1 Energy In this section the setting is as in the previous talk, i.e.

(1)

1 Energy

In this section the setting is as in the previous talk, i.e. M is a connected, closed manifold andL:T M →RTonelli.

Definition 1.1. Let Lbe a Tonelli Lagrangian on M. Recall the definition of the Hamiltonian

H :T^∗M →R, H(x, p) :=p(Leg⁻¹(x, p))−L(Leg⁻¹(x, p)).

The energy associated toL is the functionE:T M →R, E:=H◦Leg, i.e.

E(x, v) := ∂L

∂v(x, v)(v)−L(x, v)

Remark 1.2. IfLis Tonelli then its associated energyE is Tonelli as well.

Example 1.3. We consider the electromagnetic Lagrangians L(x, v) = 1

2gx(v, v) +θx(v)−U(x)

whereg is a Riemannian metric on M, θ∈Ω¹(M) a 1-form andU ∈C^∞(M) a function on M. To compute the energy and Hamiltonian we first have to compute the conjugate momentum. For that we choose local coordinates onM and get:

∂L

∂v(x, v) = ∂L

∂vⁱdxⁱ= ∂

∂vⁱ(1

2g_jkv^jv^k+θ_j(x)v^j−U(x))dxⁱ=g_x(v,·) +θ_x Therefore the energy is given by:

E(x, v) =g(v, v) +θ(v)−(1

2gx(v, v) +θx(v)−U(x)) = 1

2gx(v, v) +U(x).

This is just the sum of the kinetic and potential energy, the 1-form θ doesn’t affect the energy. Recall that the norm|·|xonTxM induces a norm also denoted by| · |xonT_x^∗M, which is given by|p|x= sup_w∈T

xM,|w|x≤1|p(w)|. Now setting (x, p) =Leg(x, v) we compute the Hamiltonian:

H(x, p) =E(x, v) =1

2|v|²_x+U(x) = 1

2|g_x(v,·)|²_x+U(x) = 1

2|p−θ_x|²_x+U(x).

Example 1.4. LetL be a Tonelli Lagrangian, θ a 1-form andU a function on M. Let ˜L(x, v) :=L(x, v) +θ_x(v)−U(x). For the associated energiesE,E˜ we get:

E(x, v) =˜ E(x, v) +U(x).

(2)

2 Tonelli Theorem and action minimizers

In this section leta, b∈Rwitha < bandM a connected manifold.

Theorem 2.1. LetM be a connected, closed manifold,La Tonelli Lagrangian.

For eachx0, x1∈M and homotopy classhof curves connectingx0andx1, there is aγh∈C_x²₀_,x₁([a, b], M;h)minimizing the action in the setC_x²₀_,x₁([a, b], M;h).

Firstly we will consider absolutely continuous curves to get better compactness properties. Secondly, the idea is to lift the problem of finding action minimizers in a fixed homotopy class to the universal cover ˜M of M. On ˜M the task will then be to find action minimizers. So far we have only consideredM compact. But we don’t know whether the universal cover ˜M is compact as well.

We therefore have to consider the non-compact case as well.

Definition 2.2. Let dbe the metric onM obtained by some fixed Riemannian metric onM.

A curve γ : [a, b] →M is called absolutely continuous if for each > 0 there exists δ > 0 such that for any familiy of disjoint intervals (]a_i, b_i[)_i=1,...,n all included in[a, b]and satisfyingP

i(b_i−a_i)< δ, we have P

id(γ(b_i), γ(a_i))< . We denote byC^ac([a.b], M)the set of absolutely continuous curves γ: [a, b]→ M.

Remark 2.3. For a curve γ : [a, b] →M the property of being absolutely continuous is independent of the chosen Riemannian metric, see [5, Proposition 3.18].

Remark 2.4. Let γ : [a, b] → M be an absolutely continuous curve. Then:

˙

γ∈T_γM exists almost everywhere on [a, b] and d(γ(a), γ(b))≤

Z b

a

||γ(s)||˙ γ(s)ds

Remark 2.5. For a Tonelli Lagrangian L and an absolutely continuous curve γ : [a, b] → M the action A_L(γ) is well defined and in R∪ {∞}.(Since L is bounded below)

Definition 2.6. Let Lbe Tonelli, x0, x1∈M. An absolutely continuous curve γL∈C_x^ac₀_,x₁([a, b], M)is called Tonelli minimizer if

A_L(γ_L) = min

γ∈C_x^ac

0,x1([a,b],M)

A_L(γ)

Theorem 2.7. (Tonelli theorem) LetM be a connected manifold,L:T M →R a Tonelli Lagrangian bounded below by a complete Riemannian metric on M, i.e. there exist a complete Riemannian metric g on M and some B ∈R such thatL(x, v)≥ |v|_g+B.

Then for eachx₀, x₁∈M there exists a Tonelli minimizer.

Remark 2.8. IfM is compact, then the assumption ofLbeing Tonelli suffices.

(3)

Proof. We only sketch the proof for the Tonelli theorem. The main idea is to show that for eachx0, x1∈M, C∈Rthe set

S_C^x⁰^,x¹:={γ∈C_x^ac₀_,x₁([a, b], M)| AL(γ)≤C}

is a compact subset ofC^ac([a, b], M) for the topology of uniform convergence.

Using this fact one can proceed as follows: SetC:= inf

γ∈C_x^ac

0,x1([a,b],M)

A_L(γ)(exists sinceL is bounded below). Then the setsS_C+^x⁰^,x1¹

n

form a decreasing sequence of non-empty compact sets. Therefore the intersectionT

n

S^x⁰^,x¹

C+_n¹ is nonempty. Each curve in this intersection is a minimizer.

We now sketch how the compactness of the setsS^x_C⁰^,x¹ in the Tonelli theorem can be proven whenM is compact:

1. The setsS_C={γ∈C^ac([a, b], M)|A_L(γ)≤C}are absolutely equicontinuous, i.e for each >0, there existsδ >0 such that for each disjoint family (]ai, bi[)i=1,...,n⊂[a, b] with

n

P

i=1

(bi−ai)< δ we have

n

P

i=1

d(γ(ai), γ(bi))<

for all γ ∈ SC.(Here one needs superlinearity) This implies cl_C0SC ⊂ C^ac([a, b], M).

2. If (γn)n ⊂SC converges uniformly toγ, thenAL(γ)≤lim inf

n→∞ AL(γn).

3. Apply the Arzela-Ascoli theorem: By 1. and 2.,S_Cis closed and equicontinuous. SinceM is compact, the sets{γ(t)|γ∈S_C} are precompact for all t ∈ [a, b]. Thus SC is compact in the C⁰ topology. S^x_C⁰^,x¹ ⊂ SC is compact as a closed subset of a compact set.

In the following we will always assumeL≥0. This is possible by adding a constant, sinceLis bounded below.

Proof. of 1.:

Set forr >0:

K(r) := inf{L(x, v)

|v|_x |(x, v)∈T M,|v|x≥r}

By superlinearity ofL

r→∞lim K(r) = +∞.

Thus for given >0 we can findr >0 with C K(r) <

2.

(4)

Letγ∈SC, J :=

N

S

i=1

[ai, bi] and E:=J ∩ {|γ|˙ γ > r}. Then

K(r)

N

X

i=1

d(γ(ai), γ(bi))≤K(r) Z

E

|γ(s)|˙ γ(s)ds+K(r) Z

J−E

|γ(s)|˙ γ(s)ds

≤ Z

E

L(γ(s),γ(s))ds˙ +K(r)rµ(J)

≤C+K(r)rµ(J) (L≥0).

Dividing byK(r) we obtain:

N

X

i=1

d(γ(ai), γ(bi))≤

2 +rµ(J),

which proves that the setS_C is absolutely equicontinuous. Hereµdenotes the lebesque measure. (This also impliescl_C0S_C⊂C^ac, since the uniform limit of an absolutely equicontinuous family of absolutely continuous curves is absolutely continuous)

Proof. of 2.: Let (γn)⊂SCconverge uniformly toγ. From the above discussion we knowγ∈C^ac([a, b], M) and want to showAL(γ)≤lim inf

n→∞ AL(γn). The main steps to show this are:

• Reduction to the caseimγ⊂U where (U, φ) is a chart onM.

We coverimγ by finitely many charts Ui such that there is a subdivision a=a0 < a1 < ... < ak =b with γ([ai−1, ai])⊂Ui. By uniform convergence of γn we can assume γn([ai−1, ai]) ⊂Ui. If the assertion holds for imγ⊂U, whereU is a chart onM, then:

A_L(γ) =X

i

A_L(γ_|[a_i−1_,a_i_])≤X

i

lim inf

n→∞ A_L(γ_n[a_i−1_,a

i])≤lim inf

n→∞ A_L(γ_n).

By the identification U =φ(U) we can from now on assume thatimγ is contained in an open subsetU ofRⁿ.

• Lemma: Let K ⊂U be compact, r > 0, >0. There exists δ >0 such that ifx∈K, y∈K,|x−y| ≤δandv, w∈Rⁿ,|v| ≤r, then

L(x, v) +∂L

∂v(x, v)(w−v)−≤L(y, w).

Proof. We define

C1:= sup{|∂L

∂v(x, v)| |x∈K,|v| ≤r}, C₂:= sup{L(x, v)−∂L

∂v(x, v)v |x∈K,|v| ≤r}.

(5)

Then we chooses >0 such that for all R≥s:

K(s)·R≥C2+C1·R, whereK(s) is from the above proof. If|w| ≥s, then

L(y, w)≥K(s)|w| ≥C₂+C₁|w| ≥L(x, v) +∂L

∂v(x, v)(w−v).

Hence we only have to find aδsuch that the asserted inequality holds if

|w| ≤s. SinceLis convex,

L(x, w)≥L(x, v) +∂L

∂v(x, v)(w−v).

By compactness of{(x, w)|x∈K,|w| ≤s} we obtain the desiredδ.

• Apply this lemma with the compact (due to uniform convergence) set K=imγ∪S

n

imγn and set Er:={|γ| ≤˙ r}to get fornbig enough:

Z

E_r

[L(γ,γ) +˙ ∂L

∂v(γ,γ)( ˙˙ γ_n−γ)˙ −]ds≤ Z

E_r

L(γ_n,γ˙_n)ds≤A_L(γ_n).

• Show

Z

Er

[∂L

∂v(γ,γ)( ˙˙ γn−γ)]ds˙ →0, asn→ ∞.

This follows from Lemma 1.3.3 in [1, Maz]. To apply this Lemma note that as in the proof of 1. we see that ( ˙γ_n)_n is uniformly integrable and therefore ˙γ_n−γ˙ is uniformly integrable.

• Letr→ ∞and get:

A_L(γ)−|b−a| ≤lim inf

n→∞ A_L(γ_n).

• Let→0, thenA_L(γ)≤lim inf

n→∞ A_L(γ_n).

This proof doesn’t work for the noncompact case, since superlinearity holds only above compact subsets ofM. Even if we could show that SC is equicontinuous, we couldn’t apply Arzela-Ascoli, because the sets {γ(t)|γ ∈ S_C^x⁰^,x¹} aren’t necessarily precompact. But we can modify this proof to obtain that for K ⊂ M compact, the sets SC,K := {γ ∈ SC|imγ ⊂ K} are equicontinuous and therefore compact. Let’s see how the fact thatL is bounded below by a complete Riemannian metric can be used to show thatS_C^x⁰^,x¹ is compact. Let γ∈S_C^x⁰^,x¹, then for eacht∈[a, b]:

d(γ(a), γ(t))≤ Z t

a

|γ|˙ γds≤AL(γ)−B·(t−a)≤C+B(b−a) =:R,

(6)

and henceS^x_C⁰^,x¹ ⊂S_C,B¯_x₀(R). Since the Riemannian metric is complete, closed metric balls inM are compact. Thus S_C^x⁰^,x¹ is compact, because it is a closed subset of the compact setS_C,B¯x0(R).

In the next talk we will see that for a Tonelli Lagrangian, Tonelli minimizers have the same regularity as the Lagrangian.

Theorem 2.9. Suppose that L is a Tonelli Lagrangian on M. Let γL be a Tonelli minimizer. IfL isC^r, then γL isC^r as well.

Let us now return to Theorem 2.1, which stated the existence of action minimizers in a given homotopy class. Letπ: ˜M →M be the universal cover.

We fix ˜x0∈π⁻¹(x0). Then we have a bijectionf : [Cx0,x1]→π⁻¹(x1) between homotopy classes of curves connectingx₀,x₁and elements of the fiber ofx₁. For [γ]∈[C_x₀_,x₁] we choose a lift ˜γof γ and setf([γ]) = ˜γ(b). For each homotopy classh∈[C_x₀_,x₁] we have a bijection h→C_x_˜₀_,f_(h), γ →˜γwhere ˜γis the lift of γwithγ(a) = ˜x₀and ˜γ(b) =f(h) .

Proof. of 2.1

The idea is to apply the Tonelli theorem to the universal cover ˜M ofM and use the 1 : 1 correspondence between curves in h and curves in ˜M with end point f(h).

First we consider the universal cover π: ˜M →M and set ˜g := π^∗g. We now show the the Lagrangian ˜L := L◦dπ on TM˜ satifies the assumptions of the Tonelli theorem. L˜ Tonelli can easily be verified.. Since M is compact g is complete. Since π is a Riemannian covering, ˜g is also complete. Since L is superlinear we can findC∈Rsuch thatL(x, v)≥ |v|g,x+Cfor all (x, v)∈T M and therefore ˜L(˜x,˜v) =L(π˜x, dπv)˜ ≥ |dπ˜v|_g,π˜_x+C=|˜v|_g,˜_˜_x+C.

By the Tonelli theorem there is a ˜γh∈C_x^ac_˜

0,f(h)([a, b],M˜) such that AL˜(˜γh) = min

γ∈C˜ ^ac_x_˜

0,f(h)([a,b],M˜)

AL˜(˜γ) = min

γ∈C^ac_x

0,x1([a,b],M;h)AL(γ).

In the second equation we used the bijection h → Cx˜₀,f(h), γ → ˜γ and the following two facts for a lift ˜γ of some curveγ∈C([a, b], M):

1)AL˜(˜γ) =A_L(γ) ifγ is absolutely continuous and

2) γ absolutely continuous iff ˜γ absolutely continuous: Let ˜U_α ⊂M , U˜ _α ⊂M such thatπ: ˜Uα :→Uαis a diffeomorphism, ˜Uα and Uα are uniformly normal neighborhoods and the ˜Uαcoveringim˜γ. There is a lebesque numberδ0>0 such that fors, t∈[a, b],|s−t|< δ0, we have ˜δ([s, t])⊂U˜αfor someα. For suchs, t we have: dM˜(˜γ(s),˜γ(t)) = dU˜α(˜γ(s),γ(t)) =˜ dU_α(γ(s), γ(t)) = dM(γ(s), γ(t)) where the first and third equality follow fromU_α,U˜_α being uniformly normal neighborhoods and the second fromπ: ˜Uα:→Uαbeing a isometry. The stated equivalence now follows if we chooseδ < δ0. This can be shown more easily if we use the definition of absolute continuity using charts.

Therefore the curveγh:=π◦˜γhhas the desired property. By the preceeding regularity theorem,γh isC².

(7)

Remark 2.10. LetM be compact andLTonelli. Fixx0, x1∈M. Then C_x²₀_,x₁([a, b], M) = a

h∈[Cx0,x1]

C_x²₀_,x₁([a, b], M;h).

Forh∈[Cx₀,x₁] there exists a minimizerγhforALinC_x²₀_,x₁([a, b], M;h). More- over there exists a minimizerγL forALin C_x²₀_,x₁([a, b], M). In particular there exists a homotopy classh∈[Cx0,x1] such that γL=γhL and

AL(γL) = min

h∈[Cx0,x1]AL(γh).

Now consider a closed 1-form θ. By talk 1 γ_h is still a minimizer for A_L+θ in C_x²

0,x₁([a, b], M;h), withA_L+θ(γ_h) =A_L(γ_h) +C_hfor some constantC_h. Ifθis exact, thenC_h is independent ofhand

{γ_L∈C_x^ac

0,x₁([a, b], M)}={γ_L+θ∈C_x^ac

0,x₁([a, b], M)}.

However, ifθis not exact, then it might happen that this is false because AL+θ(γL+θ) = min

h∈[C_x₀_,x₁](AL(γh) +Ch), AL+θ(γL) = min

h∈[C_x₀_,x₁]AL(γh) +C[γ_L]

Ifc ∈H_deRham¹ (M), we can then considerT on^c :={γL+θ ∈C_x^ac₀_,x₁([a, b], M)}, where [θ] = c. By the discussion above T on^c does not depend on the repre- sentativeθ. We will see the role ofH¹(M;R) and ofH1(M;R) in more details when we will consider minimizing measures.

2.1 Appendix

Proposition 2.11. Let(M, d)be a metric space and(Kn)a family of decreasing nonempty compact subsets ofM. Then T

i∈N

Ki is nonempty.

Proof. Letxn∈Kn. Since (xn)nis contained in the compact setK1, there is a subsequence (xn_j) converging to somex∈K1. For eachn andj with nj > n, xn_j ∈Kn. SinceKn is compact this impliesx∈Kn.

References

[1] Marco Mazzucchelli,Critical point theory for Lagrangian systems, Progress in Mathematics, 293. Birkhuser/Springer Basel AG, Basel, 2012.

[2] Albert Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version, June 2008.

[3] Gonzalo Contreras and Renato Iturriaga,Global minimizers of autonomous Lagrangians, 22 Coloquio Brasileiro de Matematica, IMPA, 1999.

(8)

[4] Alfonso Sorrentino, Action-minimizing methods in Hamiltonian dynamics.

An introduction to Aubry-Mather theory., Mathematical Notes, 50. Prince- ton University Press, 2015.

[5] Burtscher Annegret, Length structures on manifolds with continuous Rie- mannian metrics, New York Journal of Mathematics 21, 2015. http:

//nyjm.albany.edu/j/2015/21-12v.pdf