Lagrangian equations of motion

  • #1

Homework Statement



Find equations of motion (eom) of a particle moving in a D-dimensional flat space with the following Lagrangian

L = (1/2)mv2i - k/ra,

r = root(x2i), m,k,a are constants


Homework Equations





The Attempt at a Solution



The equations of motion are given by d/dt(∂L/∂vi) - ∂L/∂xi = 0

So, when I work all this out I get
ma = ka/xia+1

I have a feeling this isn;t correct though.

Am I doing the partials wrong?
 

Answers and Replies

  • #2
vela
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Yes, you're taking the partial with respect to xi incorrectly. Try writing out the potential term in terms of the xi's and differentiating.

Also why is a multiplying m? How did the exponent of r get over there?
 
  • #3
Yes, you're taking the partial with respect to xi incorrectly. Try writing out the potential term in terms of the xi's and differentiating.

Also why is a multiplying m? How did the exponent of r get over there?

U = k/ra = k/root(x2i)a
= k/xai
= kx-ai

Differentiating this with respect to xi gives -akx-a-1


As for the last part; sorry, when I wrote a on the LHS I was referring to the second derivative of xi w.r.t. time
 
  • #4
vela
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U = k/ra = k/root(x2i)a
= k/xai
= kx-ai

Differentiating this with respect to xi gives -akx-a-1
How are you going from k/root(x2i)a to k/xai?

Are you saying, for instance, that [itex]\sqrt{x_1^2+x_2^2} = x_1+x_2[/itex]?

EDIT: Oh, I see why we're getting different answers. I think there's an implied summation: [tex]r=\sqrt{x_i^2} = \sqrt{x_i x_i} = \sqrt{\sum_i x_i^2}[/tex]
 

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