An action of a Lie group G on a manifold Q is said to be of constant orbit type if the isotropy group of q₁∈Q is conjugate to the isotropy group of q₂∈Q for each q₁,q₂∈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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