Generalised Coordinates: Lagrangian/Hamiltonian Mechanics

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Generalized coordinates in Lagrangian and Hamiltonian mechanics extend beyond traditional Cartesian or polar coordinates to more efficiently describe complex systems. They can include various quantities, such as velocities or even amplitudes in a Fourier expansion, which may better capture the dynamics of a moving object. The discussion highlights a need for examples of unconventional generalized coordinates, particularly those that deviate from simple orthogonal coordinates or angles. The inquiry seeks clarity on how these exotic quantities can be applied in practical scenarios. Understanding these concepts is crucial for mastering advanced mechanics.
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Can anyone give a simple explanation to generalised cordinates in Lagarangian/hamiltonian mechanics
 
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Regular x,y,z or polar coordinates may not be the best to describe a system. For moving objects, for instance, using both position x,y,z and velocity vx,vy,vz may more efficiently describe the motion. Many other examples are possible, as well. These are the generalized coordinates.
 
Goldstein, chapter 1.3, p.14 (3rd edition):
"All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of rj may be used as generalized coordinates, or we may find it convenient to employ quantities with the dimension of energy or angular momentum."

However, most examples I have seen so far (I have just begun to struggle with all this) are either simple orthogonal coordinates or angles. Does anybody know an example where really "exotic" quantities (like the Fourier stuff mentioned above) are used as generalized coordinates ?
 

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