# Differential forms in mechanics?

I teach mechanics (including large deformation and flow of continua) to mechanical engineering students and have a continuing mission to drag the teaching of mechanics into the 20th century (I'll worry about the 21st later) by introducing 'modern' ideas (physicists may laugh!) on differential forms and the like (co- versus contra-variance in oldspeak). I have most of the standard texts and well-known papers on the subject but the applications they consider (if any) tend to be on electromagnetics, general relativity or some other topic of little interest to my students. My questions are then:

• is anyone else doing anything similar?

• does anyone know of any specifically mechanics-oriented material?

• has anyone any experience of teaching diff forms and geometric algebra to engineers or physicists. If so, what is more useful in practice?

Edit 20 Sep 15

To expand on the OP, I don't find any of the existing texts suitable for mech engineers, whether students or professionals: they all start with some formalism about diff forms and then (just maybe) do applications that are irrelevant to mech eng (Frankel's Geometry of Physics has a section on stress but it's a bit of a dangling appendage). I've argued elsewhere that, for engineers, the motivation is the application. My approach is then to start with the need to calculate the work done by a particle moving in a force field. In 'traditional' vector calculus, to compute the dot product in the the line integral, we change the column matrix that usually denotes the force 'vector' to a row matrix. This is usually glossed over but I emphasise that this isn't a trivial change - it's effectively saying that force is a 1-form. I want to then do fluid flow in $R^2$ then in $R^3$, then strain and stress, introducing 2-forms, Hodge dual, Lie derivative, etc only when needed. I've trawled the web, looked at every textbook on diff forms I can find but nothing seems to develop along these lines (though Metzler's YouTube clips are instructive). So either:

• I'm barking up the wrong tree (or possibly just barking).

• I'm breaking new ground in teaching diff forms to mech eng undergraduates.

I might be talking myself into writing Diff Geom for Mech Engineers but, if someone has already done something similar, I'd really like to see it - and put it on the 'compulsory purchase' list for Mechanics 5.

• I'm a little out of touch with this subject now (hence the comment rather than an answer), but two books that that were fairly well known in the 1970s are Advanced Calculus by Loomis/Sternberg (has a lengthy chapter on classical mechanics) and Mathematical Methods of Classical Mechanics by V. I. Arnold. And of course there's Abraham/Marsden's Foundations Of Mechanics, but this might be a bit rough going for your students. Sep 15, 2015 at 16:40
• Thanks. Truth be told, Marsden is rough going for me!! I like Arnold's book and am pulling bits out of it. I'll also have a look at Loomis. One of my favourites is Bill Burke's Diff Grad and Curl are Dead but his own untimely death was a great loss. I might edit the OP to see if others have suggestions.
– rdt2
Sep 19, 2015 at 16:08
• Have you seen Jose and Saletan's text? amazon.com/Classical-Dynamics-A-Contemporary-Approach/dp/… this might be a good fit. I almost had a course out of it once, but, apparently my peers lacked background :( Sep 21, 2015 at 13:03
• Are you including the Reynolds Transport Theorem anywhere in this approach? Dec 5, 2015 at 19:44

I'd suggest having a look at Geometric Mechanics by Darryl Holm.

The theme of developing mechanics in the framework of differential geometry has a considerable record at the graduate level, as indicated in Dave Renfro's comment. At a level suitable for undergraduates, though, the pickings may be slim. Somewhat more accessible than Abraham and Marsden might be Introduction to Mechanics and Symmetry by Marsden and Ratiu.

It is an error to begin courses on differential forms, thinking that the multilinear algebra language is easily grasped. A good sequence of knowledge is to take on vector duality and rank two tensors. Here one can develop the basis on mechanics. Try Alessandra Iozzi's notes called Multilinear algebra and applications that can be found on Notes, to see a version.

Already with those experiences you could advance on electromagnetism and both relativity's theories. Hand by hand with differential geometry of curves and surfaces à la Cartan of moving frames.

You might find Edwards' book on advanced calculus using differential forms to have useful stuff. The very first section is about work in a constant vector field - I haven't used it as a text and I'm not an engineer so I can't say whether it will fit your needs as a supplement, but at the very least it could give ideas for your proposed writing of a book...

(Borrowed from my answer at math.SX, which might be another place to ask about such things. Good luck.)