Physics 351 | Wednesday, January 14, 2015xray.hep.upenn.edu/wja/p351/2015/files/p351_notes_20150114.pdfJan 14, 2015  · Physics 351 | Wednesday, January 14, 2015 I Read Chapter 1

Physics 351 — Wednesday, January 14, 2015

I Read Chapter 1 for today.

I Two-thirds of you answered the Chapter 1 questions so far.

I Read Chapters 2+3 for Friday.

I Skim Chapter 4 for next Wednesday (1/21).

I Homework #1 due next Friday (1/23). (Hand out now.)

I Today: course overview + Chapter 1.

(We only got through about two-thirds of this material today, sosome of it will move to Friday.)

What is “analytical mechanics?”

I Course catalog: “An intermediate course in the statics anddynamics of particles and rigid bodies. Lagrangian dynamics,central forces, non-inertial reference frames, and rigid bodies.”

I We’ll review and extend the mechanics that you studied infreshman physics.

I The course re-visits mechanics in a way that takes advantageof some of the math you’ve learned since you started college.Brush up on familiar physics to gain a deeper understanding.

I Most importantly, “analytical mechanics” will give you anextended toolkit for systematically taking on morecomplicated mechanics problems.

I In particular, the Lagrangian formalism provides a systematicapproach that simplifies many tricky mechanics problems.

I The Lagrangian and (closely related) Hamiltonian approachesalso form a bridge between classical mechanics and quantummechanics.

What is “analytical mechanics?”

I Let’s look briefly at a really simple freshman physics problem.

I Then let’s look at a trickier freshman physics problem.I Then I’ll show you how much more quickly we can write down

the answer to the trickier problem, once we’ve studiedLagrangian mechanics (in the beginning of February).

I This is just to give you some hint of where we’re headed.

By the way, I chose this room in part because it makes it easy touse the projector and the blackboard side-by-side. I’ll try to useslides to display information and the board to go step-by-step. As Iget to know you, I’ll try to adapt to what works best for you.

Simple freshman physics problem: a block of mass m slides down africtionless ramp. Find the equation of motion of the block.

You write down all forces acting ON the block. You imposerelevant constraints (i.e. the forces perpendicular to the wedgesum to zero). Then you use

∑ ~F = m~a to write down theequation of motion.

Write down (or draw) the forces acting on the block: (1) gravitypoints downward; (2) the “normal force” points perpendicular tothe surface. In most freshman physics problems, we’d includefriction as well, but let’s consider the frictionless case.

In this case, it’s obvious which way the acceleration ~a will point.

Since we know the motion will be parallel to the surface, it makessense to “decompose” gravity’s m~g into “normal” and “downhill”components. I never once drew one of these diagrams until thefirst time I taught a Physics-101-like course. Have you?

If you take “x” to point in the downhill direction, then Newton’ssecond law gives you x =

∑Fx = g sin θ.

This is not what “analytical mechanics” is about!

Here is a trickier freshman physics problem, which was a favorite ofmy first physics teacher, Mr. Rodriguez. Let’s first solve it usingfamiliar methods. Block of mass mb slides down face of frictionlesswedge of mass mw. The wedge sits on a frictionless horizontaltable. Find the EOM for wedge and block. (Measure xb in Earthframe.) Here, a force diagram may be helpful — draw on board.

EOM (on board): xw, xb, yb; then constraint.

Shall I do the algebra on the board, or just show you the answer?First convince yourself that this is somewhat tedious!

Now that we have xw we can work out what I’m calling s(downhill acceleration, in rest frame of wedge), to compare withthe much simpler mw =∞ case. (On board, if time allows.)

The Lagrangian approach (Chapter 7) starts by simply writingdown the “Lagrangian” L = T − U in terms of s and xw. So webypass forces and just write down energies. (Let’s try it.)

So the “Lagrangian” (L = T − U) for this system is

L = T −U =12(mb +mw)x2

w +12mb(s2 +2sxw cos θ)+mbgs sin θ

The recipe we’ll learn (and derive!) in Chapters 6+7 is:

I First write down L = T − U , which is generally much easierthan writing down the corresponding forces, because it oftenrequires much less thinking to write down the energies.

L = T−U =12(mb+mw)x2

w+12mb(s2+2sxw cos θ)+mbgs sin θ

I Write down an EOM for each coordinate (s and xw in thiscase), using this “magic incanation” (which we’ll derive).

∂L∂s

=ddt

∂L∂s

∂L∂xw

=ddt

∂L∂xw

I This should seem completely mysterious to you at this point.I just want to give you a preview of things to come.

Let’s try it anyway!

So the Lagrangian formalism allows us to find the exact sameEOM as we get using familiar Newtonian methods, but . . .

I Requiring roughly half as much tedious algebra.

I Allowing us to bypass completely any mention of the “normalforce” that constrains the block to stay on the wedge.

I Without requiring us to think too hard about the fact thatcoordinate s is measured w.r.t. an accelerating (non-inertial)frame of reference. We just used the definition of s to writethe energies w.r.t. an inertial frame.

If you’re not yet convinced, note that the simplifications, w.r.t. theforce method, are far greater when working in polar coordinates.

Do you want to try the “trivial” version of the freshman-physicsproblem with the Lagrangian formalism?

1. Choose coordinate: s pointsdownhill.

2. Write down T in terms of s and s.

3. Write down U in terms of s and s.

4. Write downL = T − U .

5. Equation of motion isgiven by

∂L∂s

=ddt

∂L∂s

The “magic incantation” is called the Euler-Lagrange equation.We’ll meet it officially in Chapters 6 and 7.

OK, now let’s look through the list of topics we’ll cover this term,and then the course policies, etc.

positron.hep.upenn.edu/p351#Schedule

positron.hep.upenn.edu/p351

From Richard Feynman (Feynman Lectures):

From Mary Boas (Mathematical Methods):

Two representative responses to my “what did you finddifficult/interesting” question for today:

I “Since I took 150 (not 170), Newton’s laws in polarcoordinates are new to me, but they were not difficult tounderstand. I didn’t expect to encounter anything I didntknow in Chapter 1.”

I “I found the derivation of the polar coordinate form ofNewton’s second law very interesting, even though most of it Iknew from phys 150. The textbook was very clear and I amlooking forward to reading more from this book.”

So let’s go through Newton’s second law in 2D polar coordinates.(On the board, but I copied my notes into these slides.)

Constant r case (familiar):

a = r = −ω2rr + αrφ

Constant φ case (line through origin):

a = r = rr

Physics 351 — Wednesday, January 14, 2015

I Read Chapter 1 if you haven’t already.

I Two-thirds of you answered the Chapter 1 questions so far.

I Read Chapters 2+3 for Friday.

I Skim Chapter 4 for next Wednesday (1/21). Slow down forany parts that feel less familiar (e.g. 4.5, 4.7, 4.8).

I Remember online questions for each reading assignment.

I Homework #1 due next Friday (1/23). (Printed handout, alsoavailable online.)

I Did I remember to poll the class about Wednesday &Thursday study-session times?

I Is anyone interested in spending the first 3 minutes of Friday’sclass on a fun demonstration of the Coriolis effect?