Noether theorem for mu-symmetries

We give a version of Noether theorem adapted to the framework of μ-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of λ-symmetries, and connects μ-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this ‘μ-conservation law’ actually reduces to a standard one; we also note a relation between μ-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under μ-prolonged variation fields, leading to modified Euler–Lagrange equations. In this setting, μ-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.