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How to find potential energy if force depends on both position of particle and time ? Suppose force is : f(r,t) = (k/r^2) * exp(-alpha*t), k, alpha = positive constants, r = position of the particle from force-centre t = time Is this force a conservative or non-conservative ?

In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that : In Hamiltonian formulation, there can be no constraint equations among the co-ordinates. Why is this necessary ? Any simple example which will elaborate this fact ? But in Lagrangian...

But my book says these 5 generalized co-ordinates can NOT all vary independently. When the sphere rolls on perfectly rough flat,horizontal surface, at least 2 co-ordinates (out of 5) must change. Hence degrees of freedom for this case is < 5; but number of generalized co-ordinates is 5. Hence it...

Fantastic! Now I understood. Thanks for clear & logical answer. Now my next doubt is : If we consider motion of the sphere restricted to flat,horizontal plane (without slipping), can all these 5 generalized co-ordinates vary independently of each other ?

3 degrees of freedom i.e. 3 axes of rotations. And 3 axes being orthogonal to each other. Each axis passing through COM (their point of intersection).

No.

Yes, ball must roll without slipping. So do you mean 3 angles (in general) will change when ball is in motion? These 3 angles being axes of rotation, one rotation about X axis, one rotation about Y axis & one rotation about Z axis. (This XYZ system being superimposed on spherical co-ordinate...

But when you rotate sphere about line joining COM and the dot, co-ordinates of other points (on & inside) of the sphere are changing. If I move the sphere from (x1,y1) to (x2,y2) (while changing at the same time polar and azimuth angles), motion is complete. Dot on the sphere is not in motion...

But consider a fixed point on the sphere. We need 4 co-ordinates to uniquely decide its location on the sphere. (x,y) to locate COM of the sphere and altitude & azimuth to locate the point on the surface of the sphere. (2 co-ordinate systems are used : XY plane system and spherical co-ordinate...

Consider a sphere constrained to roll on a rough FLAT HORIZONTAL surface. A book on classical mechanics says it requires 5 generalized co-ordinates to specify sphere's configuration: 2 for its centre of mass and 3 for its orientation. I did not understand why 3 for orientation. I guess only 2...

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