  This paper is a contribution to the general program of embedding theories of
dynamical systems. Following our previous work on the Stochastic embedding
theory developed with S. Darses, we define the fractional embedding of
differential operators and ordinary differential equations. We construct an
operator combining in a symmetric way the left and right (Riemann-Liouville)
fractional derivatives. For Lagrangian systems, our method provide a fractional
Euler-Lagrange equation. We prove, developing the corresponding fractional
calculus of variations, that such equation can be derived via a fractional
least-action principle. We then obtain naturally a fractional Noether theorem
and a fractional Hamiltonian formulation of fractional Lagrangian systems. All
these constructions are coherents, i.e. that the embedding procedure is
compatible with the fractional calculus of variations. We then extend our
results to cover the Ostrogradski formalism. Using the fractional embedding and
following a previous work of F. Riewe, we obtain a fractional Ostrogradski
formalism which allows us to derive non-conservative dynamical systems via a
fractional generalized least-action principle. We also discuss the Whittaker
equation and obtain a fractional Lagrangian formulation. Last, we discuss the
fractional embedding of continuous Lagrangian systems. In particular, we obtain
a fractional Lagrangian formulation of the classical fractional wave equation
introduced by Schneider and Wyss as well as the fractional diffusion equation.
