Consider a multi-pendulum system , i.e. a set of points $P_0, P_1,\dots, P_n$ with masses $m_k$, connected by massless rods of length $l_k$, which thus may rotate around each other. $P_0$ is fixed.
The state of the system at time $t$ can generally be given by $n$ complex numbers
$z_1(t) = l_1e^{i\Phi_1(t)}$
$z_2(t) = z_1(t) + l_2e^{i\Phi_2(t)}$
$z_3(t) = z_2(t) + l_3e^{i\Phi_3(t)}$
$\dots$
$z_n(t) = \sum_{k=1}^n l_ke^{i\Phi_k(t)} $
In the case of massless points, i.e. $m_k = 0$ for all $k$, there is an explicit solution:
$z_1(t) = l_1e^{i(\omega_1 t + \varphi_1)}$
$z_2(t) = z_1(t) + l_2e^{i(\omega_2 t + \varphi_2)}$
$\dots$
$z_n(t) = \sum_{k=1}^n l_ke^{i(\omega_k t + \varphi_k)} = \sum_{k=1}^n c_ke^{i\omega_k t}$
with $c_k = l_ke^{i\varphi_k}$ and $\omega_k$, $\varphi_k$ reflecting the initial conditions.
Findings:
The system's trajectory in phase space is periodic iff $z_k(t)$ is periodic for all $k$.
$z_n(t)$ is periodic iff $\omega_k/\omega_n \in \mathbb{Q}$ for all $k$. (see here)
$\rightarrow z_k(t)$ is periodic for all $k$ iff $z_n(t)$ is periodic.
Any reasonable closed curve can be approximated by such a periodic system. (see here)
When the points are subject to gravity*, then
even the most simple system with $n=2$ (double pendulum), $l_1 = l_2$, $m_1 = m_2$ has no explicit solution and in general no closed trajectories in phase space
the system behaves chaotically and ergodically
there may be initial conditions that give rise to closed trajectories both in position and phase space – but it's not clear (to me) how to find them.
* as opposed to gravitation
One missing piece between these two cases – massless points and points which are subject to gravity – is the case of points with inert masses.
In the massless case (the purely "mathematical" one), any point can be arbitraryly fast, but in the non-massless case (the most simple "physical" one) the total kinetic energy of all points must be conserved – which makes the difference.
My question is:
What are the relevant findings for multi-pendulum systems of inert points?
May there be explicit formulas for the trajectories as in the case of massless points?
Is there an easy criterion for periodicity in phase space?
Moreover: When complex analysis is the framework for "massless pendulums" and the double pendulum is the prototype for "gravitational pendulums", what's the mathematical framework or prototype for "inert pendulums"?
Examples:
The final missing piece would be points with pairwise gravitational forces, i.e. the true n-body problem. But that's another story.