A proton (mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch The Hamiltonian, phase space and the variational principle in phase space

Problem

A proton (mass mp=1.673×10−27 kgm_{\rm p} = 1.673\times10^{-27}\,\mathrm{kg}, charge Qe=e=1.602×10−19 CQ_{\rm e} = \mathrm{e} = 1.602\times10^{-19}\,\mathrm{C}) moves in the xyxy-plane in a uniform magnetic field B=0.20 TB = 0.20\,\mathrm{T} along +z+z, described in the gauge A=(0,Bx,0)\mathbf{A} = (0, Bx, 0) with no scalar potential. Its Hamiltonian is H(x,y,px,py)=px2+(py−QeBx)22mp.H(x,y,p_x,p_y) = \frac{p_x^2 + (p_y - Q_{\rm e}Bx)^2}{2m_{\rm p}}. ((a)) Starting from the phase-space action S=∫t1t2[pxx˙+pyy˙−H] dtS = \int_{t_1}^{t_2}\big[p_x\dot x + p_y\dot y - H\big]\,\mathrm{d}t, with x(t)x(t), y(t)y(t), px(t)p_x(t), py(t)p_y(t) varied independently and only δx\delta x, δy\delta y required to vanish at t1t_1 and t2t_2, derive the four Hamilton's equations for this HH. ((b)) Combine them into two second-order equations for x(t)x(t) and y(t)y(t) and show they are the components of mpr¨=Qer˙×Bm_{\rm p}\ddot{\mathbf{r}} = Q_{\rm e}\dot{\mathbf{r}}\times\mathbf{B}. ((c)) Show that these equations describe uniform circular motion at the cyclotron angular frequency ωc=QeB/mp\omega_{\rm c} = Q_{\rm e}B/m_{\rm p}, and evaluate ωc\omega_{\rm c}.

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