Physics 151 - Mechanics - GitHub Pages

Physics 151 - Mechanics

Taught by Arthur Jaffe

Notes by Dongryul Kim

Fall 2016

This course was taught by Arthur Jaffe. We met on Tuesdays and Thursdays

from 11:30am to 1:00pm in Jefferson 356. We did not use a particular textbook, and there were 17 students enrolled. Grading was based on two in-class

midterms and six assignments. The teaching follow was David Kolchmeyer.

Contents

1 September 1, 2016

1.1 Three conservation laws . . . . . . . . . . . . . . . . . . . . . . .

4

4

2 September 6, 2016

2.1 More on elliptical orbits . . . . . . . . . . . . . . . . . . . . . . .

2.2 Kepler¡¯s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Hyperbolic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

6

6

7

3 September 8, 2016

3.1 The Rutherford scattering . . . . . . . . . . . . . . . . . . . . . .

8

8

4 September 13, 2016

4.1 Newton¡¯s equation in coordinates . . . . . . . . . . . . . . . . . .

4.2 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

9

10

10

5 September 15, 2016

12

5.1 Coordinate independence of the Lagrangian . . . . . . . . . . . . 12

6 September 20, 2016

14

6.1 The Lagrangian in fields . . . . . . . . . . . . . . . . . . . . . . . 14

6.2 Lagrange¡¯s equation with constraints . . . . . . . . . . . . . . . . 14

7 September 22, 2016

17

7.1 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . 17

1

Last Update: August 27, 2018

8 September 27, 2016

19

8.1 Least action principle for the oscillator . . . . . . . . . . . . . . . 19

9 September 29, 2016

21

9.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9.2 Symmetry and Noether¡¯s theorem . . . . . . . . . . . . . . . . . . 22

10 October 4, 2016

24

10.1 Legendre transformation . . . . . . . . . . . . . . . . . . . . . . . 24

11 October 13, 2016

26

11.1 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

12 October 18, 2016

27

12.1 Hamilton equations for the oscillator . . . . . . . . . . . . . . . . 27

13 October 20, 2016

29

13.1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

14 October 25, 2016

31

14.1 Examples of canonical transformations . . . . . . . . . . . . . . . 31

15 October 27, 2016

33

15.1 Symmetry in elliptical orbits . . . . . . . . . . . . . . . . . . . . 33

16 November 1, 2016

35

17 November 3, 2016

37

17.1 Lagrange equations for fields . . . . . . . . . . . . . . . . . . . . 37

17.2 Noether¡¯s theorem again . . . . . . . . . . . . . . . . . . . . . . . 38

18 November 8, 2016

39

18.1 The energy-momentum density . . . . . . . . . . . . . . . . . . . 39

19 November 10, 2016

41

19.1 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

20 November 15, 2016

43

20.1 Target space symmetry . . . . . . . . . . . . . . . . . . . . . . . 43

20.2 Topological conservation law . . . . . . . . . . . . . . . . . . . . 44

21 November 17, 2016

45

21.1 Review for the exam¡ªlots of examples . . . . . . . . . . . . . . . 45

22 November 29, 2016

47

22.1 The Feynman formula . . . . . . . . . . . . . . . . . . . . . . . . 47

2

23 December 1, 2016

50

23.1 The Heisenberg equation . . . . . . . . . . . . . . . . . . . . . . . 50

3

Physics 151 Notes

1

4

September 1, 2016

This class is about classical mechanics, but this is actually going to be an introduction to theoretical physics. There is a correspondence between conservation

laws and symmetry, which was discovered by Noether, and this has connections

with mechanics, quantum mechanics, and pure mathematics.

Let us look at a classical problem. Suppose we have a point particle ? at ~r

moving under a central force F~ = ?k/r2 . I¡¯m going to write ~r = r~n where r is

the magnitude and ~n is the unit vector. The question is: What is the orbit? The

standard answer is to use Newton¡¯s Law and solve the corresponding differential

equation. What I am going to do today is to answer the same equation with

conservation laws.

As you all know, the answer is that the orbit is an ellipse. An ellipse is the

locus of points whose sum of the distance from the two foci is constant.

~r0

~r

2a

Figure 1: Figure of an ellipse

The ellipse has a semi-major axis whose length we will denote 2a. The

distance between the two foci will be denoted 2a where  is the eccentricity

of the ellipse. Because we have ~r0 = ~r + 2a~e1 , we have

~r02 = r2 + 4a2 2 + 4ar cos ¦È = (2a ? r)2

and therefore the equation of the ellipse is given by

r(1 +  cos ¦È) = a(1 ? 2 ).

This is where we are heading to.

1.1

Three conservation laws

The first one is energy conservation. The energy E is given by

1 ¨B2 k

?~r ?

2

r

where the dot denotes time differentiation. To show that the energy is conserved,

we differentiate E to get

E=

E? =

dE

= ~r¨B ¡¤ (?~r¡§ ? F~ ) = 0

dt

Physics 151 Notes

5

by Newton¡¯s law.

Item number two is angular momentum around the origin. The angular

~ is given by ~r ¡Á p~ so

momentum L

~

dL

= ~r¨B ¡Á p~ + ~r ¡Á F~ = 0 + 0 = 0.

dt

From this we can conclude that the motion lies on the plane perpendicular to

~

L.

The third one is the highlight of this lecture. Energy and angular momentum

are general conservation laws that come over everywhere, but this is specific to

the 1/r2 potential. We define the Lenz vector ~ as

~ =

~

p~ ¡Á L

? ~n.

?k

This is a dimensionless vector. But why is this conserved? The only things that

change in this formula are p~ and ~n. We know well what the derivative of p~ is.

The derivative of ~n is given as

1

~n¨B = ? 2 ~n ¡Á (~r ¡Á ~r¨B ).

r

If this formula is correct, then

~

F~ ¡Á L

k?

~¨B =

? ~n¨B = ? 2 ~n ¡Á (~r ¡Á ~r¨B ) ? ~n¨B = 0

?k

r ?k

and hence ~ is conserved.

Now once we have this, we immediately get the orbit. The quantity

~ ¡¤ ~r =

L2

? r = r cos ¦È

?k

is conserved. This is precisely the equation of the ellipse. In fact, you even see

why I named it ; the magnitude of the Lenz vector is precisely the eccentricity

of the orbit.

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