Generalised Coordinates: Lagrangian/Hamiltonian Mechanics

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SUMMARY

Generalized coordinates in Lagrangian and Hamiltonian mechanics provide a more efficient way to describe the motion of systems than traditional Cartesian or polar coordinates. As outlined in Goldstein's "Classical Mechanics" (3rd edition, chapter 1.3, p.14), any quantity, including Fourier expansion amplitudes or dimensions of energy and angular momentum, can serve as generalized coordinates. This flexibility allows for the modeling of complex systems beyond simple orthogonal coordinates or angles. The discussion highlights the need for examples of "exotic" generalized coordinates to deepen understanding.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with Hamiltonian mechanics
  • Basic knowledge of coordinate systems (Cartesian, polar)
  • Introduction to Fourier analysis
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  • Study the application of generalized coordinates in Lagrangian mechanics
  • Explore Hamiltonian mechanics and its use of generalized coordinates
  • Investigate examples of Fourier series in physics
  • Examine case studies involving non-traditional coordinate systems in mechanics
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Students and professionals in physics, particularly those studying classical mechanics, as well as researchers interested in advanced modeling techniques using generalized coordinates.

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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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