generalized coordinates
When you describe a mechanical system, you do not have to use x, y, z Cartesian coordinates. A pendulum swinging back and forth needs only one number, the angle theta. A bead threaded on a wire needs only how far it has slid along the wire. Generalized coordinates are any set of independent quantities that completely pin down the configuration of a system.
They are usually written q_1, q_2, ..., q_n, and they need not be lengths: an angle, an area, even an electric charge can serve as a generalized coordinate. The number of them equals the system's degrees of freedom. Each particle's position is then a function r_i = r_i(q_1, ..., q_n, t), and the time derivatives q_dot are called the generalized velocities.
The power of Lagrangian mechanics is that you may write the Lagrangian in whatever coordinates make the problem simplest, and the Euler-Lagrange equations hold in all of them, with no need to resolve forces into components. Constraints are handled elegantly by choosing coordinates that already respect them.
A double pendulum is fully described by two angles theta_1 and theta_2, one for each rod.
Two generalized coordinates, matching the two degrees of freedom.
The standard Euler-Lagrange form assumes the coordinates are independent; if constraints link them, use fewer coordinates or bring in Lagrange multipliers.