Generalized coordinates Totally Explained

Everything about Generalized Coordinates totally explained

"Generalized coordinates are unspecified coordinates. By deriving equations of motion in terms of a general set of coordinates, the results found will be valid for any coordinate system that's ultimately specified." The name is a holdover from a period when Cartesian coordinates were the standard system.

Independent generalized coordinates

For any particular problem, it's advantageous to choose generalized coordinates such that they're independent, as is done in Lagrangian mechanics, because this eliminates the variables that would be required to express constraints on and among the coordinates. However, when dealing with nonholonomic constraints or when trying to find the force due to any constraint—holonomic or not, dependent generalized coordinates must be employed. Sometimes independent generalized coordinates are called internal coordinates because they're mutually independent, otherwise unconstrained, and together give the position of the system.
A system with

m

degrees of freedom and n particles whose positions are designated with three dimensional vectors,

lbrace mathbf  = f left (q_1, q_2, ..., q_n 
ight )

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