In collaboration with K. Grabowska and J. Grabowski, we have applied the recently discovered notion of a weighted algebroid to mechanics on graded bundles[1].

In our preprint “Higher order mechanics on graded bundles” We present the corresponding Tulczyjew triple for this situation and derive the phase equations from an arbitrary (maybe singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. This is all done essentially in the first order set-up of mechanics on a Lie algebroid subject to vakonomic constraints. The amazing this is that the underlying graded bundle structure gives this whole picture the flavour of higher derivative mechanics. Within this framework we recover classical higher order mechanics, but we can study some more exotic situations.

For example, we geometrically derive the (reduced) higher order Euler-Lagrange equations for invariant higher order Lagrangians on Lie groupoids. To our knowledge, not much work has been done in understanding such systems [2,3]. We hope that the example on Lie groupoids turns out to be useful, maybe in say control theory.

References
[1] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, arXiv:1409.0439 [math-ph], (2014).

[2] L. Colombo & D.M. de Diego, Lagrangian submanifolds generating second-order Lagrangian mechanics on Lie algebroids, XV winter meeting of geometry, mechanics and control, Universidad de Zaragoza, (2013). http://andres.unizar.es/ ei/2013/Contribuciones/LeoColombo.pdf

[3] M. Jozwikowski & M. Rotkiewicz, Prototypes of higher algebroids with applications to variational calculus, arXiv:1306.3379v2 [math.DG] (2014).