# Download Analytical Mechanics for Relativity and Quantum Mechanics by Oliver Johns PDF

By Oliver Johns

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S D , s˙1 , s˙2 , . . , s˙ D , t) = 2 D M j s˙j2 − U (s1 , s2 , . . 18) j=1 Then it follows that ∂ L(s1 , s2 , . . , s D , s˙1 , s˙2 , . . 19) A SIMPLE EXAMPLE 27 and ∂ ∂ L(s1 , s2 , . . , s D , s˙1 , s˙2 , . . , s˙ D , t) = − U (s1 , s2 , . . 21) for i = 1, . . , D. This is the Lagrangian form of Newton’s second law, as expressed in the s-system of coordinates. Note that we have used the usual shorthand, abbreviating L(s1 , . . , s D , s˙1 , . . , s˙ D , t) to the shorter form L(s, s˙ , t).

24) which are the correct differential equations of motion for this problem. 4 Arbitrary Generalized Coordinates The generalized coordinates of the s-system are only a trivial re-labelling of Cartesian coordinates. The real power of the Lagrangian method appears when we move to more general coordinate sets. Let q1 , q2 , . . , q D be any set of D independent variables, which we will call the q-system, such that their values completely specify all of the s-system values, and vice 28 INTRODUCTION TO LAGRANGIAN MECHANICS versa.

52) hold for all k = 1, . . , D. 52). 53) If f and g are any functions, it follows from the product rule for differentiation that f (dg/dt) = d ( f g) /dt − g (d f /dt). 54), respectively. 39). 55) is the chain rule expansion of ∂ L(q, q, ˙ t)/∂qk . 52), as was to be proved. 53). Multiplying that equation by ∂qk (s, t)/∂s j , 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.