Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

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Rigidity in topology C⁰ of the Poisson bracket for Tonelli Hamiltonians

M.-C. ARNAUD ^∗†‡

May 25, 2015

Abstract

We prove the following rigidity result for the Tonelli Hamiltonians.

LetT^∗M be the cotangent bundle of a closed manifoldM endowed with its usual symplectic form. Let(Fn) be a sequence of Tonelli Hamiltonians thatC⁰ converges on the compact subsets to a Tonelli Hamiltonian F. Let(G_n) be a sequence of Hamiltonians that thatC⁰ converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({Fn, Gn}) C⁰-converges on the compact subsets to a C¹ functionH. Then H={F, G}.

Key words: Hamiltonian dynamics, Poisson bracket, Tonelli Hamiltonians, Lagrangian functions.

2010 Mathematics Subject Classification: 37J05, 70H99, 37J50

∗ANR-12-BLAN-WKBHJ

†Avignon Universit´e , Laboratoire de Math´ematiques d’Avignon (EA 2151), F-84 018 Avignon, France.

e-mail: Marie-Claude.Arnaud@univ-avignon.fr

‡membre de l’Institut universitaire de France

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1 Introduction

For two C² functions H, K : N → R defined on a symplectic manifold (N, ω) (such functions are usually called “Hamiltonians”), their Poisson bracket {H, K} is

{H, K}=ω(XH, XK)

where X_H designates the symplectic gradient ofH defined by dH =ω(XH, .). Hence the Poisson bracket is given by {H, K} =dH(XK) and describes the evolution of H along any orbit for the Hamiltonian flow ofK. The Poisson bracket is clearly continuous for the topologies (H, K)∈C¹(N,R)×C¹(N,R)7→ {H, K} ∈C⁰(N,R).

In his PhD thesis [9], V. Humil`ere asks the following question concerning a possible C⁰ rigidity for the Poisson bracket:

Question 1. (V. Humili`ere) Let (F_k), (G_k) be two sequences of C^∞ functions such that (Fk) C⁰-converges to F ∈ C^∞(N,R), (Gk) C⁰-converges to G ∈ C^∞(N,R) and ({F_k, G_k}) C⁰-converges toH ∈C^∞(N,R). Do we have {F, G}=H?

The answer to this question is in general negative and some counter-examples due to L. Polterovitch are provided in [10].

However, F. Cardin and C. Viterbo give a positive answer in [4] when H= 0 and all the considered Hamiltonians have their support contained in fixed a compact set.

By introducing a notion of pseudo-representation, V. Humili`ere obtains in [10] results that contain the previous one. Fix a closed manifoldM and a finite-dimensional Lie al- gebra (g,[., .]), a pseudo-representation is a sequence of morphismsρ_n:g→C^∞(M,R) such that for allf, g∈g, the sequence ({ρ_n(f), ρn(g)} −ρ_n([f, g])) converges uniformly to zero. Humil`ere proves that the limit of a convergent pseudo-representation is a representation. In other word, assuming that there is a morphismρ:g →C^∞(M,R) such that any sequence (ρn(f)) uniformly converges toρ(f), he proves that

∀f, g∈g,{ρ(f), ρ(g)}=ρ([f, g]);

i.e. that{ lim

n→∞ρ_n(f), lim

n→∞ρ_n(g)}= lim

n→∞{ρ_n(f), ρ_n(g)}.

In [2], L. Buhovski proves that if (fn) and (gn) are C^∞ functions that uniformly converge on every compact subset of N to the smooth functions f and g, and if the sequences (Df_n), (Dg_n) are uniformly bounded on any compact subset of N, then we have

n→∞lim{f_n, g_n}={f, g}.

We will describe here a new case where question 1 always has a positive question: the case where one of the considered sequence is a sequence of Tonelli Hamiltonians that

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C⁰-converges to a Tonelli Hamiltonian. We will too give a simpler proof of a part of Buhovski’s result.

Notations. LetM be a closed manifold and letT^∗M be its cotangent bundle endowed with its usual symplectic formω that is the derivative of the Liouville 1-form. We use the notation (q, p)∈T^∗M and thenω=dq∧dp.

The first projection is denoted byπ : (q, p)∈T^∗M 7→q∈M.

Definition. A C² functionH:T^∗M →Ris a Tonelli Hamiltonianif it is:

• superlinear in the fiber, i.e. ∀A ∈ R,∃B ∈ R,∀(q, p) ∈ T^∗M,kpk ≥ B ⇒ H(q, p)≥Akpk;

• C²-convex in the fiber i.e. for every (q, p) ∈T^∗M, the Hessian ^∂_∂p²^H2 of H in the fiber direction is positive definite as a quadratic form.

We denote the Hamiltonian flow of H by (ϕ^H_t ) and the Hamiltonian vector-field by X_H.

Note that the class of Tonelli Hamiltonian contains all the Riemannian metrics and all the mechanical systems (Riemannian metric + potential). The class of Tonelli Hamiltonians has been recently intensely studied in the setting of weak KAM theory.

Theorem 1. Let (Fk) be a sequence of Tonelli Hamitonians defined on T^∗M such that (Fk) C⁰-converges on the compact subsets of T^∗M to a Tonelli Hamiltonian F ∈ C²(T^∗M,R), let (G_k)be a sequence of C² Hamiltonians of T^∗M that C⁰-converges on the compact subsets ofT^∗M toG∈C²(T^∗M,R)and such that ({F_k, G_k})C⁰-converges on the compact subsets of T^∗M to H ∈C⁰(T^∗M,R). Then {F, G}=H.

Remark. We prove in lemma 1 that theC⁰convergence of (Fk) toF and the convexity hypothesis imply theC⁰-convergence of ^∂F_∂p^k to ^∂F_∂p. But we don’t have the convergence of the ^∂F_∂q^k; then this theorem is not a consequence of theorem 2.

Theorem 2. Let (N, ω) be a (non necessarily compact) symplectic manifold. Let(Fk), (G_k)be two sequences ofC² Hamitonians defined onN such that(G_k)C⁰-converges on the compact subsets of N toG∈C²(N,R), (F_k) C¹-converges on the compact subsets of N to F ∈ C²(N,R) and ({F_k, G_k}) C⁰-converges on the compact subsets of N to H ∈C⁰(N,R). Then {F, G}=H.

Theorem 25.7 of [11] asserts that if (F_k) is a sequence ofC¹ and convex functions defined onRⁿthatC⁰-converge to aC¹and convex functionF on any compact subset, then the convergence isC¹ on any compact subset of Rⁿ. We deduce:

Corollary. Let (Fk), (Gk) be two sequences of C² Hamitonians defined on R²ⁿ such that (Gk) C⁰-converges on the compact subsets of R²ⁿ to G ∈ C²(R²ⁿ,R), (Fk) C⁰- converges on the compact subsets of R²ⁿ to F ∈ C²(R²ⁿ,R), the F_k are convex and

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({F_k, G_k}) C⁰-converges on the compact subsets of R²ⁿ to H ∈ C⁰(R²ⁿ,R). Then {F, G}=H.

Theorem 1 concerns Tonelli Hamiltonians and is new. Theorem 2 is a consequence of the result due to L. Buhovsky (see [2]) that we explained before, but we give here a simple proof, that uses the simple principle.

Ideas of proofs

Contrarily to what happens in the works that we mentioned above, we won’t use here any result of symplectic topology, as symplectic capacities, displacement energy. . . We use

1. the following simple principle:

Simple principle: under the hypotheses of question 1, if for any x ∈ N, there exists a T > 0 and a sequence of points (xk) such that the arcs of or- bits (ϕ^F_t^k(x_k))_t∈[0,T_] converge in some reasonable sense to(ϕ^F_t (x))_t∈[0,T_], then the answer to question 1 is positive.

If we have a good enough notion of convergence, the proof of this principle is not complicated and propositions 2 and 3 are some versions of this principle.

2. for theorem 1, we use some subtle variational arguments to prove that the hypothesis of the simple principle is satisfied; for theorem 2, we just use straightforward arguments coming from the theory of ordinary differential equations.

Plan

• In section 2, we give the precise statements of the propositions that we need to prove theorems 1 and 2.

• Section 3 is devoted to the proof of proposition 1 that asserts that every C⁰- convergent sequence of Tonelli Hamiltonians satifies the hypotheses of the simple principle.

• In section 4, we prove the simple principle, i.e. propositions 2 and 3, in the considered cases.

Acknowledgments. I am grateful to the referees for many improvements of the article.

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2 Structure of the proofs of theorems 1 and 2

2.1 How two propositions imply theorem 1

We will prove the two following propositions in sections 3 and 4. The first one uses variational arguments to state that if a sequence of Tonelli Hamiltonians (Hk)C⁰-converges to a Tonelli HamiltonianH, then the “small” arcs of orbits forH are approximated in some sense by some small parts of orbits for the H_k.

Proposition 1. Let (H_k)_k∈N, H be some C² Tonelli Hamiltonians defined on T^∗M such that(Hk) C⁰-converges on the compact subsets of T^∗M toH.

Then, for every (q, p) ∈ T^∗M, there exists T > 0 and (qk, p_k)_k∈N ∈ T^∗M such that a subsequence of the sequence of arcs (Q_k, P_k) = (ϕ^H_t ^k(q_k, p_k))_t∈[0,T] converges to (Q, P) = (ϕ^H_t (q, p))_t∈[0,T_] in the following sense (here we are in a chart because all is local and we denote the subsequence in the same way we denoted the initial sequence):

• (Q_k) C⁰-converges toQ;

• (P_k) converges to P in the L² sense;

• the (Qk, P_k) are uniformly C⁰ bounded.

The second proposition is a version of what we called previously thesimple principle.

The proof is elementary.

Proposition 2. Let (Fk) be a sequence of C² Hamiltonians defined on T^∗M and let F be a C² Hamiltonian such that:

for every (q, p) ∈ T^∗M, there exists T > 0 and (q_k, p_k)_k∈N ∈ T^∗M such that the sequence of arcs (Qk, P_k) = (ϕ^F_t^k(qk, p_k))_t∈[0,T_] converges to (Q, P) = (ϕ^F_t (q, p))_t∈[0,T_] in the following sense:

• (Qk) C⁰-converges toQ;

• (Pk) converges to P in the L² sense;

• the (Q_k, P_k) are uniformly C⁰ bounded.

Let (G_k) be a sequence ofC² Hamiltonians of T^∗M that C⁰-converges on the compact subsets of T^∗M to a C² Hamiltonian G and that is such that the sequence ({F_k, G_k}) C⁰-converges to some H ∈C⁰(T^∗M,R).

Then {F, G}=H.

Theorem 1 is clearly a consequence of propositions 1 and 2.

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2.2 How two propositions imply theorem 2

To prove theorem 2, we will prove that if (F_k) satisfy the hypothesis of theorem 2, then it satisfies the hypothesis of proposition 3 below .

Proposition 3. Let (N, ω) be a (non necessarily compact) symplectic manifold. Let (F_k) be a sequence of C² Hamiltonians defined on N and let F be a C² Hamiltonian such that:

for all x ∈N, there exists T > 0 and a sequence (xk) ∈N such that the sequence of arcs of orbit (ϕ^F_t^k(x_k))_t∈[0,T_] C⁰-converges to (ϕ^F_t(x))_t∈[0,T].

Let (Gk) be a sequence of C² Hamiltonians of N that C⁰-converges on the compact subsets of N to a C² Hamiltonian G and that is such that the sequence ({F_k, G_k}) C⁰-converges on the compact subsets of N to some H ∈C⁰(N,R).

Then {F, G}=H.

Let us be more precise.

Definition. Let X be a locally Lipschitz vectorfield on N. A C¹ arc γ : [a, b]→N is anε-solution forX if ∀t∈[a, b],kγ^′(t)−X(γ(t))k< ε.

If for example the vector fields X₁ and X₂ are ε-close, every solution for X₂ is an ε-solution for X₁.

The following proposition is classical (see for example section 6.2.2. of [8]):

Proposition 4. Assume that the vector field X is Lipschitz on Rⁿ with Lipschitz constant K. Let γ_i : [a, b]→ Rⁿ be a ε_i-solution for X for i= 1,2. We assume that for some τ ∈[a, b], we have d(γ₁(τ), γ₂(τ))≤δ. Then we have:

∀t∈[a, b], d(γ1(t), γ2(t))≤δe^K|t−τ|+ε₁+ε₂

2 (e^K|t−τ|−1).

Let now (F_k),F beC² Hamiltonians onN such that (F_k)C¹-converges toF. Using proposition 4, we will prove:

Lemma. for all x ∈ N, there exists T > 0 and a sequence (x_k) ∈ N such that the sequence of arcs of orbit(ϕ^F_t^k(xk))_t∈[0,T_] C⁰-converges to (ϕ^F_t(x))_t∈[0,T_].

Then the conclusion of theorem 2 follows from this and proposition 3.

Proof of the lemma. Let us fix x₀ ∈ N. Then we choose a chart U = B(x₀, r) at x₀, and T >0 such that sup

x∈B(x⁰,r)kX_F(x)k.T < r. As (F_k) C¹-converges to F, we have fork large enough: sup

x∈B(x0,r)kX_F_k(x)k.T < r. This implies that

∀t∈[0, T], ϕ^F_t^k(x₀)∈B(x₀, r).

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Moreover, (ϕ^F_t^k(x0))_t∈[0,T_]is aε_k-solution forX_F_kwhereε_k= sup

x∈B(x⁰,r)kX_F_k(x)−X_F(x)k. We deduce from proposition 4 that fork large enough

∀t∈[0, T], d(ϕ^F_t^k(x₀), ϕ^F_t (x₀))≤ εk

2(e^Kt−1) whereK is a Lipschitz constant for X_F_|B(x₀_,r). As lim

k→∞ε_k= 0, we obtain the wanted conclusion.

3 Proof of proposition 1

We assume that (Hk) C⁰-converges on the compact subsets ofT^∗M toH. We will use the following lemma, that is in the spirit of theorem 25.7 of [11].

Lemma 1. Let (Fk) be a sequence ofC² functions defined on T^∗M that areC² convex in the fiber direction and that C⁰-converges on the compact subsets of T^∗M to a C² function F : T^∗M → R that is C² convex in the fiber direction. Then (^∂F_∂p^k) C⁰- converges to ^∂F_∂p on the compact subsets of T^∗M.

Proof of lemma 1. If not, by possibly extracting a subsequence, we find a compact subsetK ⊂T^∗M,ε >0 and a sequence (qk, p_k)∈K such that:

∀k;k∂F_k

∂p (q_k, p_k)−∂F

∂p(q_k, p_k)k ≥2ε.

Extracting a subsequence, we can assume that lim

k→∞(qk, pk) = (q_∞, p_∞) and that we are in a chartK =K₀×B(p¯ _∞, R)⊂M×Rⁿ. Then we choose ˜K= ˜K₀×B¯(p_∞,2R) that containsK in its interior. We chooseα >0 such that ¯B_α = ¯B((q_∞, p_∞),2α)⊂K˜ and:

∀x∈B¯α,

∂F

∂p(x)−∂F

∂p(q_∞, p_∞)

< η = ε 3.

Ask^∂F∂p^k(q_k, p_k)−^∂F∂p(q_k, p_k)k ≥2ε, we can findu_k∈Rⁿ such that ku_kk= 1 and ∂F_k

∂p (qk, p_k)−∂F

∂p(qk, p_k)

.u_k≥ε;

i.e. ∂F_k

∂p (qk, pk).uk≥ ∂F

∂p(qk, pk).uk+ε.

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Using the fact that F_k isC² convex in the fiber direction, we deduce:

Fk(qk, pk+α.uk) ≥ Fk(qk, pk) +α^∂F_∂p^k(qk, pk).uk

≥ α.ε+F_k(qk, p_k) +α^∂F_∂p(qk, p_k).uk

≥ αε+F_k(qk, p_k) +α^∂F_∂p(q∞, p_∞).uk−αη

Extracting a subsequence, we can assume that (uk) converges to u_∞. We then take the limit and obtain:

F(q_∞, p_∞+α.u_∞)−F(q_∞, p_∞)−α∂F

∂p(q_∞, p_∞)u_∞≥α(ε−η) = 2αε 3 . But we have:

F(q_∞, p_∞+α.u_∞) −F(q_∞, p_∞)−α^∂F_∂p(q_∞, p_∞)u_∞

=α.R1 0

∂F

∂p(q∞, p_∞+tαu_∞)−^∂F∂p(q∞, p_∞)

u_∞dt≤α.η= ^αε₃ and this is a contradiction.

We deduce from lemma 1 that the sequence (^∂H_∂p^k) C⁰-converges on any compact subset of T^∗M.

Let us recall some well-known fact concerning Tonelli Hamiltonians that are proved for example in [7]. To any Tonelli Hamiltonian H (resp. Hk) we can associate a Lagrangian L:T M →R (resp. L_k:T M →R) that is defined by:

∀(q, v)∈T M, L(q, v) = sup

p∈Tq^∗M

(p.v−H(q, p)). Then Land L_k are C²,C²-convex and superlinear in the fiber direction.

If we denote by L^q : T_q^∗M → T_qM the Legendre map p → ^∂H∂p(q, p) (that is a C¹- diffeomorphism), we have the equivalent formula:

L(q, v) =L⁻¹q (v).v−H(q,L⁻¹q (v)).

Let us recall that the Euler-Lagrange flow associated to L is f_t^L =L ◦ϕ^H_t ◦ L⁻¹ (see the notation L just below) and that if (q_t, p_t) is an orbit for the Hamiltonian H, the corresponding orbit forL is (qt,q˙_t).

We know that L^H^k : (q, p) 7→ (q,^∂H_∂p^k(q, p)) = (q,L^Hq^k(p)) C⁰-converges on the compact subsets ofT^∗M to L: (q, p)7→(q,Lq(p)). The following lemma implies that ((L^H^k)⁻¹) C⁰-converges to L⁻¹ on any compact subset of T M and then that (Lk) C⁰-converges to Lon any compact subset of T M.

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Lemma 2. Let (hk) be a sequence of homeomorphisms from T^∗M to T M that C⁰- converges on any compact subset of T^∗M to a homeomorphism h. Then the sequence (h⁻¹_k ) C⁰-converges on any compact subset of T M toh⁻¹.

Proof of lemma 2. If not, by possibly extracting a subsequence, we can find a sequence (y_k) in T M that converge toy_∞∈T M such that:

∀k, d(h⁻¹_k (y_k), h⁻¹(y_k))≥2ε >0.

Ash is a homeomorphism, there existsR₂ > R₁ >0 such that:

∀x∈T^∗M, d(h⁻¹(y∞), x)≥R₂ ⇒d(y∞, h(x))≥4ε and

∀x∈T^∗M, d(h⁻¹(y∞), x)≤R₁ ⇒d(y∞, h(x))< ε.

As (hk) C⁰-converges on any compact subset of T^∗M, for k large enough, we have:

hk(B(h⁻¹(y_∞), R₁))⊂B(y_∞, ε) and

h_k(B(h⁻¹(y_∞),2R₂)\B(h⁻¹(y_∞), R₂))⊂T M\B(y_∞,2ε).

Hence h⁻¹_k (B(y_∞, ε)) is a connected subset of T^∗M that meets B(h⁻¹(y_∞), R₁) but doesn’t meet B(h⁻¹(y_∞),2R₂)\B(h⁻¹(y_∞), R₂). This implies that h⁻¹_k (B(y_∞, ε)) ⊂ B(h⁻¹(y_∞), R₂) .

As the sequence (yk) converges to y_∞, we may assume that d(yk, y_∞) < ε. Hence d(h⁻¹(y_∞), h⁻¹_k (yk))≤ R₂, the sequence (h⁻¹_k (yk)) is bounded, we can extract a subsequence that converges tox_∞. Then we have

• d(x_∞, h⁻¹(y_∞))≥2ε by taking the limit in the first inequality;

• d(h(x_∞), y_∞) = lim

k→+∞d(hk(h⁻¹_k (yk)), yk) = 0.

We have found two pointsx_∞6=h⁻¹(y_∞) that have the same image by the homeomor- phismh. This is impossible.

Let us fix (q, p)∈T^∗M. By Weierstrass theorem (see for example [7]), there exists τ >0 such that the arc γ : [0, τ]→M defined byγ(t) =π◦ϕ^H_t (q, p) is action strictly minimizing for the Lagrangian L, i.e. such that for every other absolutely continuous arcη: [0, τ]→M that has the same endpoints asγ, we have:

AL(γ) = Z τ

0

L(γ(t),γ(t))dt < A˙ L(η) = Z τ

0

L(η(t),η(t))dt.˙

For any k ∈ N, by Tonelli theorem (see [7]), there exists an arc γk : [0, τ] → M such that γ_k(0) = q and γ_k(τ) =γ(τ) that minimizes (non necessarily strictly) the action

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for the Lagrangian L_k among all the absolutely continuous arcs that have the same endpoints asγ_k. Then there exists a uniquep_k∈T_qM such that: γ_k(t) =π◦ϕ^H_t^k(q, pk) and we have ˙γk(t) = ^∂H_∂p^k(ϕ^H_t^k(q, pk)). We define: mτ = sup

t∈[0,τ]|L(γ(t),γ(t))˙ |. We will prove

Lemma 3. There exists a compact subset K₂ = {H ◦ L⁻¹ ≤ C + 2} of T M that contains all the (γ_k(t),γ˙_k(t)) and (γ(t),γ˙(t))for t∈[0, τ] and we have

k→+∞lim A_L(γk) =A_L(γ).

Proof of lemma 3.

As (L_k)C⁰-converges toLon the compact subsets ofT M, forklarge enough, we have:

A_L_k(γ)≤τ(m_τ+ 1) and then A_L_k(γ_k)≤τ(m_τ+ 1) becauseγ_k is minimizing forL_k. BecauseLis superlinear and convex in the fiber direction, there exists R >0 such that

∀(q, v)∈T M,kvk ≥R⇒L(q, v)> M_τ = sup (

m_τ+ 3,sup

q∈M|L(q,0)|+ 1 )

. Because L_k is convex in the fiber direction, we have forkvk ≥R

L_k(q, v)≥L_k(q,0) +

L_k(q, R v

kvk)−L_k(q,0)

kvk −R

R .

Because (Lk) C⁰-converges to L on the compact subsets of T M, for k large enough, we have

∀(q, v), Lk(q,0)≥ −M_τ and L_k(q, R v

kvk)−L_k(q,0)>1.

We deduce for klarge enough andkvk ≥R L_k(q, v)≥ kvk −R

R −M_τ and then for a certainR^∗> R , we have

∀v,kvk ≥R^∗ ⇒L_k(q, v) ≥M_τ.

Forklarge enough, we haveA_L_k(γk)≤τ(mτ+ 1) and then for at_k∈[0, τ] we have kγ˙_k(t_k)k ≤ R^∗. We introduce the constant C= sup

kvk≤R^∗

H(L⁻¹(q, v)) and the compact subsetsK₁ ={H◦L⁻¹ ≤C+1},K₂ ={H◦L⁻¹ ≤C+2}andK₃ ={H◦L⁻¹ ≤C+3} of T M. As H is convex on the fiber directions the intersection ofKi with a fiber is a topological ball. As (Hk◦(L^H^k)⁻¹)C⁰-converges toH◦L⁻¹onK₂, forK large enough we have

∀(q, v)∈K₃\K₂, H_k◦(L^H^k)⁻¹(q, v)> C+3 2