For an infinitesimal transformation in phase space, what functions are allowed for this to be a canonical transformation?

Question

Consider an infinitesimal transformation:

$$(q_{i},p_{j}) \quad\longrightarrow \quad(Q_{i},P_{j}) ~=~ \left(q_{i} + \alpha F_{i}(q,p),~p_{j} + \alpha E_{j}(q,p)\right) $$

where $α$ is considered to be infinitesimally small.

Now, if we construct Jacobian matrix, we will have:

$$ \jmath =\begin{pmatrix} \delta_{ij}+ \alpha{\frac{\partial F_{i} }{\partial q_{j}}} & \alpha{\frac{\partial F_{i} }{\partial p_{j}}} \\ \alpha{\frac{\partial E_{i} }{\partial q_{j}}} & \delta_{ij}+ \alpha{\frac{\partial E_{i} }{\partial p_{j}}} \end{pmatrix}.$$

What functions $F_{i} (q, p)$ and $E_{i} (q, p)$ are allowed for this to be a canonical transformation?

To be canonical transformation, it's required to hold: $$\jmath j \jmath^{T} = j$$ in which $ j = \begin{pmatrix} 0 & 1\\ -1&0 \end{pmatrix}$. To hold the canonical transformation, there should be: $$\frac {\partial F_{i}}{\partial q_{j}} = - \frac {\partial E_{i}}{\partial p_{j}} $$

which is true if

$$F_{i} = \frac {\partial G}{\partial p_{i}} \; \; , \; \; E_{i} = - \frac {\partial G}{\partial q_{i}} $$

for some function $G(q, p)$.

Now my problem is that by calculating everything I can't figure out how to reach to last two formulas. The formulas which shows the possibilities for $F_{i}$ and $E_{i}$?

Answer 1

1. First of all, be aware that there exist various different definitions of canonical transformations (CT) in the literature, cf. e.g. this Phys.SE post. What OP (v3) above refers to as a CT, we will in this answer call a symplectomorphism for clarity. What we in this answer will refer to as a CT, will just be a CT of type 2.

2. It is possible to show (see e.g. Ref. 1) that an arbitrary time-dependent infinitesimal canonical transformation (ICT) of type 2 with generator $G=G(z,t)$ can be identified with a Hamiltonian vector field (HVF) $$ \delta z^I~=~\varepsilon\{ z^I,G\}_{PB}~\equiv~ \sum_{K=1}^{2n} J^{IK} \frac{\partial G}{\partial z^K} , $$ $$ X_{-G}~\equiv~-\{G,\cdot\}_{PB}~\equiv~\{\cdot,G\}_{PB},\tag{1} $$ with (minus) the same generator $G$. Here $z^1,\ldots, z^{2n}$, are phase space variables, $t$ is time, $\varepsilon$ is an infinitesimal parameter, and $J$ is the symplectic unit matrix, $$\tag{2} J^2~=~-{\bf 1}_{2n\times 2n}.$$

3. A general time-dependent infinitesimal transformation (IT) of phase space can without loss of generality be assumed to be of the form $$ \tag{3} \delta z^I~=~\varepsilon \sum_{K=1}^{2n} J^{IK} G_K(z,t) ,\qquad I~\in~\{1,\ldots, 2n\}, $$ because the matrix $J$ is invertible.

4. Next consider a time-dependent infinitesimal symplectomorphism (IS), which can be identified with a symplectic vector field (SVF). It is possible to show that a SVF [written in the form (3)] satisfies the Maxwell relations$^1$ $$\tag{4} \frac{\partial G_I(z,t)}{\partial z^J}~=~(I \leftrightarrow J),\qquad I,J~\in~\{1,\ldots, 2n\}. $$

5. Eq. (4) states that the one-form $$\tag{5} \mathbb{G}~:=~ \sum_{I=1}^{2n}G_I(z,t) \mathrm{d}z^I$$ is closed $$\tag{6} \mathrm{d}\mathbb{G}~=~0. $$

6. It follows from Poincare Lemma, that locally there exists a function $G$ such that $ \mathbb{G}$ is locally exact $$\tag{7} \mathbb{G}~=~\mathrm{d}G. $$ Or in components, $$\tag{8} G_I(z,t)~=~\frac{\partial G(z,t)}{\partial z^I},\qquad I~\in~\{1,\ldots, 2n\} .$$

7. In summary we have the following very useful theorem for a general time-dependent infinitesimal transformation (IT).

   Theorem. An infinitesimal canonical transformation (ICT) of type 2 is an infinitesimal symplectomorphism (IS). Conversely, an IS is locally a ICT of type 2.

8. 2D counterexample: Consider the phase space $M=\mathbb{R}^2\backslash\{(0,0)\}$ with the symplectic 2-form $\omega =\mathrm{d}p\wedge \mathrm{d}q$. One may check that the vector field $$X=\frac{q}{q^2+p^2}\frac{\partial}{\partial q} +\frac{p}{q^2+p^2}\frac{\partial}{\partial p} $$ is SVF/IS but it is not a HVF/ICT of type 2. The problem is that the candidate ${\rm arg}(q+ip)$ for the Hamiltonian generator is multi-valued, and hence not globally well-defined.

References:

1. H. Goldstein, Classical Mechanics; 2nd eds., 1980, Section 9.3; or 3rd eds., 2001, Section 9.4.

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$^1$ OP already listed some (but not all) of the Maxwell relations (4) in his second-last equation. All of the Maxwell relations (4) are necessary in order to deduce the local existence of the generating function $G$.