FINSLER METRICS AND ACTION POTENTIALS

RENATO ITURRIAGA AND HÉCTOR SÁNCHEZ-MORGADO

Abstract. We study the behavior of Mañé’s action potential Φ_{k} associated to a convex superlinear Lagrangian, for k bigger than the critical value c(L) . We obtain growth estimates of the action potential as function of k. We prove that the action potential can be written as Φ_{k}(x, y) = D_{F} (x, y) + f(y) − f(x) where f is a smooth function and D_{F} is the distance function associated to a Finsler metric.

1. Introduction

Let M be a closed connected smooth manifold and let L : T M → ℝ be a smooth convex superlinear Lagrangian. This means that L restricted to each T_{x}M has positive definite Hessian and that for some Riemannian metric we have that

lim_{|v|→∞} \frac{L(x, v)}{|v|} = ∞,

uniformly on x ∈ M. Let H : T^{∗}M → ℝ be the Hamiltonian associated to L and let L : T M → T^{∗}M be the Legendre transform (x, v) ↦ ∂L/∂v(x, v). Since M is compact, the extremals of L give rise to a complete flow ϕ_{t} : T M → T M called the Euler-Lagrange flow of the Lagrangian. Using the Legendre transform we can push forward ϕ_{t} to obtain another flow ϕ_{t}^{∗} which is the Hamiltonian flow of H with respect to the canonical symplectic structure of T^{∗}M. Recall that the energy E : T M → ℝ is defined by

E(x, v) = \frac{∂L}{∂v}(x, v).v − L(x, v).

Since L is autonomous, E is a first integral of the flow ϕ_{t}.

Recall that the action of the Lagrangian L on an absolutely continuous curve γ : [a, b] → M is defined by

A_{L}(γ) = ∫_{a}^{b} L(γ(t), γ̇(t)) dt.

Given two points, x and y in M and T > 0 denote by C_{T} (x, y) the set of absolutely continuous curves γ : [0, T] → M, with γ(0) = x and γ(T) = y. For each k ∈ ℝ we define the action potential Φ_{k} : M × M → ℝ by

Φ_{k}(x, y) = inf{A_{L+k}(γ) : γ ∈ ∪_{T >0}C_{T} (x, y)}.

1991 Mathematics Subject Classification. 58F05 70H99 .

Both authors were partially supported by CONACYT-México grant # 28489-E.

2 R. ITURRIAGA AND H. SÁNCHEZ M.

The critical value of L, which was introduced by Mañé in [4], is the real number c(L) defined as the infimum of k ∈ ℝ such that for some x ∈ M, Φ_{k}(x, x) > −∞. For k ≥ c(L), we have that Φ_{k}(x, y) > −∞ for every x, y and it is a Lipschitz function that satisfies the triangle inequality.

We will prove two results concerning Φ_{k}(x, y). First, we establish general properties of the action potential Φ_{k}(x, y) as a function of k. Then, we fix k.

For x, y fixed and T > 0 define

S(T) = inf{A_{L}(γ) : γ ∈ C_{T} (x, y)}.

It is easily shown that S(T) is continuous. Although S(T) is not necessarily convex, its Legendre transform:

S^{∗}(e) = max_{T >0}(eT − S(T))

is a well defined convex function and coincides with the Legendre transform of the convex hull \overline{S} of S. Notice that

Φ_{k}(x, y) = −S^{∗}(−k) (1)

and so the domain of S^{∗} is dom S^{∗} = (−∞, −c(L)]. From the definition of Φ_{k}(x, y) it follows that g(k) = Φ_{k}(x, y) is nondecreasing and so is S^{∗}.

Theorem 1. For all x, y in M we have that:

(a) g grows slower than any linear function, that is

lim_{k→∞} \frac{g(k)}{k} = 0

(b) The right derivative of g at c(L) is infinite.

(c) lim_{T →∞} S(T)/T = −c(L).

Now we will consider the action potential as a function of x and y. It was proved in [2] that for any k > c(L) the flow on the energy level k is conjugate to the geodesic flow of an appropriately chosen Finsler metric on M.

Given a Finsler metric and an absolutely continuous curve γ we can define its Finsler length as

l_{F} (γ) = ∫ \sqrt{F (γ, γ̇)}

Observe that since F is homogeneous of degree two then F^{\frac{1}{2}} is homogeneous of degree one, so the definition does not depend on the parameterization of the curve. Finally we define the Finsler distance as

D_{F} (x, y) = inf{l_{F} (γ)}

where the infimum is taken over all absolutely continuous curves joining x and y.

Theorem 2. If k is bigger than the critical value then there exists F a Finsler metric and a C^{∞} real valued function f on M such that Φ_{k}(x, y) = D_{F} (x, y) + f(y) − f(x). Moreover if k is bigger than − inf L then we can choose f = 0.

As a consequence of Theorem 2 we have that there is a neighborhood V of the diagonal Δ in M × M, such that Φ_{k} is differentiable in V \ Δ.

2. Proofs

Proof of Theorem 1. It is well known that if f is a convex function of a real variable then

(1) If x ∈ int(domf), then both one side derivatives exist and f′_{−}(x) ≤ f′_{+}(x)

(2) If x ∈ domf is a boundary point then the corresponding one side derivative exists.

(3) If x < y, f′_{+}(x) ≤ f′_{−}(y).

Define

rang ∂f = ⋃_{x∈domf}[f′_{−}(x), f′_{+}(x)].

It is proved in [6] Section 24 that

int(domf^{∗}) ⊂ rang ∂f ⊂ domf^{∗}.

Therefore

rang ∂S^{∗} = dom S^{∗∗} = dom \overline{S} = (0, ∞).

Thus

lim_{e→−∞} \frac{S^{∗}(e)}{e} = 0

and

S_{−}^{∗′}(−c(L)) = lim_{e→−c(L)} \frac{S_{−}^{∗′}(e)}{e} = ∞.

From (1), items (a) and (b) follow.

By the same kind of arguments lim_{T →∞} \overline{S}(T)/T = −c(L), and then

−c(L) ≤ lim inf_{T →∞} \frac{S(T)}{T}.

To prove the other inequality, let μ be an ergodic minimizing probability. That is, an invariant ergodic probability for the Lagrangian flow ϕ_{t} such that

m := ∫ L dμ ≤ ∫ L dν

for any invariant probability ν. Mather proved that such measures exist. See [5]. Let θ ∈ T M be a regular point for μ, such that

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} L(ϕ_{t}(θ))dt = m.

Let π : T M → M be the natural projection. Comparing with the curve γ obtained by joining x and π(θ) with a short curve, then following the curve πϕ_{t}(θ) and then joining πϕ_{T} (θ) and y with a short curve, we have that for any given ε > 0 and T large enough

S(T) ≤ (m + ε)T + O(1).

4 R. ITURRIAGA AND H. SÁNCHEZ M.

So

lim sup_{T →∞} \frac{S(T)}{T} ≤ (m + ε).

Item (c) now follows from the fact due to Mañé [4, 1] that m = −c(L).

Proof of Theorem 2. We begin with the last statement. Note that L + k is bigger than zero if and only if H(x, 0) < k. Indeed

H(x, p) = max_{v∈T_{x}M}(pv − L(x, v))

then

H(x, 0) = max_{v∈T_{x}M}(−L(x, v)) = − min(L(x, v)

So if k is bigger than − inf L then H^{−1}(−∞, k) contains the zero section of T^{∗}M. Define now a new Hamiltonian G on T^{∗}M minus the zero section such that G takes

the value μ on H^{−1}(k) and such that G(x, λp) = λ^{2}G(x, p) for all positive λ. Since G is positively homogeneous of degree two and convex in p, it follows that the Legendre transform F associated to G is a Finsler metric.

Since by definition G^{−1}(μ) = H^{−1}(k) it follows that the Hamiltonian flows of G and H^{−1}(k) coincide up to reparameterization on the energy level G^{−1}(μ) = H^{−1}(k) and therefore the Euler-Lagrange solutions of L with energy k are reparameterizations of geodesics of F.

We claim that for an appropriate choice of μ and if E(x, v) = k then

\sqrt{F (x, v)} = L + k

It is proved in [4, 1] that for k greater than the critical value and for any x, y in M there exists γ such that A_{L+k}(γ) = Φ_{k}(x, y). Moreover γ is a solution of the Euler-Lagrange equation and has energy k. Also, if k > c(L), every curve can be reparameterized to have energy k and the Finsler length does not depend on the reparameterization. So for both definitions D_{F} and Φ_{k} we may restrict ourselves to curves with energy k and the Theorem follows in this case.

Proof of the claim: Since G is homogeneous of degree 2 it follows from Euler’s formula that F and G take the same value at Legendre related points.

Define λ(x, p) such that H(x, \frac{p}{λ}) = k, then G(x, p) = μλ^{2}(x, p).

We have that

\frac{∂H}{∂p} (x, \frac{p}{λ})λ^{−1} − \frac{∂H}{∂p} (x, \frac{p}{λ}) · pλ^{−2} \frac{∂λ}{∂p} = 0 (2)

and

\frac{∂G}{∂p} = 2μλ\frac{∂λ}{∂p}.

Then using (2) multiplied by λ^{2} we get

\frac{∂H}{∂p} (x, \frac{p}{λ}) · p\frac{∂G}{∂p} = 2G(x, p)\frac{∂H}{∂p} (x, \frac{p}{λ}). (3)

Suppose now that E(x, v) = k and define P (x, v) = \frac{∂L}{∂v} then by definition we have

(4)  λ(x, P (x, v)) = 1
(5)  G(x, P (x, v)) = μ
(6)  \frac{∂H}{∂p} (x, P (x, v)) = v

and so

(7)  \frac{∂H}{∂p} (x, P (x, v)) · P (x, v) = v \frac{∂L}{∂v}
(8)  = L + k
(9)  > 0.

Hence from (3) we have

\frac{∂G}{∂p}(x, P (x, v)) = \frac{2v}{v · P (x, v)}.

Then since \frac{∂G}{∂p} is homogeneous of degree one and from (7) v · P (x, v) is positive we obtain

\frac{∂G}{∂p}(x, \frac{1}{2}v · P (x, v)P (x, v)) = v

So v is related to \frac{1}{2}v · P (x, v)P (x, v) with respect to the Legendre transform of F. Hence

F (x, v) = G(x, \frac{1}{2}v · P (x, v)P (x, v))
= \frac{(v · P (x, v))^{2}}{4} G(x, P (x, v))
= \frac{(v · P (x, v))^{2}}{4} μ

So if μ = 4

\sqrt{(F (x, v))} = v · \frac{∂L}{∂v}

Let now k be bigger than c(L) then by a corollary of [2] there exists a C^{∞} real valued function on M f, such that H(x, df_{x}) < k. Define as in [2] H_{df}(x, p) = H(x, p + df_{x}). The Hamiltonian flows are conjugate by ψ(x, p) = (x, p − df_{x}) The Legendre transformation L_{df} of H_{df} is

L_{df}(x, v) = max_{p∈T_{x}^{∗}M}(pv − H_{df}(x, p)
= max_{p∈T_{x}^{∗}M}(pv − H(x, p + df_{x}))
= max_{p∈T_{x}^{∗}M}((p − df_{x})v − H(x, p))
= L(x, v) − df_{x}v

It turns out that

E(L_{df}) = E(L)
c(L_{df}) = c(L)
ϕ_{k}(L_{df})(p, q) = ϕ_{k}(L)(p, q) − f(q) + f(p)

So as the zero section is contained in H_{df}^{−1}(−∞, k), L_{df} + k is positive and there is a Finsler metric such that

ϕ_{k}(L_{df})(p, q) = D_{F} (p, q)

So

ϕ_{k}(L)(p, q) = D_{F} (p, q) + f(q) − f(p).

The results in this paper were motivated by discussions with R. Montgomery about the way the Principle of Maupertuis is presented in the book [3].

References

[1] G. Contreras, J. Delgado, R. Iturriaga, Lagrangian flows: the dynamics of globally minimizing orbits II, Bol. Soc. Bras. Mat. Vol. 28, N.2, (1997) 155–196.

[2] G. Contreras, R. Iturriaga, G. P. Paternain, M. Paternain Lagrangian graphs, minimizing measures and Mañé’s critical values Geometric and Functional Analysis vol 8 (788-809) 1998.

[3] Martin Gutzwiller Chaos in Classical and Quantum Mechanics Springer Verlag 1990.

[4] R. Mañé, Lagrangian flows: the dynamics of globally minimizing orbits, International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé), F. Ledrappier, J. Lewowicz, S. Newhouse eds., Pitman Research Notes in Math. 362 (1996) 120–131. Reprinted in Bol. Soc. Bras. Mat. Vol 28, N. 2, (1997) 141-153.

[5] J. Mather. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207, (1991), no. 2, 169-207.

[6] R.T. Rockafellar. Convex Analysis. Princeton Landmarks in Math. Princeton. Princeton University Press (1997).

CIMAT, A.P. 402, 36000, Guanajuato. Gto., México.

E-mail address: renato@@fractal.cimat.mx

Instituto de Matemáticas. UNAM, Ciudad Universitaria C. P. 04510, México, DF, México.

E-mail address: hector@@matem.unam.mx