Physics 351 — Monday, April 13, 2015

I I forgot to point out Friday that median reported HW11 timewas 6.5 hours, which seems like a good balance for theincreased reading.

I For today, you read/skimmed Chapters 1+2 of QuantumMechanics and Path Integrals by Feynman and Hibbs. Thepoint was just to show you the deep connection between QMand the Lagrangian formulation of classical mechanics.

I For Wednesday: read (mostly skim) Taylor’s Chapter 12(“Nonlinear mechanics and chaos”). HW13 will containseveral optional/XC Mathematica-based problems from Ch12.

I Today: keep working through Hamiltonian examples.

Morin 15.28. Two beads of mass m are connected by a spring(with spring constant k and relaxed length `) and are free to movealong a frictionless horizontal wire. Let their positions be x1 andx2. Find H in terms of x1 and x2 and their conjugate momenta,then write down the four Hamilton’s equations.

Morin 15.8. Two beads of mass m are connected by a spring (withspring constant k and relaxed length `) and are free to move alonga frictionless horizontal wire. Let the position of the left bead bex, and let z be the stretch of the spring (w.r.t. equilibrium). FindH in terms of x and z and their conjugate momenta, then writedown the four Hamilton’s equations.

What do you expect the general solution to the motion to looklike? (It’s similar to last week’s HW problem 2.)

Taylor 13.13. Consider a particle of mass m constrained to moveon a frictionless cylinder or radius R, given by the equation ρ = Rin (ρ, φ, z) coords. The mass is subject to force F = −krr̂, wherek is a positive constant, r is distance from the origin, and r̂ pointsaway from the origin. Using z and φ as generalized coordinates,find H, write down Hamilton’s equations, and describe the motion.

Here’s a familiar problem from HW6 and from the midterm. Let’swork through it it using Hamilton’s equations instead.

Morin 15.7. Consider the Atwood machine shownin the figure. Let x and y be the vertical positionsof the middle mass and right mass, respectively,with upward taken to be positive. Find H interms of x and y and their conjugate momenta,then write down the four Hamilton’s equations.

(The solution to this problem turns out to be not especiallyilluminating. But it does illustrate how tedious the Hamiltonianmethod can be for solving problems that are straightforward usingthe Lagrangian method.)

Morin writes: “If you want to demonstrate how the Hamiltonianmethod can be monumentally more cumbersome than theLagrangian method, you can try to solve this problem in the caseof three general masses, m1, m2, m3.” ♣

Physics 351 — Monday, April 13, 2015

I I forgot to point out Friday that median reported HW11 timewas 6.5 hours, which seems like a good balance for theincreased reading.

I For today, you read/skimmed Chapters 1+2 of QuantumMechanics and Path Integrals by Feynman and Hibbs. Thepoint was just to show you the deep connection between QMand the Lagrangian formulation of classical mechanics.

I For Wednesday: read (mostly skim) Taylor’s Chapter 12(“Nonlinear mechanics and chaos”). HW13 will containseveral optional/XC Mathematica-based problems from Ch12.

I Today: keep working through Hamiltonian examples.