Analytical mechanics is a formulation in "generalized coordinates". Meaning you are free to choose your coordinate system any way you want, e.g. axes don't have to be perpendicular to each other, you can mix and match spherical/Cartesian coordinates, or simply make up your own. So you should not assume that a theta you see in a problem is the theta that is used in spherical/cylindrical coordinates. In this example, it certainly is not.
I can see that you have copied the solution word-for-word from Fawles & Cassiday but you really did not explain why the little mass down the incline has the polar unit vector e and how you justify the mixture of cartesian and polar coordinates into one equation for the KE. Besides, what mechanism causes an incline to slide forward while the mass on it descends downwards. Your solution is too vague and unclear.
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Analytical mechanics is a formulation in "generalized coordinates". Meaning you are free to choose your coordinate system any way you want, e.g. axes don't have to be perpendicular to each other, you can mix and match spherical/Cartesian coordinates, or simply make up your own. So you should not assume that a theta you see in a problem is the theta that is used in spherical/cylindrical coordinates. In this example, it certainly is not.
Thank you very much!!!, I want to know why the direction of x prime is theta, I think that should be the radial direction.
So basically, once you have written down the Lagrangian you are pretty much done?
I can see that you have copied the solution word-for-word from Fawles & Cassiday but you really did not explain why the little mass down the incline has the polar unit vector e and how you justify the mixture of cartesian and polar coordinates into one equation for the KE. Besides, what mechanism causes an incline to slide forward while the mass on it descends downwards. Your solution is too vague and unclear.
It's not polar/cartesian, it's generic coordinates. The whole point of langrangian method is to implement generic coordinates