Lagrangian mechanics on centered semi-direct product

TL;DR

This paper introduces a new semi-direct product of a Lie group and a vector space, called the centered semi-direct product, and develops its Lagrangian mechanics framework paralleling existing theories.

Contribution

It constructs the centered semi-direct product combining left and right actions and extends Euler-Poincaré theory to this new structure.

Findings

Defined the centered semi-direct product as a sum of left and right semi-direct products.

Extended Euler-Poincaré equations to the centered semi-direct product.

Presented a toy example and a potential application in diffeomorphism groups.

Abstract

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 a right and left semi-direct product. We then 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.