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From Hamilton equations to the nonlinear resonance

Classical mechanics, which describes the motion of particles based on Newton’s laws, can be formulated using a Lagrangian $L = L(x, \dot{x})$ and the principle of least action:

$$ S[x] = \int_{x_0}^{x_1} \Di{t} L(x(t), \dot{x}(t)) \,, \quad \delta S[x] = 0$$

where the Lagrangian is considered as a function of the trajectory $x=x(t)$, and the extremum of the action functional $S = S[x]$ is over all trajectories joining the fixed points $x_0$ and $x_1$. An equivalent description is obtained, after a Legendre transformation:

$$H(x,p) = \dot{x} p - L(x, \dot{x}) $$

in terms of the Hamiltonian $H = H(x,p)$, considered as a function of the canonical variables position $x$ and momentum $p = \partial L/ \partial \dot{x}$. While the Lagrangian description is based on the particle trajectories, the Hamiltonian description is based on the distribution of states in the phase space; states are points $(x,p)$, belonging to an energy surface $E = H(x,p)$. In the Hamiltonian dynamics approach, time appears as a simple parameter: $(x(t),p(t))$ is the parametric definition of a phase space trajectory.

The particle dynamics is governed by the Hamiltonian equations (equivalent to the Newton equation of motion, but in fact more general than Newton equations):

$$\frac{\D x}{\D t} = \frac{\partial H}{\partial p},\quad \frac{\D p}{\D t} = -\frac{\partial H}{\partial x}.$$

One can take these equations as a definition of the canonical variables $(x,p)$.

Canonical transformations

A change of canonical variables that preserves the Hamiltonian structure, is called a canonical transformation. An important type of canonical transformation, the one transforming momentum $p$ and position $x$ to action $I$ and angle $\theta$ variables, is generated by the action $S = S(x,I)$:

$$p = \frac{\partial S}{\partial x}\,, \quad \theta = \frac{\partial S}{\partial I}\,, \quad I = \oint \Di{x} \cdot p\,.$$

A system of $d$ degrees of freedom is integrable if it possesses $d$ independent integrals of motion $F=\{F_1,\ldots,F_d\}$, that is phase space functions commuting with the Hamiltonian $\{H,F\}=0$, where $\{\ldots\}$ is the Poisson braket. In an integrable systems the action variables depend only on initial conditions, or equivalently, on the set $F$, allowing a trivial integration of the system:

$$I = I(F) = \text{const.}\,, \quad \theta = \theta_0 + \omega t\,, \quad \omega = \frac{\partial H}{\partial I}\,,$$

where $\omega=(\omega_1,\ldots,\omega_d)$ are the natural frequencies of the system: basically an integrable system can be transformed, by a canonical transformation, in a set of independent oscillators. Trajectories of an integrable systems fill invariant tori labeled by the actions $I$, and parametrically defined by the angles $\theta=\theta(t)$.

When the number of motion integrals is smaller than the degrees of freedom, generically the Hamiltonian system is not integrable. Let us consider a small perturbation $V$, of typical amplitude $\varepsilon$, of an integrable Hamiltonian $H_0$:

$$H(I,\theta) = H_0(I) + \varepsilon V(I, \theta) = H_0(I) + \varepsilon \sum_n V_n \E^{\I n \cdot \theta}$$

where we expanded the perturbation in a Fourier series, and $n = (n_1,\ldots,n_d)$ is a vector of integers. Using the equations of motion, we readily obtain the first order approximation:

$$I_j = I_{0j} - \I\varepsilon \sum_n \frac{n_j V_n(I_0)}{n \cdot \omega} \E^{\I n \cdot \omega t + \I n \cdot \theta_0}\,, \quad j = 1, \ldots, d\,. \tag{1}$$

We observe that this expansion breaks down when the linear combinations of frequencies vanish:

$$n \cdot \omega = n_1 \omega_1 + \ldots + n_d \omega_d = 0\,,$$

this is the so called resonance condition. In the simplest case $d=2$, this relation is always satisfied when the frequency ratio $\omega_1/\omega_2$ is rational.

Liouville theorem

The volume of a set of phase space trajectories is conserved. Equivalently, the Jacobian of a canonical transformation is one. The proof is straightforward: noting $(x,p)$ the set of original variables ($x\in\mathbb{R}^n$, $p\in\mathbb{R}^n$, for a system with n degrees of freedom), and $(X,P)$ the transformed canonical variables, such that, using the generating function $S=S(x,P)$,

$$ X = \frac{\partial S}{\partial P},\quad p = \frac{\partial S}{\partial x}$$

the Jacobian writes,

$$J=\left|\frac{\partial(x,p)}{\partial(X,P)}\right|.$$

Using the derivation chain rule one obtains,

$$J=\frac{\left|\frac{\partial(x,p)}{\partial(x,P)}\right|}{\left|\frac{\partial(X,P)}{\partial(x,P)}\right|}= \frac{\left|\frac{\partial p}{\partial P}\right|}{\left|\frac{\partial X}{\partial x}\right|}= \frac{\left|\frac{\partial^2S}{\partial P\partial x}\right|}{\left|\frac{\partial^2S}{\partial x \partial P}\right|}=1$$

As a consequence, the phase space density $\rho=\rho(x,p,t)$ (its integral gives the phase space volume) is conserved along the trajectories:

$$0=\frac{\D \rho}{\D t} = \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}\frac{\D x}{\D t} + \frac{\partial \rho}{\partial p}\frac{\D p}{\D t} = \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}\frac{\partial H}{\partial p} - \frac{\partial \rho}{\partial p}\frac{\partial H}{\partial x}$$

or using the Poisson bracket notation:

$$\frac{\partial \rho}{\partial t} + \{\rho,H\}=0,$$

which is the Liouville equation.

KAM and PB theorems

The Kolmogorov-Arnold-Moser theorem states that closed surfaces of energy (family of periodic orbits) are preserved under small perturbations. This theorem applies to irrational tori, surfaces characterized by irrational frequency ratios; for a rational frequency ratio, the Poincaré-Birkhoff theorem states that a small perturbation destroys the rational surface but creates two fixed points, at the intersection of two neighboring irrational surfaces rotating in opposite directions around the unperturbed periodic orbit.

Perturbation theory

Assume an $N$-dimensional integrable system with Hamiltonian $H_0(I)$ depending on the set $I=(I_1,\ldots,I_N)$ action variables, altered by a small perturbation $\epsilon H_1(\theta,I)$, where $\epsilon$ the perturbation strength, and $\theta=(\theta_1,\ldots,\theta_N)$ the conjugated canonical angles. Assuming that the perturbed system remains integrable, there should exist a complete set of action variables in which the hamiltonian becomes independent of the angle variables. If such is the case, exact integration of the perturbed system reduces to find a canonical transformation $(\theta,I)\rightarrow(\phi,J)$ such that the transformed Hamiltonian writes

$$H=H_0(I) + \epsilon H_1(\theta,I) \rightarrow K=K(J)\,.$$

In general it is impossible to find a closed form of $K$, however, the presence of a small parameter $\epsilon$ allows us to compute $K$ as a series in powers of $\epsilon$. The idea is to use the generating function of the canonical transformation $\theta \cdot J + S(\theta,J)$,

$$\phi=\theta + \frac{\partial S}{\partial J},\; I = J + \frac{\partial S}{\partial \theta},\quad S = \epsilon S_1 + \cdots$$

to compute the perturbation series. To first order $K(J) = K_0(J)+K_1(J)$, one obtains,

$$K_0(J) = H_0(J),\; K_1(J) = \omega_0(J) \cdot \frac{\partial S_1}{\partial \theta} + H_1(\theta,J),$$

where

$$\omega_0(I) = \frac{\partial}{\partial I} H_0(I) = \omega_0(J),$$

are a set of characteristic frequencies of the original integrable system (at this order $I=J$). Using the fact that $K_1$ do not depend on $\theta$, one can average the first order equation over the angles to get,

$$K_1(J)=\int_0^{2\pi}\frac{\D\theta}{(2\pi)^N} \, H_1(\theta,J).$$

Now, to find $S_1$ it is convenient to expand it (and $H_1$) in Fourier series:

$$S_1(\theta,J) = \sum_n s_n \E^{\I n \cdot \theta},\quad H_1(\theta,J) = \sum_n h_n \E^{\I n \cdot \theta}$$

with $n=(n_1,\ldots,n_N)$ a set of integers. After substitution of these expansions in the first order equation, one finds the form of the Fourier coefficients:

$$S_1(\theta,J) = \sum_n \frac{h_n \E^{\I n \cdot \theta}}{n\cdot \omega_0(J)}$$

Already at this order, one observes the difficulty with the perturbation series: it diverges in the case of a resonance:

$$n\cdot\omega_0=0$$

which is always the case if the unperturbed frequencies ratios are rational $\omega_n/\omega_m\in \mathbb{Q}$. Even in the absence of a resonance, there exists a combination of integers $n$ such that the denominator becomes arbritarly small, making the perturbation series meaningless: this is the problem of small divisors. In summary, general perturbations should brake the integrability of the system, leading to the appearance of chaotic regions in phase space.

Resonance

In order to illustrate the behavior of a system near a resonance, we consider a one degree of freedom Hamiltonian $H_0 = H_0(I)$ whose natural frequency is $\omega(I) = dH_0/dI$, perturbed by a time dependent potential $V(I, \theta, t)$; this can be considered as a “1+1/2” Hamiltonian system (the canonical conjugate of $t$ being trivial):

$$H = H_0 + \varepsilon V(I, \theta, t)\,, \quad V(I, \theta, t) = \frac{1}{2}\sum_{k,n} V_{kn} \E^{\I k \theta - 2\pi n t/T}$$

where $T$ is the period of the perturbation. The resonance condition, for a fixed action $I_0$ of some periodic orbit, writes

$$k_0 \omega(I_0) - n_0 \nu = 0\,, \quad \nu = 2\pi/T\,.$$

The equations of motion for this particular mode are,

$$\dot{I} = \varepsilon k_0 V_0 \sin(k_0 \theta - n_0\nu t)\,,\quad \dot{\theta} = \omega(I) + \varepsilon \frac{\D V_0}{\D I} \cos(k_0 \theta - n_0\nu t)$$

where $V_0$ is the potential amplitude of the resonant mode. To compute the dynamics near the resonance $I \approx I_0$, we use the canonical transformation (near the identity) $F(I, q, t)= - (I-I_0)(q+n_0 t)/k_0$. This transformation gives new angle and action variables

$$q=k_0 \theta - n_0 t\,, \quad p=-\partial F/\partial q= (I-I_0)/k_0\,,$$

where $q$ represents the resonant phase, and the momentum $p$ measures the distance to the unperturbed action. The motion equations in these new variables, up to small terms in $p^2$ and $\varepsilon$, are

$$\dot{p} = \varepsilon V_0 \sin q = -\frac{\partial H_r}{\partial q} \,, \quad \dot{q} = k_0 \omega(I_0 + k_0 p) - n_0 \nu = \left. \frac{\D \omega}{\D I} \right|_{I_0}\, k_0^2 p = \frac{\partial H_r}{\partial p}$$

where we expanded $\omega(I)$ in powers of $p$, and noted that the new equations can be derived from the Hamiltonian:

$$H_r(p,q) = \frac{p^2}{2M} - K \cos q\,, \quad \frac{1}{M} = \left. \frac{\D^2 H_0}{\D I^2}\right|_{I_0} = k_0^2 \omega'(I_0)\,, \; K = \varepsilon V_0\,.$$

We thus found that the effective Hamiltonian describing the phase space in the neighborhood of a resonance is given by the pendulum Hamiltonian. The characteristic oscillation frequency of the pendulum is $\sqrt{\varepsilon k_0^2 \omega'(I_0)V_0}$.

From this simple model, we can imagine a Hamiltonian system with an infinite number of resonances $p_1,\ldots,p_n,\ldots$. If neighboring resonances are overlapping, a particle can jump from one to the other, wandering in phase space between the resonances network. This discrete “time” dynamics can be modeled by a mapping of the form:

$$p_{n+1} = p_n + K \sin q_n\,, \quad q_{n+1} = q_n + p_{n+1}\,,$$

which is called the standard map.

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