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The Lazy UniverseAn Introduction to the Principle of Least Action$

Jennifer Coopersmith

Print publication date: 2017

Print ISBN-13: 9780198743040

Published to British Academy Scholarship Online: June 2017

DOI: 10.1093/oso/9780198743040.001.0001

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

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

Source:
The Lazy Universe
Author(s):

Jennifer Coopersmith

Publisher:
Oxford University Press

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

(A7.8.1)
i=1n(PiδQipiδqi)=δS

Also, we know that S is the action function, A=tatbdS. 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, occurring at nearby times, t, and, t+Δt, thereby implicitly defining the canonical transformations, CT, and, CT, respectively:

S, at time t implicitly defines the transformation:

CT, (P1,,Pn;Q1,,Qn)(p1,,pn;q1,,qn)

while satisfying

(A7.8.2)
δS=i=1n(PiδQi)(piδqi)

S, at time t+Δt implicitly defines the transformation:

CT,(P1,,Pn;Q1,,Qn)(p1+Δp1,,pn+Δpn;q1+Δq1,,qn+Δqn)

while satisfying

(A7.8.3)
δS=i=1n(PiδQi)(pi+Δpi)δ(qi+Δqi)

Note that Δt is a very small time-interval, and S, which occurs at t+Δt, is only a very slightly different function to S, which occurs at t. Therefore: p1 (p.245) is very close to p1,p2 is very close to p2,qn1 is very close to qn1,qn is very close to qn. These proximities are guaranteed only because we insist that S and S are continuous functions. The implications are that the Δqis and Δpis are small quantities whose product and squares are insignificant, and, crucially, S edges forward as t edges forward.

Now transformations that are canonical form a ‘group’, known as a Lie group. This means that, as CT and CT are canonical, then any composition of operations will also be in the group and will also be canonical. In particular, the composition CTCT1 is canonical (CT1 is the inverse of CT and is also in the group - has the property of being canonical). But the composition CTCT1 happens to perform the following transformation:

(pi,qi)(pi+Δpi,qi+Δqi),i=1 to n

while satisfying

(A7.8.4)
δ(SS)=i=1n(pi+Δpi)δ(qi+Δqi)(piδqi)

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 form1 S=S(qi,,qn;Qi,,Qn;t), and so, to first order, we have

(A7.8.5)
ΔS=i=1nSqiΔqi+SQiΔQi+StΔt

Furthermore, in our present case ΔQi=0 for all i (as we start from a fixed initial point), and therefore we obtain

(A7.8.6)
ΔS=(SS)=i=1nSqiΔqi+StΔt

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

(A7.8.7)
δ(SS)=i=1nδSqiΔqi+δStΔt

(p.246) (The Δ‎ s do not get varied.) Then, using the relations S/qi=pi (see equations (7.14) or Appendix A7.7), we obtain

(A7.8.8)
δ(SS)=i=1nδpiΔqi+δStΔt

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

(A7.8.9)
i=1n(pi+Δpi)δ(qi+Δqi)(piδqi)=i=1nδpiΔqi+δStΔt

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

(A7.8.10)
i=1n(ΔpiδqiδpiΔqi)=δStΔt

Lanczos calls this a “remarkable relation”:2 all the coordinates are relative coordinates (only Δ‎ s and δ 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=3t2+4, for example). On the other hand, the right-hand side does have a functional dependence on t (through S=S(q1,,qn;Q1,,Qn;t), and a similar form for S/t).

We have just seen how the left side of (A7.8.10) depends on pi s and qi s, while the right side depends on (q1,,qn;Q1,,Qn;t). However, we want to have both sides in the same space, that is to say, phase space. We therefore determine all the pi s from pi=S/qi (relations (7.14)), and this means that the pi s are then given as functions of (q1,,qn;Q1,,Qn;t). We then ‘invert’ these pi-functions and obtain the Qi s as functions of (q1,,qn;p1,,pn;t). Finally, we replace the Qi s in S/t by these Qi-functions, and so obtain a new function, (p.247) say, X, in phase-space coordinates:

(A7.8.11)
St=X(q1,,qn;p1,,pn;t)

Taking variations of both sides we obtain

(A7.8.12)
δSt=δX(q1,,qn;p1,,pn;t)=i=1nXqiδqi+Xpiδpi+Xtδt

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

(A7.8.13)
δSt=i=1nXqiδqi+Xpiδpi

At last, we have δS/t in (p,q) coordinates, and we substitute this form into our “remarkable relation” (A7.8.10):

(A7.8.14)
i=1n(ΔpiδqiδpiΔqi)=i=1nXqiδqi+XpiδpiΔt

The variations, δpi, are independent, and also the variations, δqi, are independent - therefore the coefficients of each δpi must be equal on both sides of (A7.8.14), and also the coefficients of each δqi must be equal on both sides of (A7.8.14). This leads to the relations:

(A7.8.15)
Δqi=XpiΔtΔpi=XqiΔti=1 to n

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

(A7.8.16)
Δqi=[St]piΔtΔpi=[St]qiΔti=1 to n

Notes:

(1) Lanczos, page 217, equation (77.4).

(2) Lanczos, page 218.