# Lagrangian formalism and dissipative systems [duplicate]

Why the central concepts of classical mechanics, viz. Lagrangian and Hamiltonian formalisms cannot address constraint forces like friction and others in dissipative systems?

• The paper by Galley here arxiv.org/abs/1210.2745 gives a natural Lagrangian and Hamiltonian formalism for non-conservative systems and thus claims it has filled a long standing gap in classical mechanics. I'm a little surprised here. How could nobody would have done it atleast a 100 years ago ? This paper in 2013 really amazes me. Jan 1 '18 at 19:37
• Jan 1 '18 at 19:48
• Hamiltonian systems cannot in principle, because, Hamiltonian systems must obey Liouville's theorem, that is, the phase space volume must be conserved. Dissipative systems violate Liouville's theorem. The only way to "make" a dissipative system Hamiltonian is by extending the phase space somehow, by adding more variables, but, then one is left with the problem of whether or not these extra variables are physical. Jan 1 '18 at 22:32

All the kinds of constraints you can have in your, say, lagrangian system must satisfy a relation called ideality of the constraint, which takes into account the so called virtual displacements, indeed it states: $$\sum_i\bar{N}_i\cdot\delta \bar{r}_i=0.$$ This is a statement you use for building your configuration space.