  We examine "dynamical similarities" in the Lagrangian framework. These are
symmetries of an intrinsically determined physical system under which
observables remain unaffected, but the extraneous information is changed. We
establish three central results in this context: i) Given a system with such a
symmetry there exists a system of invariants which form a subalgebra of phase
space, whose evolution is autonomous; ii) this subalgebra of autonomous
observables evolves as a contact system, in which the friction-like term
describes evolution along the direction of similarity; iii) the contact
Hamiltonian and one-form are invariants, and reproduce the dynamics of the
invariants. As the subalgebra of invariants is smaller than phase space,
dynamics is determined only in terms of this smaller space. We show how to
obtain the contact system from the symplectic system, and the embedding which
inverts the process. These results are then illustrated in the case of
homogeneous Lagrangians, including flat cosmologies minimally coupled to
matter; the n-body problem and homogeneous, anisotropic cosmology.
