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Action-minimizing Methods in Hamiltonian Dynamics
An Introduction to Aubryâ"Mather Theory
By Alfonso Sorrentino PRINCETON UNIVERSITY PRESS
Copyright © 2015 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-6661-8
Contents
Preface, vii,
1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds, 1,
2 From KAM Theory to Aubry-Mather Theory, 8,
3 Action-Minimizing Invariant Measures for Tonelli Lagrangians, 18,
4 Action-Minimizing Curves for Tonelli Lagrangians, 48,
5 The Hamilton-Jacobi Equation and Weak KAM Theory, 76,
Appendices,
A On the Existence of Invariant Lagrangian Graphs, 89,
B Schwartzman Asymptotic Cycle and Dynamics, 97,
Bibliography, 107,
Index, 113,
CHAPTER 1
Tonelli Lagrangians and Hamiltonians on Compact Manifolds
1.1 LAGRANGIAN POINT OF VIEW
In this section we want to introduce the basic setting that we will be considering hereafter. Let M be a compact and connected smooth manifold without boundary. Denote by TM its tangent bundle and by T*M the cotangent one. A point of TM will be denoted by (x, v), where x [member of] M and v [member of] TxM, and a point of T*M by (x, p), where p [member of] T*xM is a linear form on the vector space TxM. Let us fix a Riemannian metric g on it and denote by d the induced metric on M; let
·
x be the norm induced by g on TxM; we will use the same notation for the norm induced on T*xM.
We will consider functions L : TM -> R of class C2, which are called Lagrangians. Associated with each Lagrangian is a flow on TM called the Euler-Lagrange flow, defined as follows. Let us consider the action functional AL from the space of continuous piecewise C1 curves γ : [a, b] ->M, with a ≤ b, defined by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Curves that extremize this functional among all curves with the same endpoints are solutions of the Euler-Lagrange equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)
Observe that this equation is equivalent to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
therefore, if the second partial vertical derivative [partial derivative]2L/[partial derivative]v2(x, v) is nondegenerate at all points of TM, we can solve for [??](t). The condition
det [partial derivative]2L/[partial derivative]v2 ≠ 0
is called the Legendre condition and allows one to define a vector field XL on TM such that the solutions of [??](t) = XL(γ(t, [??](t)) are precisely the curves satisfying the Euler-Lagrange equation. This vector field XL is called the Euler-Lagrange vector field and its flow ΦLt is the Euler-Lagrange flow associated with L. It turns out that ΦLt is C1 even if L is only C2 (see Remark 1.2.2).
Definition 1.1.1 (Tonelli Lagrangian). A function L : TM -> R is called a Tonelli Lagrangian if:
i) L [member of] C2(TM);
ii) L is strictly convex in each fiber, in the C2 sense, i.e., the second partial vertical derivative [partial derivative]2L/[partial derivative]v2(x, v) is positive definite, as a quadratic form, for all (x, v);
iii) L is superlinear in each fiber, i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This condition is equivalent to asking whether for each A [member of] R there exists B(A) [member of] R such that
L(x, v) ≥ A
v
- B(A) [for all](x, v) [member of] TM.
Observe that since the manifold is compact, condition iii) is independent of the choice of the Riemannian metric g.
Remark 1.1.2. More generally, one can consider the case of a time-periodic Tonelli Lagrangian L : TM x T -> R (also called non-autonomous case), as it was originally done by John Mather [62]. In fact, as it was pointed out by Jürgen Moser, this was the right setting for generalizing Aubry and Mather's results for twist maps to higher dimensions; in fact, every twist map can be seen as the time-one map associated with the flow of a periodic Tonelli Lagrangian on the one-dimensional torus (see for instance [71]). In this case, a further condition on the Lagrangian is needed:
iv) The Euler-Lagrange flow is complete, i.e., every maximal integral curve of the vector field XL has all R as its domain of definition.
In the non-autonomous case, in fact, this condition is necessary in order to have that action-minimizing curves (or Tonelli minimizers; see section 4.1) satisfy the Euler-Lagrange equation. Without such an assumption Ball and Mizel [8] have constructed an example of Tonelli minimizers that are not C1 and therefore are not solutions of the Euler-Lagrange flow. The role of the completeness hypothesis can be explained as follows. It is possible to prove, under the above conditions, that action-minimizing curves not only exist and are absolutely continuous, but also are C1 on an open and dense full measure subset of the interval in which they are defined. It is possible to check that they satisfy the Euler-Lagrange equation on this set, while their velocity goes to infinity on the exceptional set on which they are not C1. A complete flow, therefore, implies that Tonelli minimizers are C1 everywhere and that they are actual solutions of the Euler-Lagrange equation.
A sufficient condition for the completeness of the Euler-Lagrange flow, for example, can be expressed in terms of a growth condition for [partial derivative]L/[partial derivative]t:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Examples of Tonelli Lagrangians.
• Riemannian Lagrangians. Given a Riemannian metric g on M, the Riemannian Lagrangian on (TM, g) is given by the Kinetic energy:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Its Euler-Lagrange equation is the equation of the geodesics of g:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(where [nabla][??][??] denotes the covariant derivative), and its Euler-Lagrange flow coincides with the geodesic flow.
• Mechanical Lagrangians. These Lagrangians play a key role in the study of classical mechanics. They are given by the sum of the kinetic energy and a potential U : M -> R:
L(x, v) = 1/2
v
2x + U(x).
The associated Euler-Lagrange equation is given by:
[nabla][??][??] = [nabla]U(x),
where [nabla]U is the gradient of U with respect to the Riemannian metric g, i.e.,
dxU · v = ([nabla]U(x), v)x [for all](x, v) [member of] TM.
• Mañé's Lagrangians. This is a particular class of Tonelli Lagrangians, introduced by Ricardo Mañé in [51] (see also [39]). If X is a Ck vector field on M, with k ≥ 2, one can embed its flow φXt into the Euler-Lagrange flow associated with a certain Lagrangian, namely
LX(x, v) = 1/2
v - X(x)
2x.
It is quite easy to check that the integral curves of the vector field X are solutions to the Euler-Lagrange equation. In particular, the Euler-Lagrange flow [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] restricted to Graph(X) = {(x, X(x)), x [member of] M} (which is clearly invariant) is conjugated to the flow of X on M, and the conjugation is given by π|Graph(X), where π : TM ->M is the canonical projection. In other words, the following diagram commutes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
that is, for every x [member of] M and every t [member of] R, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where γXx = φXt(x).
1.2 HAMILTONIAN POINT OF VIEW
In the study of classical dynamics, it turns often very useful to consider the associated Hamiltonian system, which is defined on the cotangent bundle T*M. Let us describe how to define this new system and what its relation is with the Lagrangian one.
A standard tool in the study of convex functions is the so-called Fenchel transform, which allows one to transform functions on a vector space into functions on the dual space (see for instance [37, 73] for excellent introductions to the topic). Given a Lagrangian L, we can define the associated Hamiltonian as its Fenchel transform (or Legendre-Fenchel transform):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where (·, ·)x denotes the canonical pairing between the tangent and cotangent bundles.
If L is a Tonelli Lagrangian, one can easily prove that H is finite everywhere (as a consequence of the superlinearity of L) and C2, superlinear and strictly convex in each fiber (in the C2 sense). Such a Hamiltonian is called a Tonelli (or optical) Hamiltonian.
Definition 1.2.1 (Tonelli Hamiltonian). A function H : T*M -> R is called a Tonelli (or optical) Hamiltonian if:
i) H is of class C2;
ii) H is strictly convex in each fiber in the C2 sense, i.e., the second partial vertical derivative [partial derivative]2H/[partial derivative]p2 (x,p) is positive definite, as a quadratic form, for any (x, p) [member of] T*M;
iii) H is superlinear in each fiber, i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Examples of Tonelli Hamiltonians
Let us see what the Hamiltonians associated with the Tonelli Lagrangians that we have introduced in the previous examples are.
• Riemannian Hamiltonians. If L(x, v) = 1/2
v
2x is the Riemannian Lagrangian associated with a Riemannian metric g on M, the corresponding Hamiltonian will be
H(x, p) = 1/2
p
2x,
where
·
represents-in this last expression—the induced norm on the cotangent bundle T*M.
• Mechanical Hamiltonians. If L(x, v) = 1/2
v
2x + U(x) is a mechanical Lagrangian, the associated Hamiltonian is:
H(x, p) = 1/2
p
2x - U(x),
sometimes referred to as mechanical energy.
• Mañé's Hamiltonians. If X is a Ck vector field on M, with k ≥ 2, and LX(x, v) =
v - X(x)
2x is the associated Mañé Lagrangian, one can check that the corresponding Hamiltonian is given by:
H(x, p) = 1/2
p
2x + .
Given a Hamiltonian, one can consider the associated Hamiltonian flow ΦHt on T*M. In local coordinates, this flow can be expressed in terms of the so-called Hamilton equations:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)
We will denote by XH(x, p) := ([partial derivative]H/[partial derivative]p(x, p) - [partial derivative]H/[partial derivative]x(x, p)) the Hamiltonian vector field associated with H. This has a more intrinsic (geometric) definition in terms of the canonical symplectic structure ω on T*M (see appendix A). In fact, XH is the unique vector field that satisfies
ω (XH(x, p), ·) = dxH(·) [for all](x, p) [member of] T*M.
For this reason, it is sometime called the symplectic gradient of H. It is easy to check from both definitions that—only in the autonomous case—the Hamiltonian is a prime integral of the motion, i.e., it is constant along the solutions of these equations.
Now, we would like to explain what the relation is between the Euler-Lagrange flow and the Hamiltonian one. It follows easily from the definition of Hamiltonian (and Fenchel transform) that for each (x, v) [member of] TM and (x, p) [member of] T*M the following inequality holds:
(p, v)x ≤ L(x, v) + H(x, p). (1.3)
This is called Fenchel inequality (or Legendre-Fenchel inequality) and plays a crucial role in the study of Lagrangian and Hamiltonian dynamics and in the variational methods that we are going to describe. In particular, equality holds if and only if p = [partial derivative]L/[partial derivative]v(x, v). One can therefore introduce the following diffeomorphism between TM and T*M, known as the Legendre transform:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)
Moreover the following relation with the Hamiltonian holds:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
A crucial observation is that this diffeomorphism L represents a conjugation between the two flows, namely the Euler-Lagrange flow on TM and the Hamiltonian flow on T*M; in other words, the following diagram commutes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Remark 1.2.2. Since L and the Hamiltonian flow ΦH are both C1, then it follows from the commutative diagram above that the Euler-Lagrange flow is also C1.
Therefore one can equivalently study the Euler-Lagrange flow or the Hamiltonian flow, obtaining in both cases information on the dynamics of the system. Each of these equivalent approaches will provide different tools and advantages, which may be very useful for understanding the dynamical properties of the system. For instance, the tangent bundle is the natural setting for the classical calculus of variations and for Mather's and Mañé's approaches (chapters 3 and 4); on the other hand, the cotangent bundle is equipped with a canonical symplectic structure (see appendix A), which allows one to use several symplectic topological tools coming from the study of Lagrangian graphs, Hofer's theory, Floer homology, among other subjects. Moreover, a particular fruitful approach in T*M is the so-called Hamilton-Jacobi method (or weak KAM theory), which is concerned with the study of solutions and subsolutions of Hamilton-Jacobi equations. In a certain sense, this approach represents the functional analytic counterpart of the above-mentioned variational approach (chapter 5). In the following chapters we will provide a complete description of these methods and their implications on the study of the dynamics of the system.
(Continues...)
Excerpted from Action-minimizing Methods in Hamiltonian Dynamics by Alfonso Sorrentino. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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