Mechanics of particles for fluid dynamics: derivation of dissipative equations of motion using the principle of least action and fractional derivatives

Equations of motion for systems with forces proportional to velocity are derived from the principle of least action using classical Euler-Lagrange equation and Lagrangian that contains terms responsible for energy dissipation. These terms are expressed through fractional derivatives of the coordinate of order $0 < \alpha < 1$.
This approach naturally leads to the introduction of non-uniform flow of time in dissipative processes. It incorporates inhomogeneous velocity without unphysical approximations. The fractional term in the Lagrangian provides correct Euler-Lagrange and, ultimately, Hamilton equations with energy dissipation rate defined by $\alpha$.
Smooth, gradual transition from classical mechanics (for example, Molecular Dynamics of point masses) to fluid dynamics (Navier-Stokes) can be realised using this approach.