Generalised Coordinates: Lagrangian/Hamiltonian Mechanics

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In summary, generalized coordinates are alternative ways of describing the motion of a system, often more efficient than regular coordinates. These can include quantities such as amplitudes in a Fourier expansion or energy and angular momentum. While most examples tend to use simple orthogonal coordinates or angles, there may be instances where more uncommon quantities are used as generalized coordinates.
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deepthir
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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.
 
  • #3
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 ?
 

1. What are generalised coordinates in Lagrangian/Hamiltonian mechanics?

Generalised coordinates are a set of independent variables used to describe the position and orientation of a system in classical mechanics. They are chosen such that the equations of motion can be expressed in terms of these coordinates, making the analysis of complex systems more efficient.

2. How do generalised coordinates differ from Cartesian coordinates?

Unlike Cartesian coordinates, which are fixed and defined by a set of axes, generalised coordinates can vary based on the specific system being studied. They are chosen based on the constraints and symmetries of the system, and can often reduce the number of coordinates needed to describe the system's motion.

3. What is the role of Lagrangian mechanics in generalised coordinates?

Lagrangian mechanics is a mathematical formalism that uses generalised coordinates to derive the equations of motion for a system. It is an alternative to Newton's laws of motion and is often more useful for solving problems involving complex systems with multiple degrees of freedom.

4. How does Hamiltonian mechanics differ from Lagrangian mechanics?

Hamiltonian mechanics is a reformulation of Lagrangian mechanics that uses the Hamiltonian function to describe the system's energy rather than the Lagrangian function. It also introduces the concept of canonical coordinates, which are a set of generalised coordinates and their corresponding momenta.

5. What are the advantages of using generalised coordinates in mechanics?

Using generalised coordinates can simplify the calculation of a system's equations of motion, especially for complex systems with multiple degrees of freedom. It also allows for a more elegant and concise description of the system's motion, making it easier to analyse and understand its behavior.

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