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Hixy

Member

Registered: 2011-09-24

Posts: 15

Expansion of multiple integrals

Last time you guys were able to see through the (to me) somewhat cluttered use of integrals and differentiation (I'm still very grateful for your amazing help). This time my question is similar. I'm currently trying to understand the following derivation of Lagrange's equations from Landau and Lifshi-tz' amazing first volume on mechanics:

The function for the action

is given.

are the coordinates and the velocities, respectively, for a system of particles.
Then the change in S when q is replaced by

is

(The following is the hard part)
When this difference is expanded in powers of

and

in the integrand, the leading terms are of the first order (1). The necessary condition for S to have a minimum is that these terms (the variation of the integral) should be zero. Thus the principle of least action may be written in the form

or, affecting the variation (2),

The conditions

show that the integrated term is zero. There remains an integral which must vanish for all values of

. This can be so only if the integrand is zero identically. Thus we have

What is meant by the sentence at (1)? Which calculations aren't shown here?
How is this translated to affect the variation at (2)?
How did we get to (3)?

Edit: Having studied this further the last couple of days, I've realized what a stupid question it was. I'm gonna read up on calculus of variations and understand this fully!

Last edited by Hixy (2011-10-18 10:39:17)

Theolog

Guest

Re: Expansion of multiple integrals

Hi I have been searching for answers to these very questions and, since you seem to have figured them out, would you mind explaining the reasoning behind (1)? I get the rest of the derivation but the expansion in powers of del(q) and del(dq/dt) is still not making sense to me! Thank you so much.