Classical Mechanics

There are different formulations of classical mechanics. There is Newtonian mechanics, Lagrangian mechanics, and Hamiltonian mechanics.

If we use s (position along a trajectory) as the independent variable, the Hamiltonian for motion in an electromagnetic field is

(1)
\begin{align} H_s (x,p_x,y,p_y,t,-E;s) = -eA_s -\left(1+\frac{x}{\rho}\right) P_s \end{align}

with

(2)
\begin{align} P_s = \left[ \left(\frac{E-e\phi}{c}\right)^2 - (p_x-eA_x)^2 - (p_y-eA_y)^2-m^2c^2\right]^{1/2} \end{align}
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