The generalized coordinates used for this system are the radial distance ($\boldsymbol{r}$) and the angle ($\boldsymbol{\varphi}$).

Here is an explanation of what generalized coordinates are and why $r$ and $\varphi$ are chosen for this specific problem:

What are Generalized Coordinates?

Generalized coordinates ($\boldsymbol{q_i}$) are the minimum set of independent variables required to completely specify the position (or configuration) of a mechanical system at any instant in time.

• Independence: They must be independent of each other.
• Minimum Set: The number of generalized coordinates equals the degrees of freedom of the system.
• Convenience: They are chosen for convenience to simplify the mathematical equations of motion (like the kinetic energy $T$ and the Lagrangian $L$). They often automatically account for constraints.

Why $r$ and $\varphi$?

The system consists of two masses ($m_1$ and $m_2$) connected by a string passing through a hole, which imposes a constraint: the total length of the string ($L$) is fixed.

1. Radial Distance ($\boldsymbol{r}$):
   • $r$ is the distance of mass $m_1$ from the central hole in the horizontal plane.
   • This single variable dictates two things:
     • The radial position of $m_1$ (its $x$ and $y$ coordinates are functions of $r$ and $\varphi$).
     • The vertical position of $m_2$ ($z_2 = r - L$), because the string length is fixed. $m_2$ only moves as $r$ changes.
   • $r$ is essential to describe the motion along the radius.
2. Angle ($\boldsymbol{\varphi}$):
   • $\varphi$ is the angle that mass $m_1$ makes with respect to a fixed axis in the horizontal plane.
   • This variable describes the rotational position of $m_1$.
   • $\varphi$ is essential to describe the rotational motion.

Since the state of both masses at any time is fully determined by knowing just $r$ and $\varphi$, the system has two degrees of freedom, and $r$ and $\varphi$ are the appropriate generalized coordinates. Using them allows the system's kinetic energy to be written concisely in terms of their velocities ($\dot{r}$ and $\dot{\varphi}$), as shown in the previous steps.

Brief audio

What are the generalized coordinates used for this system-L.mp4