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7 Jan 2019: L =1. 2. [η̇2 2η̇θ̇ cos θ θ̇2. ]g. cos θ 1. ... 2. k. mη2 ,. up to a multiplicative constant. [10]. (b) Find the equations of motion for this system.

7 Jan 2019: 6]An electromagnetic wave is represented by the 4-vector potential. Aµ = (0, A cos(kz ωt), A sin(kz ωt), 0). ... a) Evaluate the electric and magnetic fields. [6]. (b) Evaluate the Lagrangian density.

7 Jan 2019: x/a = θ sin φ (θ φ) cos φ ,y/a = 1 cos φ (θ φ) sin φ. ... A. (TURN OVER for continuation of question 4. 4. (d) By considering a wave solution of the form ϕ = C cos(kxωt) for auniform rod, find the dispersion relation and the

7 Jan 2019: m. (a, ,b ). z. M. (r, ,z). B B. (a) Show that, up to irrelevant constants and ignoring gravity, the Lagrangian forthe system is: [6]. ... 2k(r2 2ar cos(θ θB) (z bθB)2. ). A. (TURN OVER for continuation of question 1.

7 Jan 2019: E m2gl cos φ. [9]. 2 A harmonic oscillator is weakly perturbed by a cubic potential λx3 so thatits Hamiltonian has the form. ... 19]. (b) Find the temperature T (x, t) of the system if s(x, t) = cos(px)δ(t t0)and T (x, t < t0) = 0, for the two cases

7 Jan 2019: L =1. 2m(. ρ̇2 ρ2θ̇2). 12kρ2 q. (. Eρ cos θ B. ... A =B. 2. (. ρ sin θ, ρ cos θ, 0).

7 Jan 2019: The Lagrangian is. L = T V = m2. (a2ω2 l2θ̇2 2alωθ̇ cos(ωt θ). ) ... y = A cos(ω0t δ) Et. B(16). The path is a helix (free motion in z) that drifts at a rate E/B in the ydirection.

7 Jan 2019: b) The canonical momentum conjugate to x is. px =L. ẋ= (m1 m2)ẋ m2lφ̇ cos φ. ... circular motion around the z axis with constantangular velocity. In terms of the z coordinate, the particle undergoes simpleharmonic motion, z(t) = a sin κt b cos κt

7 Jan 2019: x = η sin θ y = cos θ. Correspondingly, the kinetic energy is given by [3]. ... 1. 2m2. [η̇2 θ̇2 2θ̇η̇ cos θ. ]and the potential energy by [3].

7 Jan 2019: a) The electric field is. E = Ȧ = Aω ( sin(kz ωt), cos(kz ωt), 0). ... The magnetic field is. B = A = Ak ( cos(kz ωt), sin(kz ωt), 0).