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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 :)

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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.

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  • $\begingroup$ 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, 2017 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, 2017 at 13:51
  • $\begingroup$ Provides existence and uniqueness are taken care of, of course. $\endgroup$
    – JMJ
    Sep 10, 2017 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, 2017 at 14:38
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    $\begingroup$ @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, 2017 at 14:47

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