By Bertrand Russell

First released in 1925, Bertrand Russell’s *ABC of Relativity* was once thought of a masterwork of its time, contributing considerably to the mass popularisation of technology. Authoritative and available, it offers a extraordinary introductory consultant to Einstein’s conception of Relativity for a basic readership. some of the most definitive reference courses of its type, and written by way of one of many 20th century’s so much influential philosophers, *ABC of Relativity* remains to be as proper this present day because it used to be on first publication.

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23). 2 (a) Calculate the Jacobian matrix for s, = x, s2 = y, s3 = z and q 1 = p, q2 = , and q3 = z, the transformation from Cartesian to cylindrical-polar coordinates. Show that p, , z are good generalized coordinates except on the z-axis. 23), work ont in detail the transformation to the Lagrangian L(p, tf> , z, p, ef> , z, t) for this system. 3 We are given a Lagrangian L (q, q, t). Assume that there are no non-potential forces. Let f (q, t) be an arbitrary function of q = q 1 , q2 , .

53). Multiplying that equation by oqk (s, t)/osi , summing over k = 1, . . 51), as was to be proved. 10 Relation Between Any Two Systems The q-system above is taken to be any good system of generalized coordinates. If we imagine it and any other good system, which we may call the r-system, then it follows from what we've done above that the Lagrange equations in this r-system are equivalent to the Lagrange equations in the q-system. Both of them are equivalent to the s-system, hence they are equivalent to each other.

46) 14We fullow the physics custom which uses the same letter L in both the s and q systems, and considers L(s, S,t) and L(q, 4',t) to be the same function expressed in different cooTdinates. S. 9 requires the following Lemma. 25) in which the s; = s; (q, t) depend only on q and t. 31) are functions only of q, t , so that the explicit linear term in 'lk is the only place that the variables iJ. appear. 47). 47) requires a somewhat longer proof. i;�� (q, q, � t) 8qk 1 = 1 aqk a as,(q, t ) ) � a as; (q, t ) ) .

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