Generalized Coordinates

In classical mechanics, generalized coordinates are a set of coordinates used to describe the configuration of a system in terms of its degrees of freedom.

Instead of using standard Cartesian coordinates, which may be cumbersome(PS:- idk why but like that's what old folks say about solving complex system) for systems with constraints, generalized coordinates allow us to describe the motion of a system with a smaller set of variables. These coordinates are denoted as $q_{i}$ (where $i = 1, 2, 3, \dots, N$ ), and their time derivatives ${\dot{q}}_{i}$ represent the generalized velocities.

Degrees of Freedom:

The number of generalized coordinates needed is equal to the number of degrees of freedom of the system. For example:

A particle moving in 3D space has three degrees of freedom, so we can choose $q_{1} = x$ , $q_{2} = y$ , and $q_{3} = z$ .
A pendulum constrained to move in a plane has one degree of freedom, so we might choose $q_{1} = θ$ , the angle the pendulum makes with the vertical.

Lagrangian in Generalized Coordinates:

The Lagrangian is typically expressed in terms of generalized coordinates and their velocities:

L = L (q_{i}, {\dot{q}}_{i}, t)

This generalization allows us to apply the principles of mechanics to systems with complex geometries and constraints.

look Lagrangian and Hamiltonian Dynamics for how generalized coordinates are used in the action.
look Euler-Lagrange Equation for how generalized coordinates help us to get equations of motion.