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

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marked as duplicate by Qmechanic classical-mechanics Aug 17 '18 at 7:40

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  • $\begingroup$ 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. $\endgroup$ – k.kulkarni19952 Jan 1 '18 at 19:37
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    $\begingroup$ 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. $\endgroup$ – Dr. Ikjyot Singh Kohli Jan 1 '18 at 22:32
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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.

You see that this is a requirement which excludes dynamical frictions as constraints. Nontheless, you can use pure rolling as a constraint since in that case you demonstrate it is ideal.

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  • $\begingroup$ I know that the formalisms are based on these assumptions. But I was quite confused how Lagrangian etc. formalisms turned out to be useful, say for engineers. Since it did not address the ubiquitous existence of friction and viscous forces. $\endgroup$ – k.kulkarni19952 Jan 4 '18 at 12:13
  • $\begingroup$ For viscous forces it is possible to define a "generalized force" actually. But Lagrangian formalism is extremely important because for a dynamical system it tells you much more (say in a much more easier way) than newtonian formalism, for what concerns classical mechanics. Then, lagrangian formalism can be used not only for newtonian mechanics. In physics it is everywhere, from classical mechanics to Quantum Field Theory and General Relativity... $\endgroup$ – Bellem Jan 4 '18 at 12:51

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