  Classical turnpikes correspond to optimal steady states which are attractors
of optimal control problems. In this paper, motivated by mechanical systems
with symmetries, we generalize this concept to manifold turnpikes.
Specifically, the necessary optimality conditions on a symmetry-induced
manifold coincide with those of a reduced-order problem under certain
conditions. We also propose sufficient conditions for the existence of manifold
turnpikes based on a tailored notion of dissipativity with respect to
manifolds. We show how the classical Legendre transformation between
Euler-Lagrange and Hamilton formalisms can be extended to the adjoint
variables. Finally, we draw upon the Kepler problem to illustrate our findings.
