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## Main Question or Discussion Point

My question is related to the chapter 10 of L.Sussknd's book on analytical mechanics named "The theoretical minimum". There he considers dynamics of a charged rotor in magnetic field using Hamiltonian formalism and poisson brackets.

He also introduces a treatment of the same kind for a gyroscope starting at 1:11:35 at his online lecture at http://theoreticalminimum.com/courses/classical-mechanics/2011/fall/lecture-8

In both cases, first relations for poison brackets of angualar momentum for a freely rotating body are introduced, then the very same relations are used for a hamiltonian in which coupling energy of magnetic field (first case) or potential energy in gravitational field (second case) is added. Everything looks nice and simple...

Yet as the hamiltonian formalism is originally defined, a (conjugate) momentum is a derivative of lagrangian by the velocity, which means that as soon as we introduce any potential energy (change the lagrangian), the angular momentum is no more the same one which we defined for the freely rotating body, so that we cannot just blindly reuse the formulas which we got for a case of freely rotating body. We cannot just add the coupling energy to the hamiltonian when we add the field, we need rather to reevaluate this hamilton with adjusted conjugate momentum.

So my questiin is how can the approach of Susskind be justified (it provably can). I.e. the approach in which after adding a potential field, we do not change the momentum definition at all, but are using hamilton formalism.

He also introduces a treatment of the same kind for a gyroscope starting at 1:11:35 at his online lecture at http://theoreticalminimum.com/courses/classical-mechanics/2011/fall/lecture-8

In both cases, first relations for poison brackets of angualar momentum for a freely rotating body are introduced, then the very same relations are used for a hamiltonian in which coupling energy of magnetic field (first case) or potential energy in gravitational field (second case) is added. Everything looks nice and simple...

Yet as the hamiltonian formalism is originally defined, a (conjugate) momentum is a derivative of lagrangian by the velocity, which means that as soon as we introduce any potential energy (change the lagrangian), the angular momentum is no more the same one which we defined for the freely rotating body, so that we cannot just blindly reuse the formulas which we got for a case of freely rotating body. We cannot just add the coupling energy to the hamiltonian when we add the field, we need rather to reevaluate this hamilton with adjusted conjugate momentum.

So my questiin is how can the approach of Susskind be justified (it provably can). I.e. the approach in which after adding a potential field, we do not change the momentum definition at all, but are using hamilton formalism.