Talk 4a: The Hamilton–Jacobi method

Talk 4a: The Hamilton–Jacobi method

Gabriele Benedetti

May 11, 2020

Introduction

The Hamilton-Jacobi method is a powerful way to find orbits minimizing the action. Two flavours:

I time dependent

• good for time-fixed (Tonelli) minimizers,

• we use it to prove Weierstrass’ theorem;

I time independent

• good for time-free minimizers,

• we use it for the pendulum.

Notation:

I L:TM →RTonelli Lagrangian on manifold M, I H :T^∗M →R associated Tonelli Hamiltonian.

Time-dependent subsolutions of HJ-equation

Definition (L-gradient)

LetS :M ×[a,b]→Rbe C¹ and writeSt:=S(·,t) ∀t∈[a,b].

TheL-gradientof S is the time-dependent vector field on M grad_LS_t(x) =Leg⁻¹(d_xS_t), ∀(x,t)∈M×[a,b].

Definition (Time-dependent subsolutions)

AC¹-function S :M×[a,b]→Ris a time-dependent subsolution of the Hamilton-Jacobi equation if

H(x,d_xS_t) +∂_tS_t(x)≤0, ∀(x,t)∈M×[a,b].

We denote byN_S ⊂M×[a,b] the set of pairs (x,t), where equality holds. We say thatS is a solution ifN_S =M ×[a,b].

Time-dependent subsolutions yield Tonelli minimizers

Theorem (A)

Let S:M ×[a,b]→Rbe a subsolution and x0,x1 ∈M. Then, AL(γ)≥Sb(x1)−Sa(x0), ∀γ ∈C_x^ac_0,x1([a,b],M)

with equalityiff γ is a flow line ofgrad_LS_t with (t, γ(t))∈N_S,∀t.

Each such flow line is a Tonelli minimizer.

Proof.

For all (x,v)∈TM we have by the Fenchel inequality L(x,v) +H(x,d_xS_t)≥d_xS_t·v with equality if and only ifv =grad_LS_t(x). Therefore,

L(x,v)≥dxSt·v−H(x,dxSt)≥dxSt·v+∂tSt(x)

=d_(x,t)S·(v+∂t)

with equality if and only ifv =grad_LSt(x) and (x,t)∈NS. Thus,

A_L(γ)≥ Z b

a

d_(γ(t),t)S·( ˙γ(t) +∂_t)dt = Z b

a

d dt

h

S(γ(t),t)i dt

=S(γ(b),b)−S(γ(a),a)

=S_b(x₁)−S_a(x₀).

Reminder of Weierstrass Theorem

Theorem (Part I)

Let L be bounded from below. For allK˜ ⊂TM compact there existsδ >0 such that for all(x,v)∈K the EL-solution˜

γ_(x,v) : [0, δ]→M, (γ_(x,v)(0),γ˙_(x,v)(0)) = (x,v) is well-defined and the unique minimizer in C_x,γ^ac

(x,v)(δ)([0, δ],M).

Theorem (Part II)

Let L be bounded from below. For all K ⊂M compact there exist C, δ >0 such that for all x ∈K and y ∈M with d(x,y)≤Cδ there is a (unique) EL-Solution

γ : [0, δ]→M, γ(0) =x, γ(δ) =y which is the unique minimizer in C_x,y^ac([0, δ],M).

The proof

Part I⇒ Part II.

By the implicit function theorem there existC, δ >0 such that for allx∈K

K˜_x :={v ∈T_xM | |v|_x ≤2C} →M, v 7→γ_(x,v)(δ) is an embedding whose image contains ¯B_Cδ(x). To deduce Part II, apply Part I to ˜K =∪_x∈KK˜x.

To prove Part I we use local existence of HJ-solutions.

Lemma

LetK be a compact set of TM. There are˜ δ, >0 such that for all (x,v)∈K there exists a time-dependent HJ-solution of class C˜ ² S :B(x)×[0, δ]→M with v=grad_LS0(x).

The proof

Proof of Part I.

Given ˜K ⊂TM let δ and as in the lemma:

∀(x,v)∈K˜, ∃S :B(x)×[0, δ]→R,C² solution,v=grad_LS₀(x).

Theorem (A)⇒flow lineγ_(x,v): [0, δ]→B(x) of grad_LS_t

throughx is unique minimizer inC_x,y^ac([0, δ],B(x)),y :=γ_(x,v)(δ).

γ∈C² ⇒ γ is EL-solution with initial condition (x,v).

Left to show: γ_(x,v) unique minimizer in C_x,y^ac([0, δ],M).

Takeγ in this set withγ([0, δ₁))⊂B(x),γ(δ₁)∈∂B(x) for a δ₁. WLOG:L≥0 as Lbounded from below. Then:

A_L(γ)

L≥0

≥ Z δ1

0

L(γ,γ˙)dt

LTonelli

≥d(γ(0), γ(δ₁)) +Bδ₁ ≥− |B|δ

δsmall

≥ /2,

Then: ˜K compact ⇒ L((γ_(x,v),γ˙_(x,v)))≤C for someC. Thus, A_L(γ_(x,v))≤Cδ

δsmall

< /2.

k -subsolutions of the HJ-equation

Definition (k-subsolutions)

Letk ∈R. AC¹-functionu :M →R is a k-subsolutionof the Hamilton-Jacobi equation if

H(x,dxu)≤k, ∀x ∈M.

We denote byMu⊂M the set of points x, where equality holds.

We say thatu is a k-solution ifM_u =M. Remark

I If u is a k-subsolution, S(x,t) =u(x)−kt is a time-dependent subsolution on M×R.

I If M is closed and u1 is a k1-solution, u2 is a k2 solution, then k₁ =k₂ (∃x∈M,d_xu₁ =d_xu₂).

I If L(x,v) = ¹₂|v|²_x then the geodesic radial coordinate

r :B(x)→(0,∞) is a ¹₂-solution: by Gauss Lemma|dr|= 1.

k -subsolutions yield time-free minimizers for L + k

Theorem (B)

Let u:M →Rbe a k-subsolution and x₀,x₁∈M. Then, A_L+k(γ)≥u(x₁)−u(x₀), ∀γ ∈ [

T>0

C_x^ac_0,x1([0,T],M) with equalityiff γ is a flow line of grad_Lu contained in Mu. Each such flow line is a time-free minimizer.

Hence, a flow lineγ :R→M of grad_Lu with γ(R)⊂M_u is a global time-free minimizer for L+k.

Proof.

Same ideas as in Theorem (A).

k -subsolutions with other cohomology classes

Letθ be a closed 1-form onM with c := [θ]∈H¹(M;R).

SetL_θ :=L+θ. New Hamiltonian isH_θ(x,p) =H(x,p−θx).

A functionu_θ :M →Ris a k-subsolution forL_θ iff

∀x ∈M, H(x,dxu_θ−θx)≤k.

Findinguθ is equivalent to finding ˜θ closed 1-form onM with

∀x ∈M, H(x,θ˜_x)≤k, [˜θ] =−c. Moreover,

Mu_θ ={x ∈M |H(x,θ˜x) =k}, grad_Lθu_θ =Leg⁻¹_L (˜θ).

Application to the pendulum

Consider the pendulumL:TS¹→R,L(x,v) = ¹₂|v|²+ (1−cosx).

∀k ≥0,∃θ_k^± two closed forms withc_k^±:= [θ_k^±]∈H¹(S¹;R) and H(x,−(θ^±_k)_x) =k, ∀x∈S¹.

Leg⁻¹(−θ_k^±)-flowlines are global time-free minimizers ofL_θ^±

k +k.

For allr ∈[0,1) the closed formsrθ₀^±have [rθ₀^±] =rc₀^± and satisfy H(x,−(rθ₀^±)_x)≤0, ∀x ∈S¹

with equality only atx= 0, where Leg⁻¹(−rθ^±₀) = 0. Hence, the constant orbit atx= 0 is a global time-free minimizer forL_rθ^±

0. It will follow from the general theory that these are the only global time-free minimizers for the pendulum (try direct proof).