An action of a Lie group *G* on a manifold *Q* is said to be of
*constant orbit type* if the isotropy group of
*q*_{1}∈*Q* is conjugate to the isotropy group of
*q*_{2}∈*Q* for each
*q*_{1},*q*_{2}∈*Q*. In such cases the
group orbits are each diffeomorphic to a homogenous space of the group
*G*. We thus begin by investigating simple mechanical systems (i.e.,
those whose Lagrangians are kinetic minus potential energies) whose
configuration manifold is a homogeneous space (generalising the
Euler-Poincaré equations). We then use the structure of these systems to
discuss the local geometry of general simple mechanical systems with a
symmetry group giving an action of constant orbit type.

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