Can any sum of infinitesimal canonical transforms on phase space be obtained from evolution under a static Hamiltonian?

Question

Suppose I have a canonical transformation on phase space, which is obtained by evolving a classical Hamiltonian system from time $t=0$ to $t=T$, with some arbitrary time-dependent Hamiltonian $H(t)$. That is, it's a finite canonical transformation which is a sum of infinitesimal canonical transformations. Then, can I always find some time-independent effective Hamiltonian $H_{eff}$ such that if I evolve the system from $t=0$ to $t=T$ under $H_{eff}$, this evolution gives me the original canonical transformation? Note that I don't require that the dynamics under $H(t)$ and $H_{eff}$ produce the same canonical transformation for all $t$, but only for some fixed $t=T$.

Answer 1

This can be trivially achieved, if you are willing to go to extended phase space, that is add an extra degree of freedom to your system, represented by the momentum - position pair $(p_e, q_e)$.

Then the dynamic system evolves under the effective Hamiltonian $$ H_\text{eff} = p_e + H(\mathbf{p}, \mathbf{q}, q_e) $$

The effective Hamiltonian $H_{\text{eff}}$ is time independent, i.e. autonomous, and an integral of motion.

If this does not satisfy you, and your question is about effective Hamiltonians in the original, restricted phase space, for arbitrarily large times $T\rightarrow 0$, the answer is that, in general there can be no such effective Hamiltonian. This can be demonstrated by a simple example.

Consider an autonomous Hamiltonian system with one degree of freedom evolving under the Hamiltonian $H_0(p,q)$. The Hamiltonian is conserved, therefore the system is integrable. The orbits lie on contours given by $H_0 = \text{const.}$. Suppose now that this system is perturbed by a time dependent perturbation $H_1(p,q,t)$, so that $$ H(p, q, t) = H_0(p,q) + H_1(p,q,t). $$

The question then becomes, whether there can be an autonomous Hamiltonian of one degree of freedom $H_\text{eff}(p,q)$ that describes the motion under the non autonomous Hamiltonian $H(p, q, t)$.

It is a well known fact that in general, non autonomous systems exhibit chaotic behaviour, when
(see Poincare-Melnikov theory, for example). The perturbed Hamiltonian is no longer conserved and in general there are orbits that cover densely finite areas of phase space. Such orbits cannot be described as contour levels of some well behaved function $H_\text{eff}$, therefore no autonomous effective Hamiltonian exists.

You may weaken your requirements and ask for an effective Hamiltonian that exactly describes the motion only for a finite, possibly small, time $T$, after which it becomes invalid. In other words, the question becomes whether the non autonomous Hamitonian $H$ can be expressed as a series $$ H(p, q, t) = \sum_n H_{\text{eff}, n}(p, q) U_n(t) $$ where $$ U_n = \begin{cases} &1,\quad \text{for}\ n T \leq t < (n+1) T \\ &0,\quad \text{otherwise}. \end{cases} $$ I am not aware of any theorem that says that this is impossible, but even if it is not, in my opinion it would be impractical to calculate, in most cases.

On the other hand, if you are interested in finding effective Hamiltonians that approximately describe the motion for finite, possibly small, times $T$, for perturbed autonomous Hamiltonians, then you should delve into the field of Hamiltonian mappings. This is still an active field of research. Unfortunately, no general techniques are known and even the most common ones would be hard to summarise here.

For more information, see

Abdullaev, S. S. Construction of mappings for Hamiltonian systems and their applications Springer, 2006

Answer 2

Maybe this is too naive a picture, but under your time-dependent Hamiltonian you can define a unitary time-evolution operator:

$U(T,0) = {\cal T} \left( e^{-i \int_0^T H(t') dt'} \right)$

where $\cal T$ signifies the time-ordering operator. As $U$ is a unitary operator, you can express it in terms of a hermitian operator $H_{\mathrm eff}$ as

$U(T,0) = e^{-i T H_{\mathrm eff}}$.

This effective Hamiltonian will not be unique as you can add arbitrary multiples of $2 \pi/T$ to its spectrum, but it should exist. If your Hamiltonian were $T$-periodic, this construction would correspond to employing the Floquet formalism, with the eigenvlaues of $H_{\mathrm eff}$ being the Floquet quasienergies.