Relative equilibria in dynamical systems on Lie groups

In a dynamical system with symmetry, a solution curve or its initial
point is called a relative equilibrium if the solution is also the
orbit of a one-parameter subgroup of the symmetry group. Relative
equilibria occur in various dynamical systems with symmetry and are
closely related to the geometric structure of each dynamical
system. It is already known that the system is in relative equilibrium
if and only if the initial point is a critical point of the function
called the augmented potential. However, the dynamical meanings of the
augmented potential are still worth understanding deeply.

This paper deals with simple Lagrangian systems with symmetry on Lie
groups, in which each symmetry group is a subgroup of the
configuration Lie group. For these systems, necessary and sufficient
conditions for relative equilibria are given. These conditions can be
viewed as an extension of the Lie-algebraic condition that
Hern\'{a}ndez-Gardu\~{n}o {\it et al.} gave in the case that the
symmetry group coincides with the configuration space. It is also
shown that the Euler-Lagrange equations are written out explicitly by
using the augmented potential. The expression allows of an
interpretation of the augmented potential, and further, provides the
conditions for relative equilibria in terms of the Euler-Lagrange