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I was going through the last chapter on "Mechanics" by Lev Landau and Evgeny Lifshitz. There, in the definition of Routhian, it has not considered the case if a system consists of dissipative term.

So my confusion is while writing Routhian for such systems, which definition of Lagrangian be considered?

Any answer/comments are welcome and thanks for that.

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  1. Since the Routhian (Hamiltonian) is a partial (full) Legendre transformation of the Lagrangian, respectively, the issues with a Hamiltonian formulation for non-conservative/dissipative systems (as e.g. explained in my Phys.SE answer here) also applies to a Routhian formulation.

  2. Recall that the main advantage of the Routhian formulation is that it Legendre transforms cyclic position coordinates only in order to make the corresponding momenta constants of motion. However, the latter does not hold for non-conservative/dissipative systems, which basically renders the Routhian construction moot.

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