# (p.244) Appendix A7.8 Infinitesimal canonical transformations

# (p.244) Appendix A7.8 Infinitesimal canonical transformations

From invariance of the ‘circulation’ of the phase fluid we know that

Also, we know that *S* is the action function, $A={\int}_{{t}_{a}}^{{t}_{b}}dS$. Finally, as well as satisfying (A7.8.1), and being the action function, *S* has yet another role - it implicitly defines a canonical transformation, a function that transforms from one set of coordinates to another while satisfying the canonical equations of motion, (7.9), in both the new and old coordinates. We shall now introduce one more feature: we let *S* depend explicitly on *t*, and see how this affects the canonical transformation functions.

For example, we consider two action functions, *S*, and, ${S}^{\prime}$, occurring at nearby times, *t*, and, $t+\mathrm{\Delta}t$, thereby implicitly defining the canonical transformations, CT, and, CT${}^{\prime}$, respectively:

*S*, at time *t* implicitly defines the transformation:

while satisfying

${S}^{\prime}$, at time $t+\mathrm{\Delta}t$ implicitly defines the transformation:

while satisfying

Note that $\mathrm{\Delta}t$ is a very small time-interval, and ${S}^{\prime}$, which occurs at $t+\mathrm{\Delta}t$, is only a very slightly different function to *S*, which occurs at *t*. Therefore: *p*_{1}
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is very close to ${p}_{1}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{p}_{2}$ is very close to ${p}_{2}^{\prime},\dots {q}_{n-1}$ is very close to ${q}_{n-1}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{q}_{n}$ is very close to ${q}_{n}^{\prime}$. These proximities are guaranteed only because we insist that *S* and ${S}^{\prime}$ are *continuous* functions. The implications are that the $\mathrm{\Delta}{q}_{i}\phantom{\rule{thinmathspace}{0ex}}$s and $\mathrm{\Delta}{p}_{i}\phantom{\rule{thinmathspace}{0ex}}$s are *small* quantities whose product and squares are insignificant, and, crucially, ${S}^{\prime}$ edges forward as *t* edges forward.

Now transformations that are canonical form a ‘group’, known as a Lie group. This means that, as $C{T}^{\prime}$ and *CT* are canonical, then any *composition* of operations will also be in the group and will also be canonical. In particular, the composition $C{T}^{\prime}\circ C{T}^{-1}$ is canonical ($C{T}^{-1}$ is the inverse of *CT* and is also in the group - has the property of being canonical). But the composition $C{T}^{\prime}\circ C{T}^{-1}$ happens to perform the following transformation:

while satisfying

Putting this line of enquiry to one side for the moment, we remember that *S* has yet another identity: it is a ‘generating function’ (it takes the ‘wavefront’ of common action from one position to the next to the next, and so on, in configuration space). It has the functional form^{1} $S=S({q}_{i},\dots ,{q}_{n};{Q}_{i},\dots ,{Q}_{n};t)$, and so, to first order, we have

Furthermore, in our present case $\mathrm{\Delta}{Q}_{i}=0$ for all *i* (as we start from a *fixed* initial point), and therefore we obtain

We now take the variation on both sides of (A7.8.6), and end up with:

(p.246) (The Δ s do not get varied.) Then, using the relations $\partial S/\partial {q}_{i}={p}_{i}$ (see equations (7.14) or Appendix A7.7), we obtain

Equating the right-hand sides of (7.8.4) and (7.8.8), we find

Multiplying out the brackets, throwing away variations of Δ s, and ignoring products of two Δ s, we finally arrive at:

Lanczos calls this a “remarkable relation”:^{2} all the coordinates are *relative* coordinates (only Δ s and $\delta $ s appear); moreover, all the coordinate intervals are ‘*small*’; and lastly the *t*-dependence is neatly collected together in *one* place (on the right-hand side). Why is this so - why doesn’t *t* show up on the left? This is explained in the following way. The left-hand side of (A7.8.10) arises from different ‘slices’ through the phase fluid (such as (A7.8.2) and (A7.8.3)), and we could label each slice with the time that this snapshot was taken. We would then end up with an infinite sequence of ‘photos’, each with their own *t*-number and corresponding *S*. Time thus shows up as a parameter (the photos can be put in order) but it doesn’t bring in a *functional* dependence (we cannot say that $S=3{t}^{2}+4$, for example). On the other hand, the right-hand side does have a functional dependence on *t* (through $S=S({q}_{1},\dots ,{q}_{n};{Q}_{1},\dots ,{Q}_{n};t)$, and a similar form for $\partial S/\partial t$).

We have just seen how the left side of (A7.8.10) depends on *p*_{i} s and *q*_{i} s, while the right side depends on $({q}_{1},\dots ,{q}_{n};{Q}_{1},\dots ,{Q}_{n};t)$. However, we want to have both sides in the same space, that is to say, phase space. We therefore determine all the *p*_{i} s from ${p}_{i}=\partial S/\partial {q}_{i}$ (relations (7.14)), and this means that the *p*_{i} s are then given as functions of $({q}_{1},\dots ,{q}_{n};{Q}_{1},\dots ,{Q}_{n};t)$. We then ‘invert’ these *p*_{i}-functions and obtain the *Q*_{i} s as functions of $({q}_{1},\dots ,{q}_{n};{p}_{1},\dots ,{p}_{n};t)$. Finally, we replace the *Q*_{i} s in $\partial S/\partial t$ by these *Q*_{i}-functions, and so obtain a new function,
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say, $-X$, in phase-space coordinates:

Taking variations of both sides we obtain

However we don’t allow variation of the time (that is, $\delta t=0$). Therefore, (A7.8.12) becomes

At last, we have $\delta \left(\partial S/\partial t\right)$ in ($p,q$) coordinates, and we substitute this form into our “remarkable relation” (A7.8.10):

The variations, $\delta {p}_{i}$, are independent, and also the variations, $\delta {q}_{i}$, are independent - therefore the coefficients of each $\delta {p}_{i}$ must be equal on both sides of (A7.8.14), and also the coefficients of each $\delta {q}_{i}$ must be equal on both sides of (A7.8.14). This leads to the relations:

Finally, to reach our place in the main text, equations (7.16), we substitute $-\partial S/\partial t$ back for *X* and arrive at: