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

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