I'm trying to read Goldstein's Classical Mechanics (self-study), and getting into difficulties understanding the formalism early on. I thought I had an adequate understanding of basic calculus, but apparently not! Given that q* (I'm using an asterisk to denote a dot) means the derivative of q with respect to time, what does it even mean to write something like ∂T/∂q*? Goldstein does that when deriving Lagrange's equations from Newton's laws for a general system with constraints (q is a generalized coordinate here). The final form of Lagrange's equation has this too - it contains a term ∂L/∂q*_j. I feel that I'm missing something incredibly basic here. q* is not an independent variable, q is. T or L are functions of q, not of q*. What's the mathematically rigorous way to understand the meaning of these equations?