Lambda-symmetries of Dynamical Systems, Hamiltonian and Lagrangian equations

After a brief survey of the definition and the properties of $\Lambda$-symmetries in the general context of dynamical systems, the notion of ``$\Lambda$-constant of motion'' for Hamiltonian equations is introduced. If the Hamiltonian problem is derived from a $\Lambda$-invariant Lagrangian, it is shown how the Lagrangian $\Lambda$-invariance can be transferred into the Hamiltonian context and shown that the Hamiltonian equations turn out to be $\Lambda$-symmetric. Finally, the ``partial'' (Lagrangian) reduction of the Euler-Lagrange equations is compared with the reduction obtained for the corresponding Hamiltonian equations.