# Random physics

Alberto Verga, research notebook


# 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.