The Need for Better Coordinates
Newton's equations are easy in \(x, y, z\). But for a pendulum, the position is best described by an angle \(\theta\). For a planet, it's \(r\) and \(\theta\). Generalized Coordinates (\(q_1, q_2, \dots\)) allow us to describe any system using the variables that fit its natural geometry.
Worked Examples
Example 1: The Pendulum
Instead of using \(x\) and \(y\) and having to worry about the tension in the string, we use \(\theta\). The height is \(L(1 - \cos \theta)\). By using \(\theta\) as our coordinate, the "constraint" of the string length disappears from the math.
The Bridge to Quantum Mechanics
In Quantum Mechanics, we almost never use \(x, y, z\). We use coordinates that match the Symmetry of the system. For a molecule, we might use the distances between nuclei. For a spinning particle, we use angles in "Spin Space." The choice of generalized coordinates determines the form of the Hamiltonian. Learning to pick the right \(q\) is the secret to making a quantum problem solvable.