2.2 Runge-Lenz-vector varied as 9→q=q+6a(qk,t) and (16) t→t=t+δB, (17) with some small parameter 6.Here the variation of the coordinates is a function of the original variables and time.The variation added to time 03 is taken to be constant.Since the variation of time is a constant and the Lagrangian is assumed to be time-independent,the transformation of time will not change the action integral.But the variation of time is still introduced for later calculations.Also here the variations are such that they vanish at the endpoints of integration.The velocities then become =:+dd: (18) We can as before expand the Lagrangian in a power series and write the variation of the Lagrangian as dL=∑w6 6a+入6oiy (19) We now demand that the action integral is invariant under this transformation i.e L(qk,q)dt= dt[L(g,g)+gg,] dt], (20) t t where we added a total time derivative which we can do since the integrand is determined only up to a total time derivative.The variation of the action integral then becomes 0=6S 广+割 (21) where f is defined to be the variation of g due to the coordinate transformation. We now simply focus on the integrand since it is the most important part.We can 82.2 Runge-Lenz-vector varied as qi 7→ q 0 i = qi + δαi(qk, t) and (16) t 7→ t 0 = t + δβ, (17) with some small parameter δ. Here the variation of the coordinates is a function of the original variables and time. The variation added to time δβ is taken to be constant. Since the variation of time is a constant and the Lagrangian is assumed to be time-independent, the transformation of time will not change the action integral. But the variation of time is still introduced for later calculations. Also here the variations are such that they vanish at the endpoints of integration. The velocities then become q˙ 0 i = ˙qi + δα˙ i . (18) We can as before expand the Lagrangian in a power series and write the variation of the Lagrangian as δL = X i ∂L ∂qi δα + X i ∂L ∂q˙ii δα˙ i . (19) We now demand that the action integral is invariant under this transformation i.e Z t2 t1 L(qk, q˙k)dt = Z t2 t1 dt L(q 0 k , q˙ 0 k ) + dg(q, t) dt , (20) where we added a total time derivative which we can do since the integrand is determined only up to a total time derivative. The variation of the action integral then becomes 0 = δS = Z t2 t1 dt δL + df dt  , (21) where f is defined to be the variation of g due to the coordinate transformation. We now simply focus on the integrand since it is the most important part. We can 8