2. Integrability in Hamiltonian mechanics

#### 2.1. Integrability in 1 degree of freedom

One of the central problems in classical mechanics is the integrability of the equations of motion. The classical notion of integrability is loosely related to exact solvability, and roughly corresponds to the ability to solve a system of differential equations by means of a finite number of integration steps.

2.1a. Example: Harmonic oscillator Let us take the simple harmonic oscillator, or an idealized Hookean spring-mass system, with mass m and spring constant k. If q is the displacement from equilibrium and p the momentum, then the total energy is

$$H = \frac{1}{2m}p^2 + \frac{k}{2}q^2,\quad\text{and equations of motion are}\quad \begin{cases} \dot{q} &= \ p/m, \\ \dot{p} &= -kq \end{cases}.\tag{1}$$

The change of variables <sup>ð</sup>q; <sup>p</sup>Þ¼ðQ=λ; <sup>λ</sup>P<sup>Þ</sup> transforms the system to, with <sup>λ</sup> <sup>¼</sup> ffiffiffiffiffiffi km <sup>p</sup><sup>4</sup> ,

$$H = \frac{\omega}{2} (\mathcal{P}^2 + \mathcal{Q}^2), \quad \text{and equations of motion are} \quad \begin{cases} \dot{\mathcal{Q}} &= \omega \mathcal{P}, \\ \dot{\mathcal{P}} &= -\omega \mathcal{Q} \end{cases} \tag{2}$$

where <sup>ω</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi <sup>k</sup>=<sup>m</sup> <sup>p</sup> . A second change of variables <sup>ð</sup>Q; <sup>P</sup>Þ¼ð ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> cos <sup>θ</sup>' ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> sin <sup>θ</sup>'<sup>Þ</sup> transforms the system to

$$H = \omega I,\quad\text{and equations of motion are}\quad\begin{Bmatrix}\dot{\theta} & = & \omega, \\ \dot{I} & = & 0 \end{Bmatrix}.\tag{3}$$

The differential equations in (3) are trivial to integrate since the right-hand sides are constants. Let us explain the sequence of transformations. The change of coordinates ðq; pÞ!ðQ; PÞ is an area-preserving linear transformation that transforms the elliptical level sets of H into circles. The transformation ðQ; PÞ!ðθ, IÞ is analogous to the introduction of polar coordinates– indeed the transformation <sup>ð</sup>r;θÞ¼ð ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> ;θ<sup>Þ</sup> is a transformation to polar coordinates. Because the area form dP dQ ¼ r dr dθ, we see that the transformation dP dQ ¼ dI ¼ dθ.

Therefore, the change of coordinates ðq; pÞ!ðθ' IÞ not only reveals the exact solutions of the harmonic oscillator equations, it is area preserving.

Suppose that for some reason one did not know to introduce "polar" coordinates. One might still determine the change of coordinates using only that the transformation ðQ; PÞ!ðθ'IÞ preserves area. Indeed, since d ðP dQ−I dθÞ ¼ 0, there is a function ν ¼ νðQ, θÞ such that <sup>P</sup> <sup>d</sup>Q−<sup>I</sup> <sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>d</sup><sup>ν</sup> or <sup>P</sup> <sup>¼</sup> <sup>∂</sup><sup>v</sup> <sup>∂</sup><sup>Q</sup> and <sup>I</sup> <sup>¼</sup> <sup>−</sup> <sup>∂</sup><sup>v</sup> <sup>∂</sup>θ. Then, upon substituting the identity P ¼ ν<sup>Q</sup> into (2), one obtains

$$\mathbf{v} = \int\_{0}^{Q} \sqrt{2H/\omega - Q^2} \, d\mathbf{Q} = \frac{1}{2}Q\sqrt{2H/\omega - Q^2} - (H/\omega) \, \arccos(Q/\sqrt{2H/\omega}),\tag{4}$$

where ≡ indicates that ν equals the right-hand side up to the addition of a 2π-periodic function of θ.

If ðQ;PÞ make a complete circuit around the contour fH ¼ cg then one obtains from (4) and the identity that P ¼ ν<sup>Q</sup> that

$$
\Delta \nu = \underset{\{H=c\}}{\oint} P \, \mathbf{d}Q \, = \left(H/\omega\right) 2\pi. \tag{5}
$$

On the other hand, since d2 <sup>¼</sup> 0 and <sup>I</sup> is held constant on the contour, Green's theorem implies that

$$
\Delta\nu = \underset{\{H=c\}}{\oint} \mathbf{d}\nu + I \,\mathbf{d}\theta = \underset{\{H=c\}}{\oint} I \,\mathbf{d}\theta = 2\pi I. \tag{6}
$$

Equating (5) and (6) shows that H ¼ ωI.

2. Integrability in Hamiltonian mechanics

equilibrium and p the momentum, then the total energy is

k 2 q2

One of the central problems in classical mechanics is the integrability of the equations of motion. The classical notion of integrability is loosely related to exact solvability, and roughly corresponds to the ability to solve a system of differential equations by means of a finite

2.1a. Example: Harmonic oscillator Let us take the simple harmonic oscillator, or an idealized Hookean spring-mass system, with mass m and spring constant k. If q is the displacement from

The change of variables <sup>ð</sup>q; <sup>p</sup>Þ¼ðQ=λ; <sup>λ</sup>P<sup>Þ</sup> transforms the system to, with <sup>λ</sup> <sup>¼</sup> ffiffiffiffiffiffi

<sup>H</sup> <sup>¼</sup> <sup>ω</sup>I; and equations of motion are <sup>θ</sup>\_ <sup>¼</sup> <sup>ω</sup>, \_

The differential equations in (3) are trivial to integrate since the right-hand sides are constants. Let us explain the sequence of transformations. The change of coordinates ðq; pÞ!ðQ; PÞ is an area-preserving linear transformation that transforms the elliptical level sets of H into circles. The transformation ðQ; PÞ!ðθ, IÞ is analogous to the introduction of polar coordinates–

Therefore, the change of coordinates ðq; pÞ!ðθ' IÞ not only reveals the exact solutions of the

Suppose that for some reason one did not know to introduce "polar" coordinates. One might still determine the change of coordinates using only that the transformation ðQ; PÞ!ðθ'IÞ preserves area. Indeed, since d ðP dQ−I dθÞ ¼ 0, there is a function ν ¼ νðQ, θÞ such that

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H=ω−Q<sup>2</sup>

where ≡ indicates that ν equals the right-hand side up to the addition of a 2π-periodic function

q

the area form dP dQ ¼ r dr dθ, we see that the transformation dP dQ ¼ dI ¼ dθ.

<sup>k</sup>=<sup>m</sup> <sup>p</sup> . A second change of variables <sup>ð</sup>Q; <sup>P</sup>Þ¼ð ffiffiffiffi

; and equations of motion are <sup>q</sup>\_ <sup>¼</sup> <sup>p</sup>=m;

<sup>Þ</sup>; and equations of motion are <sup>Q</sup>\_ <sup>¼</sup> <sup>ω</sup>P,

p\_ ¼ −kq � �

<sup>P</sup>\_ <sup>¼</sup> <sup>−</sup>ω<sup>Q</sup> � �

<sup>2</sup><sup>I</sup> <sup>p</sup> cos <sup>θ</sup>' ffiffiffiffi

I ¼ 0 � �

<sup>2</sup><sup>I</sup> <sup>p</sup> ;θ<sup>Þ</sup> is a transformation to polar coordinates. Because

<sup>∂</sup>θ. Then, upon substituting the identity P ¼ ν<sup>Q</sup> into (2),

<sup>−</sup>ðH=ω<sup>Þ</sup> arccosðQ<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

: (1)

, (2)

km <sup>p</sup><sup>4</sup> ,

<sup>2</sup><sup>I</sup> <sup>p</sup> sin <sup>θ</sup>'<sup>Þ</sup> transforms the

: (3)

<sup>2</sup>H=<sup>ω</sup> <sup>p</sup> <sup>Þ</sup>; (4)

2.1. Integrability in 1 degree of freedom

number of integration steps.

44 Lagrangian Mechanics

<sup>H</sup> <sup>¼</sup> <sup>1</sup>

<sup>H</sup> <sup>¼</sup> <sup>ω</sup>

where <sup>ω</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

system to

<sup>2</sup><sup>m</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup>

<sup>2</sup> <sup>ð</sup>P<sup>2</sup> <sup>þ</sup> <sup>Q</sup><sup>2</sup>

indeed the transformation <sup>ð</sup>r;θÞ¼ð ffiffiffiffi

<sup>P</sup> <sup>d</sup>Q−<sup>I</sup> <sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>d</sup><sup>ν</sup> or <sup>P</sup> <sup>¼</sup> <sup>∂</sup><sup>v</sup>

ν≡ ð<sup>Q</sup> 0

q

one obtains

of θ.

harmonic oscillator equations, it is area preserving.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H=ω−Q<sup>2</sup>

<sup>∂</sup><sup>Q</sup> and <sup>I</sup> <sup>¼</sup> <sup>−</sup> <sup>∂</sup><sup>v</sup>

<sup>d</sup><sup>Q</sup> <sup>¼</sup> <sup>1</sup> 2 Q These calculations show that one may determine H as a function of I without explicit knowledge of the coordinate transformation ðQ;PÞ!ðθ;IÞ–but one does need to solve the Hamilton-Jacobi equation

$$H(\mathbb{Q}, \nu\_{\mathbb{Q}}) = \mathfrak{c},\tag{7}$$

for ν, as performed in Eq. (4). At this point, if one wants to derive the change of coordinates from ν, Eq. (4) shows that it is easier to write ν ¼ νðQ, IÞ, in which case P dQ þ θ dI ¼ dν or

$$\theta = \nu\_l = -\arccos\left(\mathbf{Q}/\sqrt{2I}\right),\tag{8}$$

$$\text{so } Q = \sqrt{2I}\cos\left(\theta\right) \text{ and } P = \nu\_Q = \sqrt{2I}\sin\left(\theta\right).$$

Let it be observed that if, in Eq. (4), one had chosen the anti-derivative to be arcsin rather than −arccos, then Q would be ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> sin <sup>ð</sup>θ<sup>Þ</sup> and <sup>P</sup> would be � ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> cos <sup>ð</sup>θÞ. However, because dP dQ ¼ dI dθ , one would be obligated to choose the negative square root to define P; otherwise, dP dQ ¼ −dI dθ .

2.1b. Example: the planar pendulum. Let us take the idealized planar pendulum with a mass-less rigid rod of length l suspended at a fixed end with a bob of mass m at the opposite end (Figure 1). The total energy is

$$H = \frac{1}{2m}p^2 + m\lg(1 - \cos q), \quad \text{and} \quad \begin{cases} \dot{q} = & p/m, \\ \dot{p} = & -m\lg\sin q \end{cases} \tag{9}$$

To simplify the exposition, assume that the mass <sup>m</sup> <sup>¼</sup> 1 and let <sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>16</sup>lg, where <sup>ω</sup> is 4 times the frequency of the linearized oscillations at q ¼ p ¼ 0. The substitution q ¼ 2Q, p ¼ P=2 transforms the Hamiltonian to

$$\begin{aligned} 8H &= P^2 + \omega^2 \sin^2 Q, \quad \text{and} \quad \begin{Bmatrix} \dot{Q} & = & P/4\\ P & = & -\omega^2 \sin(2Q)/8 \end{Bmatrix}. \end{aligned} \tag{10}$$

If one tries to solve for a generating function ν ¼ νðQ, IÞ of a coordinate change ðQ, PÞ!ðθ,IÞ such that H ¼ HðIÞ, then one obtains from P ¼ ν<sup>Q</sup> that

Figure 1. The planar pendulum with potential energy V ¼ mlgð1− cos qÞ.

$$\nu = \int\_0^Q \sqrt{8H - \omega^2 \sin Q} dQ = \frac{\omega}{k} E(Q, k) \tag{11}$$

where <sup>≡</sup> indicates equality up to a 2π-periodic function of <sup>θ</sup>, <sup>H</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>=ð8<sup>k</sup> 2 Þ and E is the elliptic integral of the second kind defined by <sup>E</sup>ðx, <sup>k</sup>Þ ¼ <sup>ð</sup><sup>x</sup> 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−k <sup>2</sup> sin <sup>2</sup>x p dx.

If ðQ, PÞ make a complete circuit around the contour {H ¼ c}, then one obtains from Eq. (11) that

$$
\Delta \nu = 4 \frac{\omega}{k} K(k) \tag{12}
$$

where KðkÞ ¼ EðQþðkÞ, kÞ and QþðkÞ ¼ arcsinð1=kÞ if k > 1 and π=2 if k < 1 (in which case, K is the complete elliptic integral of the second kind). The area of the shaded region K in Figure 2 shows the geometric meaning of KðkÞ for k > 1. Along with the identity (6), one obtains

$$I = \frac{2}{\pi} \frac{w}{k} K(k),\tag{13}$$

which determines H ¼ HðIÞ implicitly.

Figure 3 graphs H as a function of I using the definition of I in (13) with ω ¼ 1, along with the graph for the harmonic oscillator. Although H appears to be a smooth function of I on the interval depicted, this is a numerical artifact. Indeed, there are two distinct proofs that H cannot be differentiable in I over the interval [0,1]. Without loss of generality, it is assumed that ω ¼ 1.

Figure 2. The contours of the pendulum Hamiltonian with ω ¼ 1 (9).

Figure 3. The graph of H ¼ HðIÞfor the pendulum.

ν ≡ ð<sup>Q</sup> 0

Figure 1. The planar pendulum with potential energy V ¼ mlgð1− cos qÞ.

integral of the second kind defined by Eðx, kÞ ¼

which determines H ¼ HðIÞ implicitly.

that

46 Lagrangian Mechanics

that ω ¼ 1.

where <sup>≡</sup> indicates equality up to a 2π-periodic function of <sup>θ</sup>, <sup>H</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>=ð8<sup>k</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>8</sup>H−ω<sup>2</sup> sin <sup>Q</sup> <sup>p</sup> <sup>d</sup><sup>Q</sup> <sup>¼</sup> <sup>ω</sup>

> ðx 0

If ðQ, PÞ make a complete circuit around the contour {H ¼ c}, then one obtains from Eq. (11)

ω

where KðkÞ ¼ EðQþðkÞ, kÞ and QþðkÞ ¼ arcsinð1=kÞ if k > 1 and π=2 if k < 1 (in which case, K is the complete elliptic integral of the second kind). The area of the shaded region K in Figure 2

Figure 3 graphs H as a function of I using the definition of I in (13) with ω ¼ 1, along with the graph for the harmonic oscillator. Although H appears to be a smooth function of I on the interval depicted, this is a numerical artifact. Indeed, there are two distinct proofs that H cannot be differentiable in I over the interval [0,1]. Without loss of generality, it is assumed

Δν ¼ 4

shows the geometric meaning of KðkÞ for k > 1. Along with the identity (6), one obtains

<sup>I</sup> <sup>¼</sup> <sup>2</sup> π ω p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−k

<sup>2</sup> sin <sup>2</sup>x

dx.

<sup>k</sup> <sup>E</sup>ðQ, <sup>k</sup><sup>Þ</sup> (11)

Þ and E is the elliptic

2

<sup>k</sup> <sup>K</sup>ðk<sup>Þ</sup> (12)

<sup>k</sup> <sup>K</sup>ðkÞ, (13)

The first, calculus-based, proof is this: as <sup>k</sup> ! <sup>1</sup>þð<sup>H</sup> ! <sup>1</sup>=8<sup>−</sup> Þ, ∂I=∂k ! ∞. If H is a differentiable function of I, then ∂H=∂I ¼ 0 at I ¼ 2=πðH ¼ 1=8Þ. But then the entire level set consists of fixed points, which is false.

The second, topological, proof is this: each level set {H ¼ c}, c < 1=8, is connected; each level set for c > 1=8 has exactly two connected components (c.f. Figure 2). If the generating function v were differentiable in ðQ,IÞ on any rectangle containing R=πZ · f2=πg, then Eq. (13) would determine a homeomorphism H ¼ HðIÞ, and so the level sets of H would remain connected on either side of the critical level at height 1/8. Absurd.

To derive the change of coordinates ðQ, PÞ!ðθ, IÞ from the generating function ν, one uses the identity θ ¼ ν<sup>I</sup> and properties of the elliptic integrals to deduce

$$\theta = \frac{\pi}{2} \frac{F(Q, k)}{(Q\_+(k), k)} \quad \Rightarrow \quad Q = \text{am}\_k \left(\frac{2F\_+}{\pi} \theta\right) \tag{14}$$

where <sup>F</sup>ðx, <sup>k</sup>Þ ¼ <sup>ð</sup><sup>x</sup> 0 dx= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−k <sup>2</sup> sin <sup>2</sup>x p is the elliptic integral of the first kind, F<sup>þ</sup> ¼ FðQþðkÞ, kÞ and amkðuÞ is the Jacobian amplitude function, a local inverse to F ([1], Chapter 2). Along with P ¼ νQ, (14) implies that

$$P = \frac{\omega}{k} \text{dn}\_k \left(\frac{2F\_+}{\pi} \theta\right),\tag{15}$$

where dnkðuÞis the Jacobian elliptic function.

2.1c. Example: a mechanical system. Let V ¼ VðQÞ be a smooth potential function of a 1-degreeof-freedom Hamiltonian system with

$$H = \frac{1}{2}P^2 + V(Q). \tag{16}$$

If one attempts to find the generating function ν ¼ νðQ, IÞ of an area-preserving transformation ðQ, PÞ!ðθ, IÞ that transforms H ¼ HðIÞ, then one deduces that

$$\nu = \int\_{Q\_0}^{Q} \sqrt{2(H - V(Q))} \,\mathrm{d}Q,\tag{17}$$

up to a function depending only on I. Then, in a complete circuit around the connected contour {H ¼ c}, one has 2π ¼ Δθ ¼ Δν<sup>I</sup> identically, so

$$2\pi \mathcal{I} = \oint\_{\{H=c\}} P \, \mathrm{d}Q. \tag{18}$$

and, upon solving (18) for H ¼ HðIÞ, one inverts

$$\theta = \frac{1}{\sqrt{2}} \int\_{Q\_{\mathbb{D}}}^{Q} \frac{H\_{I}}{\sqrt{H - V(Q)}} \, \mathrm{d}Q,\tag{19}$$

to obtain Q ¼ Qðθ, IÞ, and finally P ¼ ν<sup>Q</sup> yields P ¼ Pðθ, IÞ. Since the change of coordinates is area-preserving, the Hamiltonian form of the equations of motion are preserved, so the resulting equations are

$$H = H(I) \quad \text{and} \quad \begin{Bmatrix} \dot{\theta} & = & \partial H/\partial I \\ \dot{I} & = & 0 \end{Bmatrix}. \tag{20}$$

#### 2.2. The generating function

The above three examples use a generating function ν ¼ νðQ, IÞ of a mixed system of coordinates in order to create an area-preserving change of coordinates to angle-action variables ðθ, IÞ. 2.2a. Question: why do the angle-action variables exist? In order to understand the generating function, it is necessary to clarify the existence of the coordinates ðθ, IÞ, which are commonly called angle-action variables. Let H : X ! R be a smooth function from an oriented surface X to the reals. If it is assumed that <sup>A</sup>⊂<sup>X</sup> is an open, connected, saturated <sup>ð</sup>H<sup>−</sup><sup>1</sup>ðHðAÞÞ∩<sup>A</sup> <sup>¼</sup> <sup>A</sup><sup>Þ</sup> subset of the domain of H, HjAhas no critical points and HjA is proper, then HjA is a submersion onto the interval B ¼ HðAÞ⊂R. Since HjA is proper, for each b∈B, the level set ðHjAÞ −1 ðbÞ is a compact one-manifold and hence its components are circles. Since A is connected and HjA is critical-point free, the level set must be connected, so it is a circle. Therefore, the submersion theorem implies that <sup>A</sup> is diffeomorphic to <sup>A</sup>′ <sup>¼</sup> <sup>S</sup><sup>1</sup> · <sup>B</sup>. <sup>1</sup> To make this system of coordinates concrete, note that there is a complete vector field U on A such that dHðUÞ≡1. Let γ⊂A be a segment of an integral curve of U which is maximal (i.e. an integral curve that strictly contains γ intersects X−A). For each a∈A, let t ¼ tðaÞ be the time along the flow line of the Hamiltonian vector field XH beginning at the initial condition <sup>γ</sup>∩H<sup>−</sup><sup>1</sup>ðHðaÞÞ. The function <sup>t</sup> is multi-valued, since the flow line is closed, so it should be considered as a function on the universal cover of A.

Since the tangent space at a∈A is spanned by XH and U, Ω is determined by ΩðXH, UÞ. But QðXH, UÞ ¼ −dHðUÞ ¼ −1, so Ω ¼ dHdt.

Let T be such that 2πT is the least period of the function t (i.e. the first return time to γ). Then T ¼ TðHÞ is a function of H alone. Define θ by

$$\text{If } \theta = t/T(H) \pmod{2\pi} \quad \text{and } I \text{ by} \quad \text{d}I = T(H) \text{ dH.} \tag{21}$$

The function θ is the normalized time along the flow lines of Hamiltonian vector field XH, while dH=dI ¼ 1=TðIÞ is the frequency. One computes that the oriented area form Ω ¼ dHdt ¼ dIdθ. Moreover, in the coordinates ðθ, IÞ, the Hamiltonian vector field

$$X\_H = \begin{cases} \dot{\theta} &=& 1/T(H) \\ \dot{I} &=& 0 \end{cases} = \begin{array}{c} \text{d}H \\ \text{d}I \end{array},\tag{22}$$

This proves the existence of an area-preserving diffeomorphism <sup>φ</sup> : <sup>D</sup> · <sup>S</sup><sup>1</sup> ! <sup>A</sup>, where <sup>D</sup>⊂<sup>R</sup> is an open interval, such that the Hamiltonian H is transformed to a function of I alone; and φ is as smooth as H and the area form Ω are (e.g. if both are real-analytic, then φ is real-analytic).

2.2b. Question: what kind of "function" is ν? In the first instance, ν is not single-valued. Indeed, one postulates the area-preserving change of coordinates φ : ðQ, PÞ!ðθ, IÞ to deduce that

$$\mathbf{d}(P\mathbf{d}Q + \theta \mathbf{d}I) = 0,\tag{23}$$

so that locally there is a function ν such that

To derive the change of coordinates ðQ, PÞ!ðθ, IÞ from the generating function ν, one uses the

2F<sup>þ</sup> π θ � �

is the elliptic integral of the first kind, F<sup>þ</sup> ¼ FðQþðkÞ, kÞ and

, (15)

<sup>P</sup><sup>2</sup> <sup>þ</sup> <sup>V</sup>ðQÞ: (16)

<sup>2</sup>ðH−VðQÞÞ <sup>p</sup> <sup>d</sup>Q, (17)

P dQ: (18)

<sup>H</sup>−VðQ<sup>Þ</sup> <sup>p</sup> <sup>d</sup>Q, (19)

: (20)

(14)

<sup>ð</sup>QþðkÞ, <sup>k</sup><sup>Þ</sup> ) <sup>Q</sup> <sup>¼</sup> am<sup>k</sup>

amkðuÞ is the Jacobian amplitude function, a local inverse to F ([1], Chapter 2). Along with

2.1c. Example: a mechanical system. Let V ¼ VðQÞ be a smooth potential function of a 1-degree-

If one attempts to find the generating function ν ¼ νðQ, IÞ of an area-preserving transformation

up to a function depending only on I. Then, in a complete circuit around the connected contour

HI ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I ¼ 0 � �

to obtain Q ¼ Qðθ, IÞ, and finally P ¼ ν<sup>Q</sup> yields P ¼ Pðθ, IÞ. Since the change of coordinates is area-preserving, the Hamiltonian form of the equations of motion are preserved, so the

<sup>H</sup> <sup>¼</sup> <sup>H</sup>ðI<sup>Þ</sup> and <sup>θ</sup>\_ <sup>¼</sup> <sup>∂</sup>H=∂<sup>I</sup> \_

The above three examples use a generating function ν ¼ νðQ, IÞ of a mixed system of coordinates in order to create an area-preserving change of coordinates to angle-action variables ðθ, IÞ.

2πI ¼ ∮ {H¼c}

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2F<sup>þ</sup> π θ � �

identity θ ¼ ν<sup>I</sup> and properties of the elliptic integrals to deduce

FðQ, kÞ

<sup>P</sup> <sup>¼</sup> <sup>ω</sup> <sup>k</sup> dn<sup>k</sup>

> <sup>H</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>θ</sup> <sup>¼</sup> <sup>π</sup> 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−k

p

where dnkðuÞis the Jacobian elliptic function.

{H ¼ c}, one has 2π ¼ Δθ ¼ Δν<sup>I</sup> identically, so

and, upon solving (18) for H ¼ HðIÞ, one inverts

resulting equations are

2.2. The generating function

of-freedom Hamiltonian system with

<sup>2</sup> sin <sup>2</sup>x

ðQ, PÞ!ðθ, IÞ that transforms H ¼ HðIÞ, then one deduces that

ν≡ ð<sup>Q</sup> Q0

<sup>θ</sup> <sup>¼</sup> <sup>1</sup> ffiffiffi 2 p ð<sup>Q</sup> Q0

where Fðx, kÞ ¼

48 Lagrangian Mechanics

P ¼ νQ, (14) implies that

ðx 0 dx=

$$P\,\mathrm{d}Q + \theta\mathrm{d}I = \mathrm{d}\nu.\tag{24}$$

<sup>1</sup> If one prefers a purely "elementary" proof, one might apply the inverse function theorem at this point.

But since θ is an angle variable, this equation can only hold globally modulo 2πZ dI. So, in this formulation of the generating function, ν can only be defined globally modulo 2πZ I. Or, equivalently, ν<sup>I</sup> is a function with values in the circle R=2πZ.

The way to resolve these ambiguities or difficulties is simple: the domain of the change of coordinates φ must be non-simply connected (a disjoint union of open annuli, in fact, as can be deduced from the discussion above) and so one should view (24) as holding globally on the universal cover of this annulus where θ is a single-valued real function (c.f. 21). In this case, the lift of a closed contour {H ¼ c} is a path that projects to the contour and whose endpoints differ by a deck transformation–which in the angle-action coordinates is ðθ, IÞ!ðθ þ 2π, IÞ. Since I is constant along this path, the path integral of PdQ equals the path integral of dν, i.e. Δν, the change in ν from one preimage to its translate. With this understanding, Eq. (18) is correct. And, indeed, one sees that the integral in Eq. (17) is defined not on the domain of the coordinate change φ but on its universal cover; the same is true for the integral in Eq. (19), but the marvellous fact about that integral is that it is 2π-periodic: this follows from the observation that Δθ ¼ 2π identically around a closed connected contour in {H ¼ c}.

So to answer the question that started the section, the generating function ν is a function defined on the universal cover of the union of regular compact levels of H which implicitly defines a 2π-periodic change of coordinates to "angle-action" variables ðθ, IÞ.

#### 2.3. Integrability in 2 or more degrees of freedom and Tonelli Hamiltonians

Integrability in 2 or more degrees of freedom is substantially more involved than the case of 1 degrees of freedom. Of course, a sum of n distinct, non-interacting 1-degree-of-freedom Hamiltonians is a simple case; and upon reflection, a not-so- simple case, because this condition is not coordinate independent. Indeed, a necessary and sufficient condition is that the Hamiltonian vector field be Hamiltonian with respect to two distinct non-degenerate Poisson brackets {,}<sup>i</sup> that are compatible in the sense that the linear space spanned by the brackets is a space of Poisson brackets, and maximal in the sense that a "recursion" operator naturally defined from the two brackets has a maximal number of functionally independent eigenvalue fields [2].

Let us turn now to a definition which generalizes mechanical Hamiltonians.

Definition 2.1 (Tonelli Hamiltonian). Let Σ be a smooth n-manifold and T� Σ its cotangent bundle. A smooth function H : T� Σ ! R which satisfies ðT1ÞHjT� <sup>x</sup>Σ is strictly convex for each x∈Σ; and ðT2Þ Hðx, tpÞ=t ! ∞ uniformly as t ! ∞, is called a Tonelli Hamiltonian.

As noted, Tonelli Hamiltonians are natural generalizations of mechanical systems. For this reason, Σ will be referred as the configuration space of the Hamiltonian H.

If Qi are coordinates on Σ and Θ ¼ ∑iPi dQi are the coordinates of the 1-form Θ, then the canonical symplectic structure Ω ¼ dΘ ¼ ∑dPi ∧dQi on T� Σ. The symplectic form Ω equips the space of smooth functions on T� Σ with a Poisson bracket denoted {,} that satisfies

$$\{P\_i, Q\_j\} = -\{Q\_j, P\_i\} = \delta\_{ij} \quad \{Q\_i, Q\_j\} = \{P\_i, P\_i\} = 0 \tag{25}$$

for all i, j. The Poisson bracket is fundamental to Hamiltonian mechanics. For each smooth function H, one has a smooth vector field XH ¼ {H, }, and the skew symmetry of the bracket implies that H is preserved by the flow. One says that H<sup>1</sup> and H<sup>2</sup> Poisson commute if {H1, H2}≡0.

A fundamental result in Hamiltonian mechanics is the Liouville-Arnol'd theorem, which provides a semi-local description of a completely integrable Hamiltonian and the Poisson bracket.

Theorem 2.1 (Liouville-Arnol'd). Let H : T� Σ ! R be a smooth Hamiltonian. Assume there exists n functionally independent, Poisson commuting conserved quantities F ¼ ðF<sup>1</sup> ¼ H, …, FnÞ : T� <sup>Σ</sup> ! <sup>R</sup><sup>n</sup>. If L⊂F<sup>−</sup><sup>1</sup> ðcÞ is a compact component of a regular level set, then there is a neighbourhood W of L and a diffeomorphism <sup>φ</sup> ¼ ðθ, <sup>I</sup><sup>Þ</sup> : <sup>T</sup><sup>n</sup> ·B<sup>n</sup> ! W such that

$$\begin{aligned} F &= F(I) & \{I\_{i\nu} \partial\_{\bar{j}}\} &= \delta\_{\bar{i}\bar{j}\nu} & \{I\_{i\nu} \ I\_{\bar{j}}\} &= \{\Theta\_{i\nu} \ \Theta\_{\bar{j}}\} = 0, \\ X\_{F\_i} &= \Sigma \frac{\partial F\_i(I)}{\partial I\_{\bar{j}}} \frac{\partial}{\partial \theta\_{\bar{j}}}, \end{aligned}$$

that maps L to T<sup>n</sup> · {0}.

But since θ is an angle variable, this equation can only hold globally modulo 2πZ dI. So, in this formulation of the generating function, ν can only be defined globally modulo 2πZ I. Or,

The way to resolve these ambiguities or difficulties is simple: the domain of the change of coordinates φ must be non-simply connected (a disjoint union of open annuli, in fact, as can be deduced from the discussion above) and so one should view (24) as holding globally on the universal cover of this annulus where θ is a single-valued real function (c.f. 21). In this case, the lift of a closed contour {H ¼ c} is a path that projects to the contour and whose endpoints differ by a deck transformation–which in the angle-action coordinates is ðθ, IÞ!ðθ þ 2π, IÞ. Since I is constant along this path, the path integral of PdQ equals the path integral of dν, i.e. Δν, the change in ν from one preimage to its translate. With this understanding, Eq. (18) is correct. And, indeed, one sees that the integral in Eq. (17) is defined not on the domain of the coordinate change φ but on its universal cover; the same is true for the integral in Eq. (19), but the marvellous fact about that integral is that it is 2π-periodic: this follows from the observa-

So to answer the question that started the section, the generating function ν is a function defined on the universal cover of the union of regular compact levels of H which implicitly

Integrability in 2 or more degrees of freedom is substantially more involved than the case of 1 degrees of freedom. Of course, a sum of n distinct, non-interacting 1-degree-of-freedom Hamiltonians is a simple case; and upon reflection, a not-so- simple case, because this condition is not coordinate independent. Indeed, a necessary and sufficient condition is that the Hamiltonian vector field be Hamiltonian with respect to two distinct non-degenerate Poisson brackets {,}<sup>i</sup> that are compatible in the sense that the linear space spanned by the brackets is a space of Poisson brackets, and maximal in the sense that a "recursion" operator naturally defined from the two brackets has a maximal number of functionally independent

As noted, Tonelli Hamiltonians are natural generalizations of mechanical systems. For this

If Qi are coordinates on Σ and Θ ¼ ∑iPi dQi are the coordinates of the 1-form Θ, then the

∧dQi on T�

Σ with a Poisson bracket denoted {,} that satisfies

{Pi, Qj} ¼ −{Qj, Pi} ¼ δij {Qi, Qj} ¼ {Pi, Pi} ¼ 0 (25)

Σ its cotangent bundle. A

<sup>x</sup>Σ is strictly convex for each x∈Σ; and

Σ. The symplectic form Ω equips the

equivalently, ν<sup>I</sup> is a function with values in the circle R=2πZ.

tion that Δθ ¼ 2π identically around a closed connected contour in {H ¼ c}.

defines a 2π-periodic change of coordinates to "angle-action" variables ðθ, IÞ.

2.3. Integrability in 2 or more degrees of freedom and Tonelli Hamiltonians

Let us turn now to a definition which generalizes mechanical Hamiltonians.

Σ ! R which satisfies ðT1ÞHjT�

reason, Σ will be referred as the configuration space of the Hamiltonian H.

Definition 2.1 (Tonelli Hamiltonian). Let Σ be a smooth n-manifold and T�

ðT2Þ Hðx, tpÞ=t ! ∞ uniformly as t ! ∞, is called a Tonelli Hamiltonian.

canonical symplectic structure Ω ¼ dΘ ¼ ∑dPi

space of smooth functions on T�

eigenvalue fields [2].

50 Lagrangian Mechanics

smooth function H : T�

In such a situation, it is said that His Liouville, or completely, integrable. The torus T<sup>n</sup> · {I0} is a Liouville torus, the neighbourhood T<sup>n</sup> · B<sup>n</sup> is a toroidal ball and the conserved quantities are first integrals. Systems with k first integrals, of which l < k Poisson commute with all k first integrals, where k þ l ¼ 2n are called non-commutatively integrable; when k ¼ 2n−1, the system is also called super-integrable c.f. [3, 4].

There are several proofs of the Liouville-Arnol'd theorem in the literature. The basic ideas are already captured in the one-dimensional case discussed in Section 2.2.

It can be assumed, without loss, that <sup>L</sup> <sup>¼</sup> <sup>F</sup><sup>−</sup><sup>1</sup> <sup>ð</sup>cÞ. Since <sup>c</sup>∈R<sup>n</sup> is a regular value of <sup>F</sup>, the submersion theorem implies that there is an open neighbourhood C of c consisting of regular values of F and the open set F<sup>−</sup><sup>1</sup> ðCÞ is diffeomorphic to L ·C. Therefore, there is a smooth ndimensional submanifold M⊂F<sup>−</sup><sup>1</sup> ðCÞ such that M transversely intersects each level set Lf <sup>¼</sup> <sup>F</sup><sup>−</sup><sup>1</sup> ðfÞ, f∈C. Possibly by shrinking the open set C, it can be assumed that M is Lagrangian: <sup>Ω</sup>jM≡0.<sup>2</sup>

Because the functions Fi, …, Fn Poisson commute and are functionally independent, the Hamiltonian vector fields XF<sup>1</sup> , …, XFn span the tangent space TxLf , for each x∈Lf , f∈C. Because Lf is compact, each vector field is complete, so there is a well-defined flow map <sup>φ</sup>Fi : <sup>R</sup> · <sup>F</sup><sup>−</sup><sup>1</sup> ðCÞ ! <sup>F</sup><sup>−</sup><sup>1</sup> ðCÞ. Because Fi, …, Fn Poisson commute, the respective flow maps commute, so there is an action of <sup>R</sup><sup>n</sup> on <sup>F</sup><sup>−</sup><sup>1</sup>ðC<sup>Þ</sup> defined by

$$
\phi^t = \phi^{t\_n}\_{F\_n} \bullet \cdots \bullet \phi^{t\_1}\_{F\_1} \tag{26}
$$

for all t∈R<sup>n</sup>. Define a map

<sup>2</sup> The existence of M is a consequence of Darboux's theorem. Of course, a less elementary proof would appeal to Weinstein's theorem and Moser's isotopy lemma.

$$\Phi(t,m) = \phi^t(m), \quad t \in \mathbb{R}^n, m \in M. \tag{27}$$

This is a smooth map which is a local diffeomorphism of <sup>R</sup><sup>n</sup> · <sup>M</sup> with <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ. Indeed, <sup>φ</sup><sup>t</sup> carries each level Lf into itself and carries M into a submanifold φ<sup>t</sup> <sup>ð</sup>M<sup>Þ</sup> transverse to Lf at <sup>φ</sup><sup>t</sup> ðmÞ; on the other hand, the derivative of <sup>φ</sup><sup>t</sup> with respect to <sup>t</sup> is a surjective linear map onto <sup>T</sup>φtðmÞLf . Therefore, dΦ is surjective, so injective, hence Φ is a local diffeomorphism onto its image. Compactness and connectedness of the levels Lf imply that the image of <sup>Φ</sup> is <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ.

For each <sup>m</sup>∈M, let <sup>P</sup>ðmÞ⊂R<sup>n</sup> be the set of <sup>t</sup> such that <sup>Φ</sup>ðt, <sup>m</sup>Þ ¼ <sup>m</sup>. Since each level set is compact, <sup>P</sup>ðm<sup>Þ</sup> is a discrete subgroup of <sup>R</sup><sup>n</sup> isomorphic to <sup>Z</sup><sup>n</sup>. This is the "period lattice" of the action φ. If one selects a basis of PðmÞ, one obtains a map M ! GLðn; RÞ, m ! 2πTðmÞ. The implicit function theorem implies that there is a smooth map amongst these maps. Moreover, since FjM is a bijection onto its image, one can take the components of F as coordinates on M, or in other words, T ¼ TðFÞ.

Define functions θ ¼ ðθ1, …, θnÞ by

$$
\theta = T(F)^{-1} \cdot t \pmod{2\pi}, \quad \theta: \mathbf{R}^{\pi} \times \mathbb{C} \to \mathbf{R}^{\pi}/2\pi \mathbf{Z}^{\pi}. \tag{28}
$$

The flow map Φ therefore induces a diffeomorphism F<sup>−</sup><sup>1</sup> <sup>ð</sup>CÞ ! <sup>T</sup><sup>n</sup> · <sup>C</sup> : <sup>x</sup> ! ðθðxÞ, <sup>F</sup>ðxÞÞ.

To complete the proof, one might show that each vector field ∂=∂θ<sup>i</sup> is Hamiltonian with Hamiltonian function Ii and that F is functionally dependent on I so that ðθ, IÞ is a canonical system of coordinates on <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ. This is performed indirectly. Define the functions Ii <sup>¼</sup> IiðF<sup>Þ</sup> by

$$2\pi \mathcal{I}\_i = \oint\_{\Gamma\_i(\mathcal{F})} \xi,\tag{29}$$

where ξ ¼ P � dQ is the primitive of the symplectic form Ω and ΓiðFÞ is the cycle on LF on which θ<sup>i</sup> increases from 0 to 2π and the other angle variables are held equal to 0. To show that <sup>ð</sup>θ,I<sup>Þ</sup> is a system of coordinates on <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ, one computes the Jacobian <sup>½</sup>∂Ii=∂Fj�:

$$2\pi \frac{\partial I\_i}{\partial F\_j} = \lim\_{s \to 0} \frac{1}{s} \int \int \Omega,\tag{30}$$

where, in the ðt, FÞ coordinate system,

$$\mathcal{C}\_{\rangle}(F,s) = \{ (\iota T(F + \nu \iota\_{\rangle})\_{i\prime}F + \nu \iota\_{\rangle}) | \iota \in [0, 2\pi], \ \nu \in [0, s] \}$$

is the "cylinder" obtained by sweeping out the cycles ΓjðF þ vejÞ as the j-th component of Fincreases from Fj to Fj þ s, and Ti is the i-th column of the period matrix T. Since

$$
\Omega \left( \frac{\partial}{\partial F\_j}, \frac{\partial}{\partial t\_k} \right) = \frac{\partial F\_k}{\partial F\_j} = \delta\_{jk}, \tag{31}
$$

which implies

$$\frac{\partial I\_i}{\partial F\_j} = T\_{ji}.\tag{32}$$

Since the period matrix T is non-singular, the transformation ðθ, FÞ!ðθ, IÞ is a diffeomorphism.

Finally, the functions I1, …, In Poisson commute and since M is Lagrangian, the functions ti, …, tn Poisson commute, which implies θ1, …, θ<sup>n</sup> Poisson commute. And, since {Fi, tj} ¼ δij, this implies that {Ii, θj} ¼ δij.

The remainder of the theorem follows from the fact that the angle-action coordinates ðθ, IÞ are canonical and F ¼ FðIÞ.
