PHY411. PROBLEM SET 3 1. Conserved Quantities - Astro Pas ...

6 PHY411. PROBLEM SET 3(b) Draw a closed or periodic orbit with φ = θ r −2θ = π and with φ ′ = θ z −2θ = 0.One of these types of orbits is called banana shaped.5. On Creating a Map for a separable Hamiltonian with the Dirac CombSuppose we have a separable HamiltonianThe equations of motionH(p, q) = P (p) + Q(q)ṗ = − ∂H∂q = −∂Q ∂q˙q = ∂H∂p = ∂P∂pOver a small time τ∆p = ṗτ = − ∂Q∂q τWe can approximate the equations of motion using the Hamiltonianwhere D τ is the Dirac comb orH(p, q, t) = P (p) + Q(q)τD τ (t) (4)D τ =∞∑n=−∞Denote p n , q n as p, q at times t = nτ.δ(t + nτ)(a) Create a map for p n+1 and q n+1 as a function of p n , q n that is consistent withthe equations of motion for the approximate Hamiltonian in equation 4.(b) Show that your map is area preserving.Hint: Because H is independent of q between delta functions, p is conservedand only changes at times t = nτ. The Hamiltonian implies that ˙q is constant attimes other than t = nτ.