, ) = w(, J.) it is easy to seeProposition 1.2. Let M be a symplectic manifold with compatible triple (w, J,g).Then Vf = JEf.Proof. We havedf = g(Vf, ) =w(Vf, J.) =w(-JVf, ).口Itfollowsthat-JVf=-三f,i.e.Vf=J(E).So if one think of an almost complex structure on M as “rotation by 90 degreescouterclockwise", then the usual gradient vector field can be obtained from thesymplectic gradient vector fields via“rotation by 90degrees couterclockwise"!Symplectic form v.s.Hamiltonian vector field.Now suppose 三f is the Hamiltonian vector field associated with f.The followingproperties are easily seen from the definition. One should be aware of the use of thethree parts of the definition of a symplectic form.Proposition 1.3. Let (M,w) be a symplectic manifold, and f e Co(M)(1) C=,f = 0.(2) C=,w = 0.(3) IfE Symp(M,w), then 三* =p*三f

LECTURE 6: GEOMETRY OF HAMILTONIAN SYSTEMS 3 Remark. Obviously the formula (4) also gives the local expression of Ξf on an arbi￾trary symplectic manifold if one uses the Darboux coordinates. The vector field Ξf is also called the symplectic gradient of f. ¶ Gradient vector field v.s. Hamiltonian vector field. Back to the R 2n example. Note that the usual gradient vector field of f is ∇f = X ∂f ∂xi ∂ ∂xi + ∂f ∂ξi ∂ ∂ξi  . So the symplectic gradient and the usual gradient of f are related by J(Ξf ) = ∇f. where J is the usual complex structure on R 2n , i.e. J( ∂ ∂xi ) = ∂ ∂ξi , J( ∂ ∂ξi ) = − ∂ ∂xi . This observation is easily extended to K¨ahler manifolds, or more generally any symplectic manifold M with compatible triple (ω, J, g). In this the usual gradient vector field of f is the vector field ∇f on M so that g(∇f, ·) = df(·). Using the fact g(·, ·) = ω(·, J·) it is easy to see Proposition 1.2. Let M be a symplectic manifold with compatible triple (ω, J, g). Then ∇f = JΞf . Proof. We have df = g(∇f, ·) = ω(∇f, J·) = ω(−J∇f, ·). It follows that −J∇f = Ξf , i.e. ∇f = J(Ξf ). So if one think of an almost complex structure on M as “rotation by 90 degrees couterclockwise”, then the usual gradient vector field can be obtained from the symplectic gradient vector fields via “rotation by 90 degrees couterclockwise”! ¶ Symplectic form v.s. Hamiltonian vector field. Now suppose Ξf is the Hamiltonian vector field associated with f. The following properties are easily seen from the definition. One should be aware of the use of the three parts of the definition of a symplectic form. Proposition 1.3. Let (M, ω) be a symplectic manifold, and f ∈ C ∞(M). (1) LΞf f = 0. (2) LΞfω = 0. (3) If ϕ ∈ Symp(M, ω), then Ξϕ∗f = ϕ ∗Ξf .

4LECTURE6:GEOMETRYOFHAMILTONIANSYSTEMSProof. (1) follows from the skew-symmetry of w:Ca,f =ldf=w=0.(2) follows from the closeness of w:C=,w = di=jw = d(df) = 0.(3) follows from the non-degeneracy of w:l=W=d(0"f)=0df=0W=bp*,0'w=l0*w2.THEPOISSON STRUCTURE The Poisson bracket.Applying the identity (2) to Hamiltonian vector fields 三f and g, we get(5)[3f,三9] = =-w(=,3g)-Definition 2.1. For any f,g E Co(M), we call[f,g) =w(三f,三g)the Poisson bracket of f and gSo by definition,(f,g) = L=,w(Xg) =df(=g) = L=gf =三g(f)As a consequence we seeCorollary 2.2. (f,g] = 0 if and only if g is constant along the integral curves ofEf, i.e. the Hamiltonian vector field Ef is tangent to the level sets g = c.In particular, the hamiltonian vector field E, is always tangent to the level setsf=c.The poisson bracket also behaves well under symplectomorphisms:Proposition 2.3. If E Symp(M,w), then (*f,*g) =*[f,g).Proof. Using the fact 三*f = p*f we get[p*f, Φ*g) =w(三p*f,三p*g) =w(β*f,Φ*三g) = p*(w(三f,三g)) = p*[f,g)口

4 LECTURE 6: GEOMETRY OF HAMILTONIAN SYSTEMS Proof. (1) follows from the skew-symmetry of ω: LΞf f = ιΞf df = ιΞf ιΞfω = 0. (2) follows from the closeness of ω: LΞfω = dιΞfω = d(df) = 0. (3) follows from the non-degeneracy of ω: ιΞϕ∗fω = d(ϕ ∗ f) = ϕ ∗ df = ϕ ∗ ιΞfω = ιϕ∗Ξfϕ ∗ω = ιϕ∗Ξfω. 2. The Poisson structure ¶ The Poisson bracket. Applying the identity (2) to Hamiltonian vector fields Ξf and Ξg, we get (5) [Ξf , Ξg] = Ξ−ω(Ξf ,Ξg) . Definition 2.1. For any f, g ∈ C ∞(M), we call {f, g} = ω(Ξf , Ξg) the Poisson bracket of f and g. So by definition, {f, g} = ιΞfω(Xg) = df(Ξg) = LΞg f = Ξg(f). As a consequence we see Corollary 2.2. {f, g} = 0 if and only if g is constant along the integral curves of Ξf , i.e. the Hamiltonian vector field Ξf is tangent to the level sets g = c. In particular, the hamiltonian vector field Ξf is always tangent to the level sets f = c. The poisson bracket also behaves well under symplectomorphisms: Proposition 2.3. If ϕ ∈ Symp(M, ω), then {ϕ ∗ f, ϕ∗ g} = ϕ ∗{f, g}. Proof. Using the fact Ξϕ∗f = ϕ ∗Ξf we get {ϕ ∗ f, ϕ∗ g} = ω(Ξϕ∗f , Ξϕ∗g) = ω(ϕ ∗Ξf , ϕ∗Ξg) = ϕ ∗ (ω(Ξf , Ξg)) = ϕ ∗ {f, g}.

5LECTURE6:GEOMETRYOFHAMILTONIANSYSTEMS The Poisson bracket in local coordinates.In local Darboux coordinates, (4) gives(of ogof g)(f,g] =OroaoIn particular, the Poisson bracket of the Darboux coordinate functions are simple:[,c] =[,]=0,[]=uConversely, we haveProposition 2.4. A coordinate system [r1,, n,Si,...,En) on M is a Darbourcoordinate system if and only if they satisfies[r,,] =[,] =0,[c,] =oProof. We can rewrite these set of equations asd(三r)=0,d(三)=0,dr(三)=-de(三)=It followsaaEa =EeaOriand thusaa0000=0,=SiiwrEEOrOri口T The Poisson bracket as a Lie bracket.Using Poisson bracket we can rewrite the equation (5) as(6)E(f.g) = -[Ef,Eg].In particular, if (f, g] = o, then [Ef,=g] = o, thus the Hamiltonian flows of f andg commute.Theorem 2.5. Let (M,w) be a symplectic manifold.(1) The Poisson bracket [, ↓ is a Lie algebra structure on Co(M).(2) The mapCo(M)→ VectHam(M,w), f →Efis a Lie algebra anti-homomorphism.Proof. (1) Obviously (, J is bilinear and anti-symmetric. To show it is a Lie algebrastructure it remains to check the Jacobi identity:0 =(C=,)(三g,三h)=(dt=,)(三g,三h)=三g(w(三f,三h)) - 三h(w(三f,g)) -w(三f,[Eg,三hl)={{f,h),g) -{{(f,g),h) +w(三f,三(gh))={(f,h),g) +{(g, f],h) +{(h,g), f]

LECTURE 6: GEOMETRY OF HAMILTONIAN SYSTEMS 5 ¶ The Poisson bracket in local coordinates. In local Darboux coordinates, (4) gives {f, g} = X ∂f ∂xi ∂g ∂ξi − ∂f ∂ξi ∂g ∂xi  . In particular, the Poisson bracket of the Darboux coordinate functions are simple: {xi , xj} = {ξi , ξj} = 0, {xi , ξj} = δij . Conversely, we have Proposition 2.4. A coordinate system {x1, · · · , xn, ξ1, · · · , ξn} on M is a Darboux coordinate system if and only if they satisfies {xi , xj} = {ξi , ξj} = 0, {xi , ξj} = δij . Proof. We can rewrite these set of equations as dxi(Ξxj ) = 0, dξi(Ξξj ) = 0, dxi(Ξξj ) = −dξi(Ξxi ) = δij . It follows Ξxi = ∂ ∂ξi , Ξξi = − ∂ ∂xi and thus ω( ∂ ∂xi , ∂ ∂xj ) = ω( ∂ ∂ξi , ∂ ∂ξj ) = 0, ω( ∂ ∂xi , ∂ ∂ξj ) = δij . ¶ The Poisson bracket as a Lie bracket. Using Poisson bracket we can rewrite the equation (5) as (6) Ξ{f,g} = −[Ξf , Ξg]. In particular, if {f, g} = 0, then [Ξf , Ξg] = 0, thus the Hamiltonian flows of f and g commute. Theorem 2.5. Let (M, ω) be a symplectic manifold. (1) The Poisson bracket {·, ·} is a Lie algebra structure on C ∞(M). (2) The map C ∞(M) → VectHam(M, ω), f 7→ Ξf is a Lie algebra anti-homomorphism. Proof. (1) Obviously {·, ·} is bilinear and anti-symmetric. To show it is a Lie algebra structure it remains to check the Jacobi identity: 0 = (LΞfω)(Ξg, Ξh) = (dιΞf )(Ξg, Ξh) = Ξg(ω(Ξf , Ξh)) − Ξh(ω(Ξf , Ξg)) − ω(Ξf , [Ξg, Ξh]) = {{f, h}, g} − {{f, g}, h} + ω(Ξf , Ξ{g,h}) = {{f, h}, g} + {{g, f}, h} + {{h, g}, f}.