Physics 105A
Using the Lagrangian and
the Euler-Lagrange Eqs
Forces of Constraint
Small Oscillations
28 June 2016
Manuel Calderón de la Barca Sánchez
Forces of Constraint
 Revisit particle sliding off frictionless hemisphere.
 Solve using Lagrangian, leaving r = R from beginning.
 Solve using Lagrangian and find Normal force
Force of Constraint
Lagrange Multipliers
m
Rq
28 June 2016
MCBS
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
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
Homework 8 …
Homework 8
 A classic problem.


6.19 The Double Pendulum **** .
We did two falling sticks, now you do
6.28 Three falling sticks ***
6.43 Oscillating Hoop **
 Problems to try out
ℓ1
m1
ℓ2
m2
6.24 The Brachistochrone ***
– The problem that started the calculus of variations.
– http://www.math.usma.edu/people/Rickey/hm/CalcNotes/brachistochrone.pdf
6.44 Oscillating Hoop with a pendulum ***
28 June 2016
MCBS
qq
a