Nekhoroshev theorem for the periodic Toda lattice

and constants γ, δ, V 1 , V 2 ∈ R (γ, δ = 0). The Toda lattice has been introduced by Toda [24] and studied extensively in the sequel. It is an FPU lattice, i.e. a Hamiltonian system of particles in one space dimension with nearest neighbor interaction. Models of this type have been studied by Fermi-Pasta-Ulam [FPU]. In numerical experiments they found recurrent features for the lattices 1 INTRODUCTION they considered. Despite an enormous effort from the physics and mathematics community in the past fifty years, by and large, these numerical experiments still defy an explanation. For a recent account of the fascinating history of the FPU problem, see e.g. [1] or [6]. At least in the case of the periodic Toda lattice, the recurrent features can be fully accounted for. In fact, Flaschka [5], Hénon [8], and Manakov [17] independently proved that the periodic Toda lattice is integrable. In this paper, we show that on the open dense subset of the phase space where all action variables are strictly positive, the Nekhoroshev theorem [19,20] applies. It means that the action variables of the Toda lattice vary slowly over an exponentially long time interval along solutions of a Hamiltonian system with Hamiltonian sufficiently close to H T oda .

To continue, let us note that in (1), without loss of generality, we can assume that V 1 = V 2 = 0. When expressed in the canonical coordinates (δq j , 1 δ p j ) 1≤j≤N , the Hamiltonian H T oda is, up to a scaling factor δ -2 , of the form

where a = |γδ|. Moreover, notice that the total momentum N n=1 p n is conserved. Hence the motion of the center of mass 1 N N n=1 q n is linear and therefore unbounded. However, the orbits of the system relative to the center of mass all lie on tori. To describe these orbits, consider the relative coordinates v n := q n+1q n (1 ≤ n ≤ N -1) and their canonically conjugate ones, u n := nβ -n j=1 p k (1 ≤ n ≤ N -1), where β = 1 N N j=1 p n . In the sequel, we view the Toda lattice as a two parameter family of integrable systems with the two parameters α > 0 and β ∈ R. For α > 0 and β ∈ R arbitrary, denote by H β,α the Toda Hamiltonian when expressed in the canonical coordinates (v k , u k ) 1≤k≤N -1 ∈ R 2N -2 and the parameters α and β.

In [10], we proved the following result.

Theorem 1.1. The periodic Toda lattice admits Birkhoff coordinates. More precisely, there exist globally defined canonical coordinates (x k , y k ) 1≤k≤N -1 ∈ R 2N -2 so that for any β ∈ R and α > 0, the Toda Hamiltonian H β,α , when expressed in these coordinates, takes the form N β 2 2 + H α (I), where H α (I) is a real analytic function of the action variables

In particular, Theorem 1.1 states that the action variables (I n ) 1≤n≤N -1 are independent of β ∈ R and α > 0. Note that each of the N -1 frequencies

of the Toda lattice H β,α is independent of the parameter β.

The main result of this paper says that the Hamiltonian H α , introduced in Theorem 1.1, is a convex function of the action variables (I k ) 1≤k≤N -1 : Theorem 1.2. In the open quadrant R N -1 >0 , the Hamiltonian H α is a strictly convex function of the action variables (I k ) 1≤k≤N -1 . More precisely, for any compact subset U ⊆ R N -1 >0 and any compact interval [α 1 , α 2 ] ⊆ R >0 , there exists m > 0, such that

for any I ∈ U , any β ∈ R, and any α 1 ≤ α ≤ α 2 .

Theorem 1.2 implies that Nekhoroshev’s theorem holds for the Toda lattice on

an open and dense subset of R 2N -2 by Theorem 1.1.

Corollary 1.3. For any β ∈ R and α > 0, Nekhoroshev’s theorem applies to (sufficiently small) Hamiltonian perturbations of the Toda Hamiltonian H β,α on all of P • . (See [15], [16], [19], [20], [21], [22] for various versions of Nekhoroshev’s theorem and their proofs.)

In practice, it is difficult to verify for an integrable system with a given Hamiltonian H whether the convexity (or steepness) condition of Nekhoroshev’s theorem is satisfied as this condition refers to H, when expressed in action variables, and is not invariant under canonical transformations. Typically one does not know the Hamiltonian as a function of the action variables explicitly enough to derive the convexity property.

To prove Theorem 1.2, we make use of the Birkhoff normal form of the Toda lattice H β,α on R 2N -2 near the elliptic fixed point (v, u) = (0, 0), established in [11].

Theorem 1.4. Let α > 0 be arbitrary. Near I = 0, the function H α (I), introduced in Theorem 1.1, has an expansion of the form

with

In particular, the Hessian of H α (I) at I = 0 is given by

As an immediate consequence of Theorem 1.4 we get Corollary 1.5. Near I = 0, H α (I) is strictly convex for any α > 0.

Outside of I = 0, we argue differently. For any α > 0, consider the frequency map

In view of Coro

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