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The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian Mechanics, this formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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Help finding equations of motion from Hamiltonian with integral of motion

I've given this Hamiltonian with one degree of freedom: $$ H(x,p)=\frac{p^2}{2}+\frac{\omega_0^2x^2}{2}+\lambda\left(\frac{p^2}{2}+\frac{\omega_{0}^2x^2}{2}\right)^2 $$ I need to find the general solu …
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