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?
Generalized coordinates in Goldstein's Classical Mechanics are a set of variables that describe the position and orientation of a system in terms of its degrees of freedom. These coordinates are independent of each other and can be used to fully describe the system's configuration.
Generalized coordinates are used in classical mechanics because they simplify the mathematical description of a system. They allow for a more compact and efficient representation of the system's configuration, making it easier to solve equations of motion and analyze the dynamics of the system.
The number of generalized coordinates needed for a system is equal to the number of degrees of freedom of that system. Degrees of freedom refer to the number of independent ways a system can move or change its configuration. For example, a point particle in three-dimensional space has three degrees of freedom, so we would need three generalized coordinates to fully describe its position.
No, not all sets of variables can be used as generalized coordinates. The chosen variables must be independent of each other and must fully describe the position and orientation of the system. In addition, they should be easy to work with mathematically, making the analysis of the system simpler.
Generalized coordinates are related to other coordinate systems through a transformation known as a coordinate transformation. This transformation allows us to express the generalized coordinates in terms of other coordinate systems, such as Cartesian or polar coordinates. This can be useful when analyzing a system from different perspectives or when solving equations of motion in different coordinate systems.