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Calculus of Variations in the Convex Case : an Introduction to Fathi's

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Calculus of Variations in the Convex Case : an Introduction to Fathi's Calculus of variations in the convex case : an introduction to Fathi’s weak KAM theorem and Mather’s theory of minimal invariant measures Alain Chenciner, Barcelona july 2004 1st lecture. Calculus of variations in the convex case (local struc- tures). From Euler-Lagrange equations to the Poincar´e-Cartan integral invariant, the Legendre transform and Hamilton’s equations. Exercices. Flows, differential forms, symplectic structures 2nd lecture. The Hamilton-Jacobi equation. The solutions of Hamilton’s equations as characteristics. Lagrangian sub- manifolds and geometric solutions of the Hamilton-Jacobi equation. Caus- tics as an obstruction to the existence of global solutions to the Cauchy problem. Exercices. The geodesic flow on a torus of revolution as an example of a completely integrable system 3rd lecture. Minimizers. Weierstrass theory of minimizers. Minimizing KAM tori, Existence of min- imizers (Tonelli’s theorem) and the Lax-Oleinik semi-group. Exercices. Examples around the pendulum 4th lecture. Global solutions of the Hamilton-Jacobi equation Weak KAM solutions as fixed points of the Lax-Oleinik semi-group; con- vergence of the semi-group in the autonomous case. Conjugate weak KAM solutions. Exercices. Burger’s equation and viscosity solutions. 5th lecture. Mather’s theory. Class A geodesics and minimizing mea- sures. The α and β functions as a kind of integrable skeleton Exercices. The time-periodic case as a generalization of Aubry-Mather the- ory, Birkhoff billiards, Hedlund’s example in higher dimension. 1 1st lecture. Calculus of variations in the convex case (local struc- tures). General convexity hypotheses. M = TI n = IRn/ZZ n is the n-dimen- sional torus (the theory works with an arbitrary compact manifold but the torus will allow us to work with global coordinates). The C∞ (C3 would be enough) Lagrangian L(q, q,˙ t) L : TTI n × IR = TI n × IRn × IR → IR will be assumed to satisfy the “Mather” hypotheses (the third one will be explained later : it is only in case L depends effectively on the time variable t that it is not automatically satisfied) : 1) L is strictly convex in q˙, that is (in the sense of quadratic forms) : ∂2L ∀q, q,˙ t, (q, q,˙ t) > 0; ∂q˙2 2) L is superlinear in q˙ : ∀C ∈ IR, ∃D ∈ IR, ∀q, q,˙ t, L(q, q,˙ t) ≥ C||q˙|| − D, L(q,q,t˙ ) ∞ that is lim||q˙||→∞ ||q˙|| =+ uniformly in (q, t). 3) the Euler-Lagrange flow associated to L is complete. Path. A C0 and piecewise C1 mapping γ :[a, b] → TI n. When only minima of the action are concerned, it is more natural to work with absolutely continuous paths. Action. To a path γ, one associates its action b AL(γ)= L(γ(t), γ˙ (t),t) dt. a Variation. A variation of γ is a mapping from ] − , [×[a, b]toTI n, (u, t) → Γ(u, t)=γu(t), such that 1) γ0 = γ; 2) ∀u, γu is a path; 3) ∂Γ/∂u is continuous; 4) there exists a subdivision of [a, b] into subintervals [τi,τi+1] such that ∂2Γ/∂u∂t and ∂2Γ/∂t∂u are continuous (and hence equal) on the rectangles ] − , [×[τi,τi+1]. Infinitesimal variation. It is the vector-field on TI n along γ defined by ∂Γ X(t)= (0,t). ∂u 2 Figure 1 It is C0 and piecewise C1, and vanishes at a and b. The set of all these infinitesimal variations plays the rˆole of the tangent space to the “manifold of paths”. Computing the derivative of the function u →AL(γu) via an integration by parts, one gets b ∂L t ∂L dAL(γ)X = γ(t), γ˙ (t),t − γ(s), γ˙ (s),s ds · X˙ (t) dt. a ∂q˙ a ∂q The following lemma is classical : → Lemma (Erdmann). Let ϕ :[a, b] R be continuous except possibly at b ˙ 0 1 a finite set of points. If a ϕ(t)ψ(t)dt =0for every C and piecewise C → function ψ :[a, b] IR, which vanishes at a and b, the function ϕ coincides with the constant 1 b ϕ(t)dt at each point of continuity. b − a a Extremals. The paths γ such that dAL(γ)X = 0 for any infinitesimal variation X. Euler-Lagrange equations (integral form). One deduces from the Erdmann lemma that a path γ is an extremal iff there exist constants Ci ,i=1, 2 ···n, such that, for i =1, 2 ···n, t ∂L ∂L γ(t), γ˙ (t),t = γ(s), γ˙ (s),s ds + Ci . (E ) ∂q˙i a ∂qi Legendre mapping. The “general hypotheses” we made on L imply that the Legendre mapping Λ:TTI n × IR = TI n × IRn × IR → (IRn)∗ ×TI n × IR = T ∗TI n × IR defined by ∂L Λ(q, q,˙ t)=(p, q, t),p= (q, q,˙ t), ∂q˙ is a global diffeomorphism ( strict convexity for all p ofq ˙ → L(q, q,˙ t) − p · q˙ implies the injectivity of Λ and surlinearity implies that it is proper, hence surjective). One says that L is globally regular. Using equations (E), this implies immediately the 3 Regularity lemma. Any extremal is as regular as L, and the following form of equations (E): Euler-Lagrange equations (differential form). d ∂L ∂L γ(t), γ˙ (t),t = γ(t), γ˙ (t),t ,i=1, ···,n. (E) dt ∂q˙i ∂qi This amounts to computing dAL(γ)·X by the “other” integration by parts, which is permitted because γ is regular. Intrinsic character of equations (E) : the Euler-Lagrange flow. It follows from the fact that Λ is a diffeomorphism that these equations define n ∗ ∗ n (time-dependant if L is) vector-fields XL in TTI and XH in T TI (the ∗ notation XH will be explained below). These vector-fields are intrinsically defined (i.e. they do not depend on the choice of local or global coordinates on TI n). Their flows will both be called the Euler-Lagrange flow. Indeed, their variational origin implies that the Euler-Lagrange equations (E) take exactly the same form in any local or global coordinate system. In other words, the mapping [L] :[a, b] → T ∗TI n defined by γ ∂L d ∂L [L] (t)= γ(t), γ˙ (t),t − γ(t), γ˙ (t),t ∈ T ∗ TI n γ ∂q dt ∂q˙ γ(t) is an intrinsically defined field of covectors tangent to TI n “along γ” and the derivative of the action can be written b dAL(γ) · X = [L]γ (t) · X(t) dt. a Unconstrained variations and the Poincar´e-Cartan integral in- variant. The main structures of classical mechanics can be deduced from a single computation : the variations of the action when no constraints are imposed on the extremities of the paths γu or on their intervals of definition ∂q [a(u),b(u)]. We note X (t)= = ∂Γ(u, t) the infinitesimal variations. u ∂u ∂u d d b(u) AL γu = L (γu(t), γ˙ u(t),t) dt du du a(u) b(u) ∂L d ∂L = − γ (t), γ˙ (t),t · X (t) dt ∂q dt ∂q˙ u u u a(u) ∂L ∂L + γu(t), γ˙ u(t),t · Xu(t) − γu(t), γ˙ u(t),t · Xu(t) ∂q˙ t=b(u) ∂q˙ t=a(u) db da + L γu(t), γ˙ u(t),t (u) − L γu(t), γ˙ u(t),t (u) , du t=b(u) du t=a(u) a formula that we shall abreviate in dA b ∂L d ∂L ∂q ∂L ∂q dt b L = − · dt + · + L . du a ∂q dt ∂q˙ ∂u ∂q˙ ∂u du a 4 If, in particular, γu is a family of extremals of Ldt,weget dA ∂L ∂q dt b L = · + L . du ∂q˙ ∂u du a ∂q ∂Γ We replace now the partial derivative ∂u (that is ∂u), deprived of geometric meaning, by the “effective variation” d dq ∂q ∂q dt ∂q dt Γ u, t(u) = = + = +˙q ,t(u)=a(u)orb(u), du du ∂u ∂t du ∂u du of the extremities of the path γu as a fonction of u. (figure 2.1). This d A transforms the expression of du ( L(γu)) for a family of extremals into an identity between differential 1-forms on the interval U of definition of the parameter u : Figure 2 A ∗ − ∗ d L = δb L δa L, U→ ∗ n × where δa,δb : T TI IR denote the mappings ∂Γ δ (u)= Γ u, t(u) , u, t(u) ,t(u) ,t(u)=a(u)orb(u), t ∂t n × and L is the differential 1-form on TTI IR defined by ∂L ∂L = (q, q,˙ t) · dq − (q, q,˙ t) · q˙ − L(q, q,˙ t) dt. L ∂q˙ ∂q˙ Finally, we can simplify the formulas by transporting everything on the cotangent side with the Legendre diffeomorphism Λ. The function on T ∗TI n × IR defined by H(p, q, t)=p · q˙ − L(q, q,˙ t), whereq ˙ is expressed in terms of p, q, t via Λ is called the Legendre transform of L, or the Hamiltonian associated to the Lagrangian L.If H denotes the 1-form on T ∗TI n × IR defined by H = p · dq − H(p, q, t)dt, the formula for the unconstrained variations of extremals becomes ∗ ∗ dAL =(Λ◦ δb) H − (Λ ◦ δa) H . The 1-form H is the Poincar´e-Cartan integral invariant (tenseur impul- sion-´energie in Cartan’s terminology). 5 Rewriting the action. The action istself can now be written as the integral of = p · dq − Hdt on the lift Γ∗(t)= ∂L(γ(t), γ˙ (t),t),γ(t),t H ∂q˙ to T ∗TI n × IR of the path γ(t)inTI n : AL(γ)= H .
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