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Hi, Im getting started on analytical mechanics but need some help understanding some of the things that is said in my book. Therefore I have attached a section of the derivation and for that I have some questions:
1) First of all the author says that in cartesian coordinates, i.e. (x,y,z), the derivative of T=kinetic energy with respect to the generalized coordinate q (which I assume is now taken to be a coordinate in a cartesian coordinate system, right?) vanishes. On the other hand it doesn't for angular coordinates. Can someone explain this? Why can't the kinetic energy depend on the position of the object? Certainly it can, so I must be misunderstanding something.
2) Secondly, he later says that it is possible to find a set of generalized coordinates (q1,q2...qn) which are independent of each other such that any virtual displacement, δj, is independent of any other δk. I don't understand this, does this not only hold for orthogonal coordinates? Or is it not that property which assures that the above can be assumed?
1) First of all the author says that in cartesian coordinates, i.e. (x,y,z), the derivative of T=kinetic energy with respect to the generalized coordinate q (which I assume is now taken to be a coordinate in a cartesian coordinate system, right?) vanishes. On the other hand it doesn't for angular coordinates. Can someone explain this? Why can't the kinetic energy depend on the position of the object? Certainly it can, so I must be misunderstanding something.
2) Secondly, he later says that it is possible to find a set of generalized coordinates (q1,q2...qn) which are independent of each other such that any virtual displacement, δj, is independent of any other δk. I don't understand this, does this not only hold for orthogonal coordinates? Or is it not that property which assures that the above can be assumed?
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