A NOTE ON THE NORMALIZATION OF ...

JOURNAL OF GEOMETRIC MECHANICS c

American Institute of Mathematical Sciences Volume 10, Number 2, June 2018

doi:10.3934/jgm.2018008 pp. 209–215

A NOTE ON THE NORMALIZATION OF GENERATING FUNCTIONS

Simone Vazzoler Previnet S.p.A. Via E. Forlanini, 24 Preganziol (TV), Italy

(Communicated by Manuel de Le´on) Abstract. In the present note, I will propose some insights on the normalization of generating functions for Lagrangian submanifolds. From the literature (see, for example [4], [6], [7], [3] and [1]), it is clear that a problem exists concerning the nonuniqueness of generating functions and, in particular, of the generating functions quadratic at infinity (GFQI). This problem can be avoided introducing a normalization on the whole set of generating functions that will allow us to (i) choose an unique GFQI for Lagrangian submanifolds of the form ϕ(L), where L is a Lagrangian submanifold and ϕ is an Hamiltonian isotopy; (ii) compare the critical values c(α, S1 ) and c(α, S2 ) of two GFQI generating the submanifolds, ϕ1 (L) and ϕ2 (L), where ϕ1 and ϕ2 are Hamiltonian isotopies relative to two Hamiltonians H1 and H2 , respectively.

1. Introduction. Let M be a path connected compact manifold; consider its cotangent bundle T ∗ M = {(q, p) | p ∈ Tq∗ M }, endowed with its canonical symplectic ∗ form ω = dp ∧ dq, meaning that in local Pn coordinates (q1 , . . . , qn , p1 , . . . , qn ) ∈ T M , the two form can be written as ω = j=1 dpj ∧dqj . It can be also useful to introduce Pn the Liouville form λ = pdq = j=1 pj dqj . Definition 1.1. (1) A Lagrangian submanifold L is a submanifold of dimension n such that ω vanishes on L. This is equivalent to the condition that the restriction of λ to L is closed. If, moreover, this restriction is exact, it can be said that the submanifold is an exact Lagrangian. (2) A generating function for a Lagrangian submanifold L ⊂ T ∗ M is a smooth function S : M × Rk → R such that (i) the map (q, η) 7→ ∂η S(q, η) has zero as regular value; (ii) L = {(q, ∂x S(q, η)) | ∂η S(q, η) = 0}. (3) A generating function is quadratic at infinity (i.e. GFQI) if there is a nondegenerate quadratic form Q such that S(q, η) = Q(η) for |η| large enough, uniformly in q ∈ M . Three standard operations for generating functions preserve the manifold L: e η) = S(q, ϕ(q, η)), where (q, η) 7→ (q, ϕ(q, η)) is a fiber preserving diffeo(a) S(q, morphism from M × Rk to itself; 2010 Mathematics Subject Classification. Primary: 53D05; Secondary: 53D12. Key words and phrases. Symplectic geometry, Lagrangian submanifolds, generating functions.

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e η, ξ) = S(q, η) + F (ξ), where F is a non degenerate quadratic form on Rl (b) S(q, and e η) = S(q, η) + C, where C ∈ R is a constant. (c) S(q, Two generating functions S1 and S2 are equivalent if there is a sequence of the three operations (a), (b) and (c) that transforms S1 in S2 (or vice-versa). In what follows, I will consider Lagrangian submanifolds L Hamiltonian isotopic to the zero section OM (i.e. for which there exists a Hamiltonian H whose flow ϕt is such that ϕ1 (OM ) = L). The main theorem that ensures uniqueness of generating function for this particular class of Lagrangian submanifolds (modulo the three operations above) is the following. Theorem 1.2 ([5], [6], [4]). If L is Hamiltonian isotopic to the zero section OM , then any two GFQI for L are equivalent. Let S λ = {(q, η) | S(q, η) ≤ λ}, let E ± be the positive and negative eigenspaces of Q, and let D± be large discs in E ± . Thus, for c large enough, S ±c = M × Q±c will give us H ∗ (S c , S −c ) = H ∗ (M × Qc , M × Q−c ) = H ∗ (M ) ⊗ H ∗ (D− , ∂D− )

(1)

so that, to each cohomology class α ∈ H ∗ (M ), we may associate a class, that is the image of α by the K¨ unneth isomorphism, denoted by T α. To the class T α ∈ H ∗ (S c , S −c ), we may associate a minimax critical level c(α, S) = inf {T α ∈ / Ker(H ∗ (S c , S −c ) → H ∗ (S λ , S −c ))}. λ

(2)

Theorem 1.2 shows that given L, the generating function S is essentially unique up to adding a constant, and more precisely, up to a global shift; the numbers c(α, S) depend only on L, not on S and they can thus denoted by c(α, L). Here I will propose a method to fix the ambiguity in the selection of the generating function (in particular, I will find a normalization constant for point (c) above) that will allow a comparison of the critical values c(α, S1 ) and c(α, S2 ) of the two GFQI generating the submanifolds ϕ1 (L) and ϕ2 (L) (where ϕ1 and ϕ2 are Hamiltonian isotopies relative to two Hamiltonians H1 and H2 , respectively). 2. Critical points and action functional. I will use the following notations: with Sϕ(L) (q1 ; η), I denote a GFQI for the Lagrangian submanifold ϕ(L), where ϕ will be the time one flow of an Hamiltonian with compact support, and SL (q0 ; ξ) will identify a generating function for L. I will proceed by selecting a critical value, c = c(α, Sϕ(L) ), and corresponding to it we will find a set of critical points, Kc = {(q1∗ , η ∗ ) | q1∗ ∈ M, η ∗ ∈ Rk }. Those points are precisely the solutions of the following system of equations  ∂Sϕ(L) ∗ ∗   (q1 , η ) = 0    ∂q ∂Sϕ(L) ∗ ∗ (3) (q1 , η ) = 0   ∂η    Sϕ(L) (q1∗ , η ∗ ) = c. If we define K = ∪c∈R Kc as the set of all critical points for Sϕ(L) , then there exists a bijection ψ : K → ϕ(L) ∩ OM . In fact, this leaves


∂S n o ∂Sϕ(L) ϕ(L) ϕ(L) = (q1 , p1 ) ∈ T ∗ Q (q1 ; η) = 0, p1 = (q1 ; η) ∂η ∂q o 1 n ϕ(L) ∩ OM = (q1 , 0) (q1 , 0) ∈ ϕ(L) .

211

(4) (5)

Because Kc ⊂ K, then we can consider the restriction of the previous bijection to ψ(Kc ) = I ⊂ ϕ(L) ∩ OM . Introducing the action functional will allow us to normalize Sϕ(L) : Z 1 S(q0 , q1 ; γ, ξ) = SL (q0 ; ξ) + p(t)q(t) ˙ − H(t, q(t), p(t))dt (6) 0

where q0 , q1 ∈ M , γ(t) = (q(t), p(t)) is a curve in M and ξ ∈ Rk . The following lemma will be useful to show that the critical points of S are precisely the points of L that are sent on ϕ(L) ∩ OM by the flow ϕ. Lemma 2.1. The set of critical points PS = {(q0∗ , q1∗ ; γ ∗ , ξ ∗ )} of S is made by the points such that  ∂S L ∗ ∗  (q ; ξ ) = 0    ∂ξ 0       ∂SL (q ∗ ; ξ ∗ ) = p∗ 0 ∂q0 0 (7)  ∗ t  γ (t) = ϕ      q1∗ = πq (ϕ1 (q0∗ , p∗0 ))    ∗ p1 = 0. where γ ∗ (0) = (q0∗ , p∗0 ). Proof. The proof is quite straightforward. The set of the critical points of S is given by the points for which δS = 0. Because ∂S ∂SL L (q0 ; ξ)δξ+ (q0 ; ξ) − p0 δq0 + p1 δq1 + δS = ∂q0 ∂ξ Z 1 h i ∂H ∂H q(t) ˙ − (t, q(t), p(t)) δp(t) − p(t) ˙ + (t, q(t), p(t)) δq(t) dt, ∂p ∂q 0 one must set to zero all the terms of the last expression. To summarize, the bijections are as follows: (q1∗ ; η ∗ ) ∈ Kc O (q1∗ , 0) ∈ I o

(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) ∈ PS O / (q0∗ , p∗0 ) ∈ L o

(8)

/ (q0∗ ; ξ ∗ ) ∈ M × Rk

3. Main result. Once critical value c(α, Sϕ(L) ) has been choosen, a critical point of the action functional S can be uniquely associated to every critical point of Sϕ(L) . At this point, the following definition can be given. Definition 3.1. Let (q1∗ ; η ∗ ) ∈ Kc , a critical point such that Sϕ(L) (q1∗ ; η ∗ ) = c = c(α, Sϕ(L) ), and let (q0∗ , q1∗ ; γ ∗ , ξ ∗ ) be the correspondent critical point for the action functional S. The normalization constant relative to α is defined as lα = S(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) + Sϕ(L) (q1∗ ; η ∗ )

(9)

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Consequentely, Seϕ(L) will be the normalized generating function (with respect to α) for ϕ(L) defined by Seϕ(L) (q1 ; η) = Sϕ(L) (q1 ; η) − lα .

(10)

The normalization depends only on c(α, Sϕ(L) ) and not on the critical point of Kc . In fact, the following proposition holds true. Proposition 1. Let (q1∗ , η ∗ ), (¯ q1 , η¯) ∈ Kc (i.e. two critical points relative to the same critical value c = c(α, Sϕ(L) )). Define ∗ lα = S(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) + Sϕ(L) (q1∗ ; η ∗ )

(11)

¯lα = S(¯ ¯ + Sϕ(L) (¯ q0 , q¯1 ; γ¯ , ξ) q1 ; η¯),

(12)

∗ lα = ¯lα .

(13)

and then

Proof. Because the two critical points correspond to the same critical value c(α, Sϕ(L) ), we have Sϕ(L) (q1∗ ; η ∗ ) = Sϕ(L) (¯ q1 ; η¯) = c = c(α, Sϕ(L) ). To prove the proposition, we need to show that ¯ S(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) = S(¯ q0 , q¯1 ; γ¯ , ξ),

(14)

which is equivalent to SL (q0∗ ; η ∗ )

Z +

¯ + pdq − Hdt = SL (¯ q0 ; ξ)

γ

Z pdq − Hdt,

(15)

β

where γ = ϕt (q0∗ , p∗0 ) and β = ϕt (¯ q0 , p¯0 ). If we define [ Λ= ϕt (L)

(16)

t∈[0,1]

this will be a set contained in the preimage of zero for the extended Hamiltonian p0 + H. In particular we will have Λ ⊂ (p0 + H)−1 (0) ⊂ T ∗ (M × R),

(17)

and it is clear that we will have dim Λ = dim M + 1. We define P i = (q0∗ , p∗0 ), P f = (¯ q0 , p¯0 ), Qi = (q1∗ , p∗1 ) and Qf = (¯ q1 , p¯1 ). Moreover: SLi = SL (q0∗ , ξ ∗ ), SLf = f i ∗ ∗ ¯ S SL (¯ q0 ; ξ), q1 ; η¯), where ϕ(L) = Sϕ(L) (q1 ; η ) and Sϕ(L) = Sϕ(L) (¯ ∂SL ∗ ∗ (q ; ξ ); ∂q 0 ∂Sϕ(L) ∗ ∗ p∗1 = (q1 ; η ); ∂q p∗0 =

∂SL ¯ (¯ q0 ; ξ) ∂q ∂Sϕ(L) p¯1 = (¯ q1 ; η¯). ∂q

p¯0 =

(18) (19)

and p∗1 = p¯1 = 0. We choose a path α, inside L, joining P i and P f , and another path δ, inside ϕ(L), between Qi e Qf , in such a way that α + β = γ + δ (as in Figure 1). Λ is a Lagrangian submanifold with the Liouville 1-form λ = pdq − Hdt. Using the Poincar´e-Cartan Theorem, we must have Z Z pdq − Hdt = pdq − Hdt. (20) α+β

γ+δ


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γ Qi

Pi

α

δ

Pf Qf

β

ϕ

L

ϕ(L)

Figure 1. The two paths α + β and γ + δ

Working on the right hand side of the equation, we get Z Z Z pdq − Hdt = pdq − Hdt + pdq − Hdt γ+δ γ δ Z Z Z f i = pdq − Hdt + pdq = pdq − Hdt + Sϕ(L) − Sϕ(L) . γ

δ

γ

f i Because Sϕ(L) = Sϕ(L) = c, the last equation becomes Z Z pdq − Hdt = pdq − Hdt. γ+δ

(21)

γ

Using the same method for the left hand side of equation (20) we have Z Z pdq − Hdt = SLf − SLi + pdq − Hdt α+β

β

which is equivalent to SLf

−

SLf +

SLi

Z

Z pdq − Hdt =

+

pdq − Hdt

β

Z

pdq − Hdt = SLi +

β

γ

Z pdq − Hdt, γ

which is exactly what we needed to prove. To conclude, the definition 3.1 of renormalization of a generating function is well posed; it does not depend on the point but only on the critical value we have chosen. The generating functions renormalized with respect to a class α, have the following property: Seϕ(L) (q1∗ ; η ∗ ) = c(α, Seϕ(L) ) = −S(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) (22) We conclude this section with the following definition. Definition 3.2. A generating function Sϕ(L) is said to be normalized with respect to α if c(α, Sϕ(L) ) = −S(q0∗ , q1∗ ; γ ∗ , ξ ∗ ) (23)

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4. Properties of the normalization. As an application of the previous construction we will look at the comparison of critical values. This topic has already been discussed extensively in [5] and [2]. Let us consider two Hamiltonians, H0 and H1 , and the corresponding generating functions S0 and S1 (normalized relatively to the class α) of the Lagrangian submanifolds ϕ0 (L) and ϕ1 (L). Moreover, let S be a generating function for L. The following lemma allow us to compare the critical values c(α, S0 ) and c(α, S1 ). Lemma 4.1. If H0 ≤ H1 then c(α, S0 ) ≤ c(α, S1 ). Proof. Define Hλ (t, q, p) = (1 − λ)H0 (t, q, p) + λH1 (t, q, p). Let ϕλ and Sλ be the flow at time one relative to the Hamiltonian Hλ and a generating function of ϕλ (L) re-normalized as before, respectively. To ease the notations, let us pose q(0) = q0 and q(1) = q1 . From the normalization of Sλ , it follows that Sλ (qλ (1); ηλ ) = −Sλ (qλ (0), qλ (1); γλ , ξλ ), where (qλ (1); ηλ ) is a critical point relative to the value c(α, Sλ ). We can write d d Sλ (qλ (1); ηλ ) = − Sλ (qλ (0), qλ (1); γλ , ξλ ) = (24) dλ dλ Z d d 1 − (S(qλ (0); ξλ )) − pλ (t)q˙λ (t) − Hλ (t, qλ (t), pλ (t))dt , (25) dλ dλ 0 Clearly, from the first derivative of (25), we get ∂S dqλ ∂S dξλ ∂S dqλ − (qλ (0), ξλ ) (0) − (qλ (0), ξλ ) = − (qλ (0), ξλ ) (0). ∂q dλ ∂ξ dλ ∂q dλ From the second one, we get Z 1 d − pλ (t)q˙λ (t) − (1 − λ)H0 (t, qλ (t), pλ (t)) − λH1 (t, qλ (t), pλ (t)) dt = 0 dλ Z 1h dq˙λ dpλ (t) + pλ (t) (t) + H0 (t, qλ (t), pλ (t)) − H1 (t, qλ (t), pλ (t))+ − q˙λ (t) dλ dλ 0 ∂H0 dqλ ∂H0 dpλ −(1 − λ) (t, qλ (t), pλ (t)) (t) − (1 − λ) (t, qλ (t), pλ (t)) (t)+ ∂q dλ ∂p dλ i dqλ ∂H1 dpλ ∂H1 (t, qλ (t), pλ (t)) (t) − λ (t, qλ (t), pλ (t)) (t) dt = −λ ∂q dλ ∂p dλ Z 1 h dp ∂Hλ ∂Hλ dqλ λ − q˙λ (t) − (t, qλ (t), pλ (t)) (t) − (t, qλ (t), pλ (t)) (t)+ ∂p dλ ∂q dλ 0 i dq˙λ +pλ (t) (t) + (H0 − H1 )(t, qλ (t), pλ (t)) dt dλ and, integrating by parts, we have Z 1 Z 1 dq˙λ dqλ 1 dqλ − pλ (t) (t)dt = −pλ (t) (t) + (t)dt, p˙λ (t) dλ dλ dλ 0 0 0 which is Z 1h dqλ dqλ (1) + pλ (0) (0) − (H0 − H1 )(t, qλ (t), pλ (t))+ = −pλ (1) dλ dλ 0 dp ∂Hλ λ + q˙λ (t) − (t, qλ (t), pλ (t)) (t)+ ∂p dλ ∂Hλ dqλ i − p˙λ (t) + (t, qλ (t), pλ (t)) (t) dt. ∂q dλ


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This implies that, along the critical points, we have Z 1 d Sλ (qλ (1); ξλ ) = − (H0 − H1 )(t, qλ (t), pλ (t))dt, dλ 0 and, using the hypothesis H0 ≤ H1 , we obtain d Sλ (qλ (1); ξλ ) ≥ 0, dλ which implies that c(α, S0 ) ≤ c(α, S1 ). Acknowledgments. I want to thank Luca Rizzi, Nadir Pajaro, Federica Brugnolo and all my colleagues from “Ufficio Architettura IT” at Previnet S.p.A.: Thomas Basciutti, Jurij Bellin, Paolo Biasuzzi, Alberto Bitto, Mirko Bonotto, Alberto Bovo, Daniele De Rosa, Gianmaria Parigi Bini, Nicola Moretto, Paolo Soleni and Christian Voltini. REFERENCES [1] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Mathematical Journal, 144 (2008), 235–284. [2] F. Cardin, Elementary Symplectic Topology and Mechanics, Springer, 2015. [3] A. Monzner, N. Vichery and F. Zapolsky, Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization, J. Mod. Dyn., 6 (2012), 205–249, arXiv:1111.0287. [4] D. Th´ eret, A complete proof of Viterbo’s uniqueness theorem on generating functions, Topology and its Applications, 96 (1999), 249–266. [5] C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685–710. [6] C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 439–459, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006. [7] C. Viterbo, Symplectic Homogenization, 2014, arXiv:0801.0206v3.

Received December 2016; revised December 2017. E-mail address: [email protected]