- #1
H_man
- 143
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I shall give a short example to illustrate where I am confused.
If we take the K.E. of a particle in spherical polar coords.
<br /> T = \frac{1}{2}m( \dot r^2 + r^2 \dot\theta^2 + r^2 sin^2 \theta \dot \phi^2) <br />
And
<br /> T' = \frac{1}{2}( \dot r^2 + r^2 \dot\theta^2 + r^2 sin^2 \theta \dot \phi^2) <br />
Now plugging this in Lagrange's equation:
<br /> \frac {d} {dt} \frac {\partial T'} {\partial \dot \theta} - \frac {\partial T'} {\partial \theta} = r^2 \ddot \theta + 2r \dot r \dot \theta - r^2 \dot \phi^2 sin \theta cos \theta<br /> --------- Line 3
Now, as far as I understand the above equation is a_\theta. That is, the \theta component of acceleration.
However, it seems I am wrong.
My book tells me I have to divide the expression by h_\theta
where
<br /> h_\theta = \left ( \left ( \frac {\partial x} {\partial \theta} \right )^2 <br /> + \left ( \frac {\partial y} {\partial \theta} \right )^2 <br /> + \left ( \frac {\partial z} {\partial \theta} \right )^2 \right )^\frac {1}{2} = ( (r cos \theta cos \phi )^2 + ( (r cos \theta sin \phi )^2 + r^2 sin^2 \theta )^\frac {1}{2} = r<br />
Producing a_\theta = r \ddot \theta + 2 \dot r \dot \theta - r \dot \phi^2 sin \theta cos \theta
So making the reasonable assumption that the book is correct and I am not. What does the expression (Line 3) that I thought was the acceleration represent?
This is especially confusing as I know that if we do not divide T by m then line 3 should produce the force F_\theta??
If we take the K.E. of a particle in spherical polar coords.
<br /> T = \frac{1}{2}m( \dot r^2 + r^2 \dot\theta^2 + r^2 sin^2 \theta \dot \phi^2) <br />
And
<br /> T' = \frac{1}{2}( \dot r^2 + r^2 \dot\theta^2 + r^2 sin^2 \theta \dot \phi^2) <br />
Now plugging this in Lagrange's equation:
<br /> \frac {d} {dt} \frac {\partial T'} {\partial \dot \theta} - \frac {\partial T'} {\partial \theta} = r^2 \ddot \theta + 2r \dot r \dot \theta - r^2 \dot \phi^2 sin \theta cos \theta<br /> --------- Line 3
Now, as far as I understand the above equation is a_\theta. That is, the \theta component of acceleration.
However, it seems I am wrong.
My book tells me I have to divide the expression by h_\theta
where
<br /> h_\theta = \left ( \left ( \frac {\partial x} {\partial \theta} \right )^2 <br /> + \left ( \frac {\partial y} {\partial \theta} \right )^2 <br /> + \left ( \frac {\partial z} {\partial \theta} \right )^2 \right )^\frac {1}{2} = ( (r cos \theta cos \phi )^2 + ( (r cos \theta sin \phi )^2 + r^2 sin^2 \theta )^\frac {1}{2} = r<br />
Producing a_\theta = r \ddot \theta + 2 \dot r \dot \theta - r \dot \phi^2 sin \theta cos \theta
So making the reasonable assumption that the book is correct and I am not. What does the expression (Line 3) that I thought was the acceleration represent?
This is especially confusing as I know that if we do not divide T by m then line 3 should produce the force F_\theta??
