Variable Doubling in Lagrangian Mechanics

Question

I have been looking into how we handle non-conservative forces in Lagrangian mechanics. They don't form a potential $V$ quite as easily as conservative forces do. I have read in several places that there are approaches that involve doubling the degrees of freedom in order to construct potentials, allowing non-conservative forces to appear in variational problems. However, the references I find don't provide direction as to how this is done.

How can one construct the potentials needed to solve physics problems with variation when non-conservative forces are present?

Answer 1

This is e.g. explained in C.R. Galley, arXiv:1210.2745. For a (possible non-conservative) generalized force $Q_j(q,\dot{q},t)$, the potential is
$$ -Q_j(q_+,\dot{q}_+,t)q^j_-,$$ cf. eq. (10) in my Phys.SE answer here.