The most
beautiful,
deepest result
in theoretical
physics....
28 June 2016
Manuel Calderón de la Barca Sánchez
Physics 105A
Analytical Mechanics
Noether’s Theorem:
Symmetries and
Conservation Laws.
28 June 2016
Manuel Calderón de la Barca Sánchez
Conserved Quantities
 Before:
Used F=ma and assumed momentum conservation (3d Law)
Proved that Energy is conserved.
 Lagrangian formalism:
Do we need to assume momentum conservation?
Can we prove that Energy is conserved in the Lagrangian
formalism?
28 June 2016
Manuel Calderón de la Barca Sánchez
Symmetry




1: balanced proportions ; also : beauty of
form arising from balanced proportions
2: the property of being symmetrical ;
especially : correspondence in size, shape,
and relative position of parts on opposite
sides of a dividing line or median plane or
about a center or axis — compare bilateral
symmetry , radial symmetry
3: a rigid motion of a geometric figure that
determines a one-to-one mapping onto
itself
4: the property of remaining invariant
under certain changes (as of orientation
in space, of the sign of the electric
charge, of parity, or of the direction of
time flow) —used of physical
phenomena and of equations describing
them
28 June 2016
Manuel Calderón de la Barca Sánchez
Cyclic Coordinates
 Thrown ball in 3-D
1
2
2
2
L = m(x + y + z ) - mgz
2
 3-D Cylindrical Coords: Linear and Angular Momenutm
1
2
2 2
2
L = m(r + r q + z ) -V (r)
2
 Spherical coordinates with V(r,q).
28 June 2016
Manuel Calderón de la Barca Sánchez
Translational and Rotational Invariance
 What if I told you, that if you moved a few meters, you
couldn’t tell the difference?
 Or that if you looked turned by any angle, you couldn’t tell
the difference either?
28 June 2016
Manuel Calderón de la Barca Sánchez
The Hamiltonian: Energy Conservation
 Useful for Lagrangians which don’t explicitly depend on
time.
 Definition of the Hamiltionian:
 Theorem: If the Lagrangian doesn’t depend explicitly on
time, then the Hamiltonian is a conserved quantity.
If
, then
 For systems with no external forces: Hamiltonian is the
total energy!
28 June 2016
Manuel Calderón de la Barca Sánchez
Emmy Noether: Symmetries and
Conservation Laws

In the judgment of the most competent living
mathematicians, Fräulein Noether was the
most significant creative mathematical genius
thus far produced since the higher education
of women began. In the realm of algebra, in
which the most gifted mathematicians have
been busy for centuries, she discovered
methods which have proved of enormous
importance in the development of the
present-day younger generation of
mathematicians.
Albert Einstein, in a letter to New York Times
28 June 2016
Manuel Calderón de la Barca Sánchez
Noether’s Theorem

For each differentiable symmetry of the action, there is a
corresponding conservation law.
Allows a generalization on all constraints of motion.

Examples:
Observation: system looks the same regardless of how it is oriented in
space, i.e. Lagrangian is rotationally symmetric.
Consequence: Angular momentum is conserved.

Doubly Important:
Provides an explicit way to calculate and derive conservation laws.
Allows one to consider general properties of Lagrangians that preserve
given symmetries: insight!!!
28 June 2016
Manuel Calderón de la Barca Sánchez
Noether’s Theorem:

Stewie, old chap! Is your Lagrangian
symmetric?

Pray, differentially, of course!

As your peers say nowadays: Fo shizzle!
So, you down with that?

Symmetric? In what way?
Ah, yes, you mean to say invariant to first
order under the infinitesimal change of
coordinates
qi  qi   Ki (q)
Of course! I always have what is
intellectually necessary. I am NOT one of
those... simpleton, anorexic babies from
the diaper commercials
What the deuce?! That was just what I
needed to bring dozens of kinematical
objects under control. Victory is mine!
Very well! Then the quantity
N
¶L
K i (q)
i=1 ¶qi
P(q,q) º å

is conserved. Can you dig it?
28 June 2016
MCBS
Example: Find the Conserved Momenta
 Lagrangian with linear potentials in x and y…
1
2
2
L = m(5x - 2xy + 2 y ) + C(2x - y)
2
 Mass-Spring in x-y plane
28 June 2016
Manuel Calderón de la Barca Sánchez
Discussion
 In general, the K’s can depend on the coordinates, but
they don’t have to. They can be constants.
 Cyclic coordinates: special case of Noether’s Theorem.
Theorem requires invariance to 1st order.
Cyclic coordinates: invariance to all orders.
 Symmetry: when we change the coordinates (geometry),
and the system (Lagrangian) does not change.
 Conservation Laws
Useful to know if something doesn’t change with time.
– Useful for quantities that depend on coords, and velocities only.
Integrated form of E-L equations.
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 or down. Assume that the
string always remains taut.
Find the Lagrangian.
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
Frequency of small oscillations: procedure
 Write down Lagrangian and E-L equations.
 Find the equilibrium point
q = q = 0 in appropriate coordinate.
 Set q(t) = q + d (t) where q0 is the equilibrium point.
0
let
Taylor expand, with d (t)
 Obtain (and solve) d equation…
 Insert into E-L equation.
q0
If q0 =0, simpler procedure: ignore terms in Eq of motion higher
than first order.
28 June 2016
MCBS
Limits:
 If m>> M then
3M g
w »
»0
m r0
2
Inertia from the mass m dominates.
– Oscillation has an infinite period (no oscillation at all).
All time scales are bigger 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
wr = q
?
MCBS
ℓ-r
M
q
For 2m=M, Top view of motion
28 June 2016
MCBS