This question is about the dynamics (in classical mechanics) of a rigidly linked chain of $N$ point masses, see figure. Let us say that the masses $m_1,\ldots,m_N$ have initial positions ($\mathbf{x}_1,\ldots,\mathbf{x}_N$) and initial velocities (velocities $\dot{\mathbf{x}}_1,\ldots,\dot{\mathbf{x}}_N$) are known and a homogeneous gravitational field is present. The lengths of the links between masses are rigid and must stay the same for all time. Given this information, we are able to compute the dynamics of the chain for all time.
How is it possible to numerically iterate the dynamics of the beads without fully describing the system as ODEs? I would like retain the ability to add/remove masses without changing without having to rewrite the Lagrangian and re-deriving the equations of motion. This includes changing the topology of the "chain" (by making loops like in the figure on the right).
Which numerical methods are used to iterate the dynamics without writing long Euler-Lagrange equations? For example, how do physics engines in games do it?
