# Lagrangian mechanics on centered semi-direct product

There exists two types of semi-direct products between a Lie group \$G\$ and a vector space \$V\$. The left semi-direct product, \$G \ltimes V\$, can be constructed when \$G\$ is equipped with a left action on \$V\$. Similarly, the right semi-direct product, \$G \rtimes V\$, can be constructed when \$G\$ is equipped with a right action on \$V\$.

In this paper, we will construct a new type of semi-direct product, \$G \Join V\$, which can be seen as the `sum' of right and left semi-direct products. We then proceed to the parallel existing semi-direct product Euler-Poincar\'e theory. We find that the group multiplication, the Lie bracket, and the diamond operator can each be seen as a sum of the associated concepts in right and left semi-direct product theory. Finally, we conclude with a toy example and the group of 2-jets of diffeomorphisms above a fixed point. This final example has potential use in the creation of particle methods for problems on diffeomorphism groups.

Article:
Lagrangian mechanics on centered semi-direct product
Authors:
Leonardo J. Colombo, Henry O. Jacobs
Journal:
ArXiv preprint
Year:
2013
URL:
http://arxiv.org/abs/1303.3883