Analytical (rational) mechanics is the mathematical structure of Newtonian
deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi,
and many other luminaries of applied mathematics. Diffusion as a stochastic
process of an overdamped individual particle immersed in a fluid, initiated by
Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is
nowhere differentiable. In this exposition, we illustrate how analytical
mechanics arises in stochastic dynamics from a randomly perturbed ordinary
differential equation dXtβ=b(Xtβ)dt+Ο΅dWtβ where Wtβ is a Brownian
motion. In the limit of vanishingly small Ο΅, the solution to the
stochastic differential equation other than xΛ=b(x) are all rare events.
However, conditioned on an occurence of such an event, the most probable
trajectory of the stochastic motion is the solution to Lagrangian mechanics
with L=β₯qΛββb(q)β₯2/4 and Hamiltonian equations with
H(p,q)=β₯pβ₯2+b(q)β p. Hamiltonian conservation law implies that the
most probable trajectory for a "rare" event has a uniform "excess kinetic
energy" along its path. Rare events can also be characterized by the principle
of large deviations which expresses the probability density function for Xtβ
as f(x,t)=eβu(x,t)/Ο΅, where u(x,t) is called a large-deviation
rate function which satisfies the corresponding Hamilton-Jacobi equation. An
irreversible diffusion process with βΓbξ =0 corresponds to a
Newtonian system with a Lorentz force qΒ¨β=(βΓb)ΓqΛβ+1/2ββ₯bβ₯2. The connection between stochastic motion and
analytical mechanics can be explored in terms of various techniques of applied
mathematics, for example, singular perturbations, viscosity solutions, and
integrable systems.Comment: 23 pages, 2 figure