The paper presents an algorithm of exit time estimation for Lagrangian systems with linear dissipation and weak noise. The reference domain of operation is associated with the domain of attraction of a stable equilibrium in the noiseless system.

Small additive noise induces small oscillations around the stable point with a very large time until escape. Large deviations theory reduces the mean exit time problem for rare escape to the minimization of the action functional and the solution of the associated Hamilton-Jacobi equation. Under some non-restrictive assumptions, the solution of the Hamilton-Jacobi equation is found in the explicit form, as transformations of kinetic and potential energy of the system.

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