Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds (Q2812794)

The paper presents a rather formal summary of the physicists' way of proceeding with the rather subtle issue of the calculus of variations for Brownian motion on smooth manifolds. It may be used as a reference to basic equations like, e.g., the Langevin, Fokker-Planck, Smoluchowski, Kramers equations in their covariant form. The main computational ``trick'' is highly questionable from the mathematical point of view (first the authors perform standard variational manipulations, next they replace the force term in the Euler-Lagrange equation by a random force driven by a white noise (Assumption 8). Mathematically oriented readers should consult at this point for example [\textit{E. Nelson}, Quantum fluctuations. Princeton, New Jersey: Princeton University Press (1985; Zbl 0563.60001)], where both the random dynamics on manifolds and principles of the properly defined variational calculus on Riemannian manifolds were neatly formulated. Links with the Malliavin stochastic calculus of variations are exhibited there as well. This also includes Lagrangian and Hamiltonian variational principles for stochastic processes on manifolds. A reference to [\textit{K. D. Elworthy}, Stochastic differential equations on manifolds. Cambridge etc.: Cambridge University Press (1982; Zbl 0514.58001)] is useful as well. There is a multitude of other approaches, concentrating merely on Fokker-Planck and Kramers equations, dating back to the 60's, see for example [\textit{H. Hasegawa}, Prog. Theor. Phys. 58, No. 1, 128--146 (1977; Zbl 1098.82510)].