Physics 105A
Analytical Mechanics
The Euler-Lagrange Equations
The Principle of Stationary Action
28 June 2016
Manuel Calderón de la Barca Sánchez
Particle’s Motion: Free fall
Drop a ball starting from rest.
 Analyze the motion between
t1=0 sec, t2=1 sec
 Endpoints:

y(t=0) = 0 meters
y(t=1) = -4.9 meters
y(t)
1
0
Infinitely many curves
connecting endpoints
 Each one will yield a different
Action, S.

28 June 2016
MCBS
-4.9
t
Hamilton’s Principle of “Least” Action
Ok, it should really be of Stationary Action…
 The “Action” thingy in Mechanics is another functional:

Take the Lagrangian, L = T – V, and integrate it over time:
S is a functional, i.e. a function of a function.

Classical Mechanics:
– It is typically a minimum, but not necessarily.
28 June 2016
MCBS
Fundamental Lemma of
Calculus of Variations
x=x0(t)+ a b(t)
x=x0(t)
t2
t1
 The first order variation in the action given by a small
change in x0(t), parameterized as a b(t), will vanish iff x0(t)
satisfies the Euler-Lagrange equations.
28 June 2016
MCBS
S:Stationary Point → E-L Equations

The first order variation on the
action S vanishes...
S=
t2
ò L(x, x,t) dt
t1
æ ¶L ¶x ¶L ¶x ö
¶S
= òç
+
÷ dt = 0
¶a t1 è ¶x ¶a ¶x ¶a ø
t2

…if and only if the E-L
equations are satisfied.
28 June 2016
d æ ¶L ö ¶L
ç ÷- = 0
dt è ¶x ø ¶x
MCBS
6.2 Two Falling Sticks **

Two massless sticks of length 2r,
each with a mass m fixed at its
middle, are hinged at an end. One
stands on top of the other. The
bottom end of the lower stick is
hinged on the ground. They are held
such that the lower stick is vertical
and the upper stick is tilted at a
small angle e with respect to the
vertical. They are then released.
Find the Lagrangian of the system
For small angles, find the equations of
motion and determine the angular
acceleration of the sticks at the instant
they are released.
28 June 2016
MCBS
e
r
m
r
r
m
r
Principle of Least Action in QM

Particle takes ALL possible paths.
Sum amplitude for each path: Feynman Path Integral

Quantum Mechanical amplitude for each path:

Stationary action: phases add constructively.
28 June 2016
MCBS
Adding complex amplitudes…
 Illustration of a “path
integral”.
 Each path has a phase
(angle of the arrow)
 Paths with similar phase
add constructively.
 Similar phase: change in
path causes little change
in phase
28 June 2016
MCBS
QED Lagrangian

y, y : electrons, positrons

Dirac (bi)spinor, for spin ½ particles
(fermions).
† 0 :Dirac adjoint


Dm = ¶m + ieAm +ieBm
Bm : External EM field.
 Fmn = ¶m An -¶n Am

e: coupling constant, electric
charge of bispinor field
Am : Covariant four-vector potential
of EM field generated by the
electron itself.
28 June 2016
E = ¶A / ¶t - Ñf
B = Ñ´ A
y =y g
gauge covariant derivative

Am = (f / c, A)
MCBS
EM Field tensor