one of the things i loved most in my engineering undergraduate education was learning kinematics — all kinds of cool mechanisms. the pendula that we’ve been looking at have some interesting joints. the planar pendulum has a simple hinge joint, easily built using a high-quality low-friction bearing. the spherical pendulum, on the other hand, has a ball-and-socket joint, which in practice could be effected by a spherical bearing. these are beautiful mechanisms that use lubrication and/or tiny ball bearings to get a nice clean swivel.
but this is not the only joint type with multiple degrees of freedom. consider the universal joint, built by fusing two transverse hinges, as illustrated below.
wouldn’t it be fun to simulate a double-pendulum with universal joints and compare it to the analogous system with spherical joints? what a good idea…
this is so much fun to play with… with the bottom views in particular, you sense that the universal joints give very different behavior than the spherical joints. the universals are more… jerky… constrainted. the spherical joints seem more free.
but are they? are they equivalent in some way? how would you tell? i like to think in terms of the configuration space of a joint (or system of joints). a universal joint has configuration space a 2-dimensional torus (one circle for each hinge joint). of course, this ignores physical constraints such as rigid bars colliding, but there you have it. the spherical joint, on the other hand, as a full SO(3) configuration space, which is three-dimensional, not 2-d like the universal. so, in one sense, this is why a spherical joint “feels” more free. it’s also the reason why a spherical joint cannot transmit torque the way that universal joints can. (universals — or rather pairs of them — are very helpful in automobiles to transmit power from the axles to the wheels…)
i like pendula in all forms as a way to visualize interesting dynamical systems. i’m most interest in the mathematics behind it all, but occasionally, i let my initial training in mechanical engineering get the best of me. this is one of those times.
consider what happens when you have a double-pendulum, but where the joints are spherical as opposed to planar (or “hinge”) in nature. this is really beautiful.
this can be a little difficult to visualize properly, since there is motion out of the plane of view. let’s take a look at the exacy same simulation, but from below…
all of the pendula below are “planar” or have “hinge” joints that constrict motion to a plane. if we replace said hinge joints with a “ball-and-socket” joint, then we have what one calls a spherical pendulum. these have an increased range of motion and lead to some lovely behavior when viewing multiple initial conditions.
you might rightly complain that these “spherical” pendula do not take advantage of their increased degrees of freedom. if you look carefully, each pendulum above moves within a single plane. this is a consequence of there being no initial velocity.
so what happens if we build a multi-pendulum with structures other than rigid rods? using hoops (solid tori) is especially mesmerizing, and leads one to believe that the dynamics is very different that the rigid-rod case…
a more careful analysis would show that using rings is equivalent to using rods, as long as you are careful about the equivalent mass (becuase of moment of inertia, the mass of the solid rod and that of the corresponding ring are not the same).
of course, this is be taken to foolish extremes. i do not recommend that you build the following system:
of course, instead of shaking a pendulum horizontally or vertically, one can go meta and hang a pendulum from a pendulum. let’s see what happens with a such a double pendulum.
that looks chaotic, doesn’t it? but what is meant by chaos? that takes a long time to learn properly. for now, let’s focus on one crucial requirement for chaos: sensitive dependence on initial conditions. this is the idea that nearby initial conditions diverge in behavior exponentially quickly. let’s take a look:
why is the double pendulum chaotic when the single pendulum is not (unless shaken in the right way)? it’s all about topology. the configuration space of a single pendulum is two-dimensional — one for the angle and one for the angular velocity. the configuration space is the tangent bundle of a circle, homeomorphic to an annulus. that’s not enough room for a flow to get chaotic. but the double pendulum has twice the dimension — it’s a 4-d tangent bundle of a torus — and that is plenty of room for chaos to creep in.
several commenters on twitter/youtube liked the horizontally shaken pendula and asked about what was being represented. there were many pendula with different initial positions — all initial velocities were set to zero. there were several queries as to what would happen with different lengths or different masses… would resonances with the vibration and the natural frequencies kick in?
dynamical systems studies *interesting* behaviors in systems. one such class of interesting behaviors is equilibria, with stable and unstable varieties. but there’s more. consider a slight variation on the shaken inverted pendulum, where the shaking is happening horizontally. this leads to interesting and varied behaviors, depending (somewhat sensitively) on the initial conditions. some are *periodic* (they repeat over time); others seem to keep evolving and keep getting more & more complex. what is happening? how do we make sense of such? oh, there’s much more to learn about dynamical systems…
dynamical systems begins with the notion of equilibria — states that remain constant over time. In the simplest cases, one distinguishes between stable and unstable equilibria, though to specify what exactly these mean requires care (and math).
one simple example of stable and unstable equilibria is found in a rigid-rod pendulum, rotating about a pivot point (with or, as here, without, friction). letting the system evolve from a variety of initial conditions reveals a stable equilibrium (bottom) and an unstable equilibrium (top).
there’s a lot to discuss about this system. skipping ahead (for the sake of seeing something wonderful), let’s consider what happens when you give this system a shake, vibrating the pivot point up & down. the simulation indicates that past a criticla threshold of amplitude/frequency, shaking the pendulum converts the unstable equilibrium to a stable equilibrium.
that’s swell. but does the mathematics prove that this is the expected behavior? well, that’s one of the reasons why you take a dynamical systems course.