Dynamical Equations and Lagrange--Ricci Flow Evolution on Prolongation Lie Algebroids

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. $\mathbf{\text{49}}$ (2008) 043504 & Rep. Math. Phys. $\mathbf{\text{63}}$ (2009) 95] is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometric mechanics and analogous gravity can be modelled as gradient flows characterized by generalized Perelman functionals if an equivalent geometrization of Lagrange mechanics [J. Kern, Arch. Math. (Basel) $\mathbf{\text{25}}$ (1974) 438] is considered. The R. Hamilton equations on Lie algebroids describing Lagrange-Ricci flows are derived. Finally, we show that geometric evolution models on Lie algebroids are described by effective thermodynamical values derived from statistical functionals on prolongation Lie algebroids.