Physics 351 — Monday, February 9, 2015

I You read §7.1-7.7 of Chapter 7 for today. (We’ll read the restof Ch7 for next Monday.)

I Remember quiz #2 (a HW2 problem) Wednesday. One sheetof your own handwritten notes is OK.

I Today: we’ll work through some examples of using theLagrangian approach to find EOM.

I It was pointed out to me that solutions to many HW problemscan be found online. One of last year’s students had told meabout this, which is how he and I decided that weekly quizzeswould provide a good incentive for people to do the HWhonestly rather than to copy answers (which by the way ischeating). He also suggested that I retype the problems, editthem where feasible, and use alternative sources of problems.Tanner and I facilitate your working with us or with eachother. We try our best to help you to do the right thing. Didyou get into Penn by copying answers in high school? I don’tlike the Physics 150 (100% exams) grading scheme — do you?

I “I definitely found this chapter to be the most interestingyet.”

I “The most interesting part of today’s reading was how simpleit is to write down the Lagrangian before some of theproblems. It is interesting that such a powerful approach tomechanics is not introduced (or even mentioned!) in earliermechanics courses.”

Lagrangian mechanics: Use the Euler-Lagrange equation to findthe trajectory x(t) for which the “action” S[x] is stationary.

S[x] =∫L(t, x, x) dt =

∫ tf

ti

(T − U) dt

As several of you quoted this weekend:

(1) Write down the K.E. and P.E. and hence the LagrangianL = T − U using any convenient inertial reference frame.

(2) Choose a convenient set of n generalized coordinates qi andsolve for original coords (from step 1) in terms of q1 . . . qn.

(3) Rewrite L in terms of qi and qi.

(4) Write down the n Lagrange equations.

∂L∂qi

=ddt

∂L∂qi

Let’s work through several examples together, starting from somebasic one-variable cases, then becoming more complicated.

How about a block of mass m moving horizontally on a frictionlesstable, under the influence of Hooke’s-Law potential

U =12kx2

so x = 0 when spring is at its equilibrium length.

Try writing down L, then use the Lagrange equations to find EOM.

After writing down L, use Lagrange eqns to write EOM for x(t),i.e. the expression for x.

Reading question: “Does the Lagrangian method still work if onechooses generalized coordinates relative to a non-inertial referenceframe? If so, is there some precaution one needs to take in writingdown the Lagrangian?”

Quoting one of you: “The Lagrangian method still works forgeneralized coordinates in a noninertial frame, so long as the framein which the Lagrangian was originally written is inertial.”

A cart of mass m1 rolls horizontally without friction. The cart’sposition is x1. Inside the cart, a mass m2 is attached to the wall ofthe cart with a spring (constant k). The position of m2 w.r.t. thespring’s relaxed position is x2. So x2 is w.r.t. the cart, not w.r.t.the ground. Write L(t, x1, x1, x2, x2).

By the way, notice that x1 is an “ignorable” (a.k.a. “cyclic”)coordinate, i.e. ∂L/∂x1 = 0. The corresponding conservedquantity is the momentum of the CM, m1x1 +m2(x1 + x2).

I’m unsure about that last statement now — need to think it over.

Consider a pendulum made of a spring with a mass m on the end.The spring is arranged to lie in a straight line (e.g. by wrappingthe spring around a massless rod). The equilibrium length of thespring is `. Let the spring have length `+ x(t), and let its anglew.r.t. vertical be θ(t). Assuming the motion takes place in avertical plane, write Lagrangian and find EOM for x and θ.

Physics 351 — Monday, February 9, 2015

I Remember quiz #2 (a HW2 problem) Wednesday. One sheetof your own handwritten notes is OK.