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A free particle with coordinates as shown has kinetic energy ##T = \frac{1}{2}m\left(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2\theta\dot\phi^2\right)##

So we see ##T## depends on ##\theta##.

Now suppose we rotate our coordinate system such that only one coordinate ##\theta## changes from ##\theta## to

##\theta'## and fix it there as is shown in figure below.

In this coordinate system the kinetic energy should be the same as before since the kinetic energy should be same in all inertial frames.

However if we substitue values of ##r, \theta',\phi## in the formula for ##T## we will get a different value of ##T## hence a contradiction.

Can anyone please tell me what is wrong. Thank you.

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