Understanding Generalized Coordinates in Goldstein's Classical Mechanics

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SUMMARY

Generalized coordinates encompass any coordinate system, including Cartesian, polar, spherical, and cylindrical systems. In classical mechanics, as outlined in Goldstein's text, one can utilize any convenient coordinate system to solve problems using Lagrange's equations. Polar and spherical coordinates are indeed considered generalized coordinates for the position of a single particle, while Cartesian coordinates also qualify as generalized coordinates in a broader sense.

PREREQUISITES
  • Understanding of Lagrange's equations
  • Familiarity with coordinate systems (Cartesian, polar, spherical, cylindrical)
  • Basic knowledge of classical mechanics principles
  • Ability to interpret mathematical representations of motion
NEXT STEPS
  • Study Lagrange's equations in detail
  • Explore the application of generalized coordinates in various mechanical problems
  • Learn about the advantages of using non-Cartesian coordinate systems
  • Investigate examples of motion described in polar and spherical coordinates
USEFUL FOR

Students of classical mechanics, physics educators, and anyone seeking to deepen their understanding of generalized coordinates and their applications in problem-solving within mechanics.

radou
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I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates? the way I got it, every coordinate system different from the standard cartesian-one is a set of generalized coordinates...?
 
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No. Think about the 1D movement along the "x" axis. Which is the generalized coordinate...?

Daniel.
 
Generalized coordinates refer to any coordinate system. i.e. a statement about generalized coordinates holds for cartesian, spherical, cylindrical, etc. coordinate systems. In particular, one is free to choose any convenient coordinate system for a problem and solve the problem using Lagrange's equations for that coordinate system.
 
radou said:
I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates?

Yes, polar and spherical coordinates are generalized coordiantes for the position of a single particle. But general coordinates are a lot moe general. And cartesian coordinates are, technically at least, also "general coordinates".

Carl
 

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