  Fractional equations appear in the description of the dynamics of various
physical systems. For Lagrangian systems, the embedding theory developped by
Cresson ["Fractional embedding of differential operators and Lagrangian
systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain
such equations, stemming from a least action principle. However, no matter how
equations are obtained, the dimension of the fractional derivative differs from
the classical one and may induce problems of temporal homogeneity in fractional
objects. In this paper, we show that it is necessary to introduce an extrinsic
constant of time. Then, we use it to construct two equivalent fractional
embeddings which retain homogeneity. The notion of fractional constant is also
discussed through this formalism. Finally, an illustration is given with
natural Lagrangian systems, and the case of the harmonic oscillator is entirely
treated.
