Are My Partial Derivatives Correct in Finding the Equations of Motion?

[Maybe_Memorie]

Homework Statement

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

L = (1/2)mv^2_i - k/r^a,

r = root(x²_i), m,k,a are constants

Homework Equations

The Attempt at a Solution

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

So, when I work all this out I get
ma = ka/x_i^a+1

I have a feeling this isn;t correct though.

Am I doing the partials wrong?

 

[vela]

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

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

 

[Maybe_Memorie]

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

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

U = k/r^a = k/root(x²_i)^a
= k/x^a_i
= kx^-a_i

Differentiating this with respect to x_i 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 x_i w.r.t. time

 

[vela]

U = k/r^a = k/root(x²_i)^a
= k/x^a_i
= kx^-a_i

Differentiating this with respect to x_i gives -akx^-a-1

How are you going from k/root(x²_i)^a to k/x^a_i?

Are you saying, for instance, that $\sqrt{x_1^2+x_2^2} = x_1+x_2$?

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

 

[paranormal]

Your approach is correct, but there may be a mistake in your calculation of the partial derivatives. Let's break it down step by step:

1. First, we need to find the partial derivative of L with respect to vi.

∂L/∂vi = mvi

2. Next, we need to find the partial derivative of L with respect to xi. To do this, we need to use the chain rule since xi appears in the denominator of the second term.

∂L/∂xi = -k/xi^2 * ∂(1/ra)/∂xi
= -k/xi^2 * (∂(1/ra)/∂r * ∂r/∂xi) [using chain rule]
= -k/xi^2 * (-1/a * 1/r^2 * ∂r/∂xi) [since ∂(1/ra)/∂r = -1/a * 1/r^2]
= k/xi^3 * ∂r/∂xi

3. Now, we need to find the partial derivative of r with respect to xi. This is simply equal to 1/xi.

∂r/∂xi = 1/xi

4. Substituting this back into our expression for ∂L/∂xi, we get:

∂L/∂xi = k/xi^3 * (1/xi)
= k/xi^4

5. Now, we can plug these expressions back into the equation for the equations of motion:

d/dt(mvi) - k/xi^4 = 0

This is the correct equation of motion for a particle moving in a D-dimensional flat space with the given Lagrangian. Hope this helps!