  Motivated by mechanical systems with symmetries, we focus on optimal control
problems possessing symmetries. Following recent works, which generalized the
classical concept of static turnpike to manifold turnpike, we extend the
exponential turnpike property to the exponential trim turnpike for control
systems with symmetries induced by abelian or non-abelian groups. Our analysis
is mainly based on the geometric reduction of control systems with symmetries.
More concretely, we first reduce the control system on the quotient space and
state the turnpike theorem for the reduced problem. Then we use the group
properties to obtain the trim turnpike theorem for the full problem. Finally,
we illustrate our results on the Kepler problem and the Rigid body problem.
