As a personal exercise, I was planning on implementing computationally some of the Lie group methods. So I'd like your input about differential equations where the space of configurations is a Lie group.

I know about the isospectral flow on the Toda lattice, and the QR flow on orthogonal matrices. These two examples do the job, but they are sooooo well-worn that I wonder if there are other interesting (this is opinion-based, but anyway) examples.

I thought about the Heisenberg group, whose Lie algebra is nilpotent, so the exponential can be exactly computed, and there is a global chart. Yet this global chart is so trivial, that any ODE on $H_3(\mathbb{R})$ could be easily rewritten on $\mathbb{R}^3$ itself, and there is no point in emphasizing the group structure.

I can think of some sources of inspiration, where I am no expert though:

  • Mechanics, e.g. non-holonomic systems.
  • Machine learning, e.g. low-rank factorization considering the Stiefel manifold.
  • Matrix nearness problems and similar topics in numerical analysis or optimization.

Arguably there is non-empty intersection among these fields.

Using an exceptional group, e.g. $G_2$, would be a bonus, because it would be... well, exceptional.

To summarize: I would like to see explicit examples of ODEs, where trajectories lie on a manifold, with some appealing physical/mathematical/computational interpetation.

Disclaimer: if the question has some obscure point, it is because of my lack of expertise. Please, comment nicely before downvoting :)


Rigid body rotation is the biggest practical example I know of. We have the equations

$$q' = \omega q, \\ I\omega' + \omega\times I\omega = M(q),$$

on $SU(2)\times R^3$, where $q$ is a unit quaternion, $\omega$ the angular velocity, $I$ is the inertia tensor, and $M$ is the net external moment on the body. Note that in the first equation we are representing $\omega$ as a pure quaternion, the set of which is isomorphic with $R^3$, so the notation $\omega q$ indicates quaternion multiplication.

  • $\begingroup$ Interesting... but may I invent $M(q)$ anyway?, or is it somehow restricted to guarantee that the state remains on the manifold? On second thoughts, since $w\in\mathbb{R}^3$ whose tangent is (isomorphic to) $\mathbb{R}^3$ itself, my guess is yes, $M(q)$ is any matrix function, correct? $\endgroup$
    – Miguel
    Sep 10 '17 at 11:15
  • $\begingroup$ @Miguel physically meaningful solutions come from torques, but you can use any function which maps S3 into R3. $\endgroup$
    – JMJ
    Sep 10 '17 at 13:51
  • $\begingroup$ Provides existence and uniqueness are taken care of, of course. $\endgroup$
    – JMJ
    Sep 10 '17 at 13:52
  • $\begingroup$ Existence and uniqueness of ODEs do not seem a big issue, it is enough that $M(q)$ is smooth, or even Lipschitz continuous, isn't it? What about time-varying inertia $I(t)$? $\endgroup$
    – Miguel
    Sep 10 '17 at 14:38
  • 1
    $\begingroup$ @Miguel It's enough to have $M$ Lipschitz but almost all classical torques are smooth. You can absolutely have $I$ time-varying. This would be the case for a rocket or spacecraft with thrusters for instance. $\endgroup$
    – JMJ
    Sep 10 '17 at 14:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.