PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics
Action and Symmetry, Ch Symmetry, Noether's theorem and what is conserved
Problem
Three short derivations, each in a system not worked in full elsewhere.
((a)) A charged bead of mass m=0.50 kg slides on a frictionless straight wire under a uniform electric field, which exerts a constant force F0=4.0 N along the wire, so V(x)=−F0x and L=21mx˙2+F0x. Show that the shift x→x+ϵt changes L by ϵdf/dt with f=mx+21F0t2, and hence write down the conserved quantity Q=pt−f. The bead is released from rest at x0=2.0 m at t=0. Evaluate Q at t=0, then confirm that it has the same value at t=3.0 s using the equation of motion.
((b)) Two particles of mass m1=1.0 kg and m2=3.0 kg move along a frictionless line and interact only through a spring joining them, an isolated pair with no external force. At t=0 their positions are x1=0 and x2=1.0 m and their velocities are v1=2.0ms−1 and v2=−1.0ms−1, whatever state the spring is in. Derive the boost constant K=Pt−MXcm for this pair, evaluate it at t=0, and use K together with the (separately conserved) total momentum P to predict the centre-of-mass position Xcm at t=2.0 s, without solving for the motion of either particle.
((c)) A mass m=0.20 kg on a spring of stiffness ks=50Nm−1 oscillates with amplitude A=0.10 m. Starting from the stationarity of the action under the stretch r(t)→(1+ϵ)r(t) applied over one exact period, derive 2⟨T⟩=ν⟨V⟩ for ν=2, then confirm it directly for this oscillator by computing ⟨T⟩ and ⟨V⟩ from x(t)=Asin(ωt).