This article uses physical arguments to derive variational integration schemes for dissipative mechanical systems. These integration algorithms find utility in the solution of the equations of motion and optimal control problems for these systems. Engineers usually represent dissipation effects using phenomenological devices such as "dampers." In this article, we replace these dampers with a lossless transmission-line in order that the equations of motion are derivable from a variational principle. The associated system Lagrangian can then be discretized and used to develop low-order variational integration schemes that inherit the advantageous features of their conservative counterparts. The properties of a lossless spring-inerter based transmission system are analyzed in detail, with the resulting variational integration schemes shown to have excellent numerical properties. The article concludes with the analysis of a dissipative variant of the classical Kepler central force problem.