So
who
is
coming
to
my
party?
Might stay at rest, but if there is a critical mass then
I’ll be forced to attend. –Isaac Newton
Been looking for some action, have a lot of energy.
–William Rowan Hamilton
Party? I’m buoyant! Will bring pi. –Archimedes
Right on! We’re radiating with enthusiasm! –The
Curies.
How special! It should be relatively easy to go, in
general. -Albert Einstein.
I’m electrified about it! -Alessandro Volta
Ok, but might not be up to current fashion. –André
Marie Ampére
I resisted at first, but I think I’ll go. –Georg Ohm.
Can’t, I’m under so much pressure I feel squashed
–Bob Boyle
Should be illuminating. -Thomas Edison
Sure, want to let off some steam. –Jimbo Watt
The party has very high potential. Might bring a few
gadgets, you got wireless? –Nikola Tesla
I’ll think about it. –René Descartes
My brother and I will go, if we can get a flight. –
Wilbur Wright
Be there on the dot. Can’t say more, gotta dash. –
Sam Morse
I’m uncertain if I’ll be there... might be somewhere
else. –Werner Heisenberg
Will miss, but I promise to attend with greater
frequency. –Heinrich Hertz
Your parties are pretty warped! I’ll try to string a
hole in my schedule. –Stephen Hawking
Dunno. Will see what evolves. –Chuck Darwin
Have to take my cat to the vet. Wait.. maybe not.
–Erwin Schrödinger
Potluck? I’ll mix some veggies together and see
what comes out. –Greg Mendel
Just heard, already salivating. –Ivan Pavlov
Absolutely cool! –Lord Kelvin
Things are in flux, but could be induced to go. –
Michael Faraday
Feeling hot, hot, hot! Work it! –Jimmy Joule
Physics 105A
Analytical Mechanics
Small oscillations, many
variables.
Applications of calculus of
variations.
28 June 2016
Manuel Calderón de la Barca Sánchez
Small Oscillations: Many deg. of freedom

A mass m is free to slide on a
frictionless table. It is connected via
a string which passes through a hole
in the table to a mass M that hangs
below the table. Assume that M only
moves up-down only. Assume that
the string always remains taut.
Find the Lagrangian and EOM.
Under what condition does m undergo
circular motion?
What is the frequency of small
oscillations about the circular motion?
28 June 2016
MCBS
r
m
ℓ-r
M
q
Limits:
 If m>> M then
3M g
w 
0
m r0
2
Inertia from the mass dominates. Oscillation has an infinite
period (no oscillation at all).
All time scales are shorter than w: everything moves slowly
compared to the oscillation frequency.
r
g
2
 If M>>m then w  3
m
r0
 Can we make
28 June 2016
w q ?
MCBS
ℓ-r
M
q
Frequency of small oscillations: procedure
 Write down Lagrangian and E-L equations.
 Find the equilibrium point
let
 Set
q=q=0
in appropriate coordinate.
q(t )  q0  d (t )
where q0 is the equilibrium point.
 Expand the equations. You should get a SHO equation
for d.
If q0 =0, simpler procedure: ignore terms in Eq of motion higher
than first order.
28 June 2016
MCBS
For 2m=M, Top view of motion
28 June 2016
MCBS
Calculus of Variations
28 June 2016
Manuel Calderón de la Barca Sánchez
Applications of the Calculus of variations

Catenary: shape of an ideal string hanging from two points.
“ideal”: perferctly flexible, inextensible, no thickness, uniform density.
Surface of Revolution
 Minimal surface:

find the curve from a point
(x1,y1) to a point (x2,y2) which,
when revolved around the x-axis,
yields a surface of
smallest surface area
28 June 2016
MCBS
Minimal Surface of Revolution
y(x2)=y2
y(x1)=y1
r=y(x)
ds 2  dx 2  dy 2
28 June 2016
MCBS
The Brachistochrone

Birth of the calculus of variations
Bracistos: shortest, cronos: time.
 History:

Johann Bernoulli solved the problem first.
Posed it for general readers in Acta
Eruditorium on June 1696.
To determine the curved line joining two given
points situated at different distances from the
horizontal and not on the same vertical along
which a mobile body, running down by its own
weight and starting from the upper point, will
descend most quickly to the lowest point.
Five solutions: Jakob Bernoulli (Johann’s
Brother), Gottfried Leibnitz, von Tscirnhaus,
Guillaume de l’Hopital… and an anonymous
entry...
28 June 2016
MCBS
You can’t handle
my curves…
Especially those
of my hair!
Problem 6.24: The Brachistochrone
 A bead is released from rest at the origin and
slides down a frictionless wire that connects
the origin to the given point.
 You wish to shape the wire so that it reaches
the point in the shortest possible time.
 Let the desired curve be given by y(x), with
axes as in the figure. Show that y(x) satisfies:
B
1+ y' =
y
2
 And that the solution can be written as:
x = a(q - sinq ), y = a(1- cosq )
28 June 2016
Manuel Calderón de la Barca Sánchez
From Acta Eruditorium, May 1697

Figs I, II, III:
Johann Bernoulli.

Figs IV, V, VI, VII, VIII:
Jacob Bernoulli.

Figs: IX, X:
l’Hopital

Figs: XII, XIII:
Anonymous entry…
“We know indubitably that the author is
the celebrated Mr. Newton; and
besides, it were enough to
understand so by this sample, Ex
ungue Leonem.”
“From the paw of the lion.”
28 June 2016
MCBS
Least time, Graphically
ds
v=
dt
T=
28 June 2016
ò
x2 ,y2
x1 ,y1
MCBS
ds
v
More to Explore:
Symmetry in Physical Law
 “The role of symmetry in fundamental physics”
D. Gross, Proceedings of the National Academy of Science,
93, 14256 (1996).
– Classical Mechanics, Quantum Mechanics, Symmetry Breaking,
Gauge Symmetry.
 Want to see a lecture by Feynman?
R. Feynman, The Character of Physical Law, Part 4, Symmetry
in Physical Law
– http://www.youtube.com/watch?v=zQ6o1cDxV7o
28 June 2016
MCBS
From Stationary Action to F=ma

Hamilton’s Principle of Stationary Action is equivalent to Newton’s Laws
Stationary Action
E-L equations
Newton’s 2nd Law F=ma
We showed it in cartesian coordinates
We showed that if it E-L eqs. hold in one coordinate system, they are true in all other
(sensible) coordinates.

Comments:
Deals only with scalars, makes problems easier.
– Recall inclined plane, problem 3.8 and problem 6.1.
d æ ¶L ö ¶L
=0
ç ÷dt è ¶x ø ¶x
Stationary against Local, not global, variations.
Multiple coordinates? Multiple E-L equations.
Better suited for generalizations.
Feynman’s approach to Quantum Mechanics: Based on Hamilton’s Principle.
Maxwell’s Equations: Can be derived applying stationary action to Quantum
Electrodynamics.
28 June 2016
MCBS