Scaling Symmetries, Contact Reduction and Poincar\'e's dream

A symplectic Hamiltonian system admitting a scaling symmetry can be reduced to an equivalent contact Hamiltonian system in which some physically-irrelevant degree of freedom has been removed. As a consequence, one obtains an equivalent description for the same physical phenomenon, but with fewer inputs needed, thus realizing "Poincar\'e's dream" of a scale-invariant description of the universe. This work is devoted to a thorough analysis of the mathematical framework behind such reductions. We show that generically such reduction is possible and the reduced (fundamental) system is a contact Hamiltonian system. The price to pay for this level of generality is that one is compelled to consider the coupling constants appearing in the original Hamiltonian as part of the dynamical variables of a lifted system. This however has the added advantage of removing the hypothesis of the existence of a scaling symmetry for the original system at all, without breaking the sought-for reduction in the number of inputs needed. Therefore a large class of Hamiltonian (resp. Lagrangian) theories can be reduced to scale-invariant contact Hamiltonian (resp. Herglotz variational) theories.