HW2_prob1_GPS08_G14

HW2_prob1_GPS08_G14 - 2 F q i q j q i + 2 F tq j . (7)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Goldstein 1.14 (3 rd ed. 1.8) If L is the Lagrangian for a system of n degrees of freedom, the Lagrange EOM are given by d dt ∂L · q j ∂L ∂q j =0 , (1) where q j and · q j are the j th generalized coordinate and velocity, respectively. De f ne a new function L 0 as L 0 = L + dF dt , (2) where F is only a function of the generalized coordinates and time. We want to show that the new EOM are formally identical to the old, d dt ∂L 0 · q j ∂L 0 ∂q j = d dt ∂L · q j ∂L ∂q j . (3) To do this, we should prove that d dt · F · q j · F ∂q j =0 , (4) since this is what remains if we substitute Eq.(2) into the lhs of Eq.(3). We f rst calculate · F, · F = n X i =1 ∂F ∂q i · q i + ∂F ∂t . (5) From Eq.(5) we see that · F · q j = ∂F ∂q j , (6) because the partial derivatives ∂F/∂q i and ∂F/∂t do not depend on · q j . Now we take the total time derivative of Eq.(6) to obtain d dt · F · q j = d dt ∂F ∂q j = n X i =1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 F q i q j q i + 2 F tq j . (7) Because it is immaterial in which order partial di f erentiation is carried out, we can rewrite Eq.(7) as d dt F q j = q j " n X i =1 F q i q i + F t # = F q j , (8) 1 using Eq.(5) to replace the term in square brackets. From this result, we see that Eq.(4) is indeed satis f ed and that, therefore, Eq.(3) is also valid. Thus, Lagranges EOM are invariant under the operation of adding to the Lagrangian the total time derivative of a function F that depends only on the generalized coordinates and time. A more general result of this sort will be useful to us in Chapters 9 and 10. 2...
View Full Document

This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

Page1 / 2

HW2_prob1_GPS08_G14 - 2 F q i q j q i + 2 F tq j . (7)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online