We study the problem of sampling from a probability distribution $\pi $ on $\mathbb{R}^{d}$ which has density w.r.t. Lebesgue measure known up to normalization factor $x\mapsto \mathrm{e}^{-U(x)}/\int _{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. analyze method based Euler discretization Langevin stochastic differential equations under assumptions that potential $U$ is continuously differe...

The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in orbital mechanics. This methodology assumes that certain "constants" associated with a homogeneous problem will vary in time in response to an external force. It also introduces one or more constraint equations. We show ...

A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. Our main result establishes that both the Euler-...

In this Comment, we point out that the Euler-Lagrange equations, which are referred to as the general equilibrium shape equations presented by Zhang et al. [Phys. Rev. E 70, 051902 (2004)] are incorrect, along with equations derived from them. The correct equations are provided here and they are cross-checked using certain energy functions previously presented in the literature. Further, with t...