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

Abstract

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{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) \textbf{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.

latex2e, 11pt, 25 pages, v3 substantially modified with a changed title and new co-author - accepted to Can. J. Phys

References in corpus (5)

• Finsler and Lagrange Geometries in Einstein and String Gravity
• Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
• Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in $R^2$ and Einstein Gravity Theories
• Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids
• Covariant Renormalizable Anisotropic Theories and Off-Diagonal Einstein-Yang-Mills-Higgs Solutions