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This paper concerns Lagrangian systems with symmetries, near points with configuration space isotropy. Using twisted parametrisations corresponding to phase space slices based at zero points of tangent fibres, we deduce reduced equations of motion, which are a hybrid of the Euler-Poincare and Euler-Lagrange equations. Further, we state a corresponding variational principle.

Euler-Poincaré reduction for systems with configuration space isotropy