Physics 105A
28 June 2016
Manuel Calderón de la Barca Sánchez
Giuseppe Lodovico Lagrangia/
Joseph Louis Lagrange


One of the creators of the calculus of variations
Motion subject to constraints:
Lagrange Multipliers

Method of “variation of parameters”
Used for solving differential equations

Theorie des fonctions analytiques
Some of the foundations of group theory

Gravitational three-body problem (Earth, Sun, Moon)
Special case solutions, “Lagrange points”

Transformed mechanics into a branch of “analysis” (calculus).
Lagrangian mechanics: Consequence of Hamilton’s Principle of “Least”
Action. (or rather: Principle of Stationary Action).
28 June 2016
MCBS
Lagrange Approach

Write down the Lagrangian of the system:
L=T–V

Write down the Euler-Lagrange Equation:
E-L Equation:

d æ ¶L ö ¶L
ç ÷- = 0
dt è ¶x ø ¶x
If there is more than one coordinate, apply E-L equation to
each coordinate.
They do NOT have to be cartesian: generalized coordinates.
28 June 2016
MCBS
Example: Spring-Pendulum
Consider a pendulum made of a spring
with mass m attached at its end.
 The spring is arranged to lie in a straight
line from pivot to mass (say, by wrapping
the spring around a frictionless, rigid,
massless rod).
 The equilibrium length of the spring is ℓ.
Let the spring have length ℓ + x(t). Let its
angle with the vertical be q(t). Assume the
motion is only in a vertical plane.
 Find the equations of motion for x and q.

28 June 2016
MCBS
q
m
Solution:
mx = m( + x)q + mg cosq - kx
2
m( + x)q = -2mq x - mg sin q

For x equation (radial part of motion):
Centripetal acceleration: ~mv2tangential/r
Weight along radial direction: ~mgcos(q)
Spring force along radial direction: ~kx

For q equation (tangential part of motion):
Angular momentum : m(ℓ+x)2dq/dt
Torque: r x F ~ |r| |F| sin(q) ~ (ℓ+x) mg sin (q)
– Rate of change of angular momentum = Torque
Automatically includes Coriolis Force
– ~ 2m(dq/dt)(dx/dt) or 2m (w x v)
Tangential component of weight ~ mg sin(q)
28 June 2016
MCBS
Functionals and Calculus of Variations
 In the usual calculus, for a number x, get a number f:
f(x) means the number f depends on the number x.
 In the Calculus of Variations, for a function x(t), get a
number S.
S[x(t)] means the number S depends on the function x(t).
28 June 2016
MCBS
Functionals
 Area under the curve: A =
x2
ò f (x)dx
x1
 Surface area of revolution:
28 June 2016
MCBS
General problem of the Calculus of
Variations
 The million dollar question, the general problem of the
Calculus of Variations:
Consider functions where the endpoints are restricted to have
the same value.
Can you find a function x(t) that makes the value S be a local
minimum, maximum or saddle point?
28 June 2016
MCBS
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