  The extended Lagrangian molecular dynamics (XLMD) method provides a useful
framework for reducing the computational cost of a class of molecular dynamics
simulations with constrained latent variables. The XLMD method relaxes the
constraints by introducing a fictitious mass $\varepsilon$ for the latent
variables, solving a set of singularly perturbed ordinary differential
equations. While favorable numerical performance of XLMD has been demonstrated
in several different contexts in the past decade, mathematical analysis of the
method remains scarce. We propose the first error analysis of the XLMD method
in the context of a classical polarizable force field model. While the dynamics
with respect to the atomic degrees of freedom are general and nonlinear, the
key mathematical simplification of the polarizable force field model is that
the constraints on the latent variables are given by a linear system of
equations. We prove that when the initial value of the latent variables is
compatible in a sense that we define, XLMD converges as the fictitious mass
$\varepsilon$ is made small with $\mathcal{O}(\varepsilon)$ error for the
atomic degrees of freedom and with $\mathcal{O}(\sqrt{\varepsilon})$ error for
the latent variables, when the dimension of the latent variable $d'$ is 1.
Furthermore, when the initial value of the latent variables is improved to be
optimally compatible in a certain sense, we prove that the convergence rate can
be improved to $\mathcal{O}(\varepsilon)$ for the latent variables as well.
Numerical results verify that both estimates are sharp not only for $d' =1$,
but also for arbitrary $d'$. In the setting of general $d'$, we do obtain
convergence, but with the non-sharp rate of $\mathcal{O}(\sqrt{\varepsilon})$
for both the atomic and latent variables.
