Considerations for formulating Lagrangians in general

My question albeit trivial is something I wish to understand better. Given a system either say 3 masses, 2 masses and a pulley (with mass), and 2 masses attached via a string with mass $$m$$ (and uniform mass density $$\lambda$$).

• In these cases do we have to consider 9 generalized coordinates (excluding constraints for now) since there are 3 mass-full objects and assuming we are in 3D-space.

• Is this always true? Is it therefore a good practice when formulating Lagrangians to identify objects with mass and finding their respective kinetic and potential energies?

EDIT: Assume stationary lab frame of reference and no rotations (other than the pulley/disk)

• Take the case of 2D space with out rotation, each particle has 2 degrees of freedom, and n particles has 2n degrees of freedom. Now each constraint equation reduce the degrees of freedom by one, so the numbers of the generalized coordinates for n particles are $2n-n_c$ – Eli Apr 10 at 7:20
• I understand that for a system of n particles in k-space we have nk - $n_c$ degrees of freedom where $n_c$ is the number of constraints in the system. However I guess another way to ask my question, are the 'particles' of interest that we must consider are always just any object with mass? Such as the string with mass m, or the pulley with mass m, or just point masses m. Thank you! – QuantumPanda Apr 10 at 17:13