3.1 Variational Calculus and Lagrange Multipliers
17
We discuss the Lagrange multipliers by considering the simple mechanical
model of a falling rock and then constraining the rock’s motion to an inclined
plane. Figure 3.1 illustrates the geometry. We describe the unconstrained motion
of the rock in two dimensions x and y by the Lagrangian L = T − U, where
T = (m/2)( ˙
x
2
+ ˙
y
2
) is the kinetic energy and U = mgy is the potential energy. Constraining the “falling” motion to the inclined plane, given by the equation y = x tan α,
results in the adapted Lagrange function
L = T − U + λ(y − x tan α)
=
1
2
m( ˙
x
2
+ ˙
y
2
) − mgy + λ(y − x tan α)
(3.4)
with the Lagrange multiplier λ, which we treat as an additional dynamic variable.
There is no term with ˙
λ, such that the Euler-Lagrange equations of motion are given
by
0 =
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
=
d
dt
m ˙
x − [−λ tan α] = m ¨
x + λ tan α
0 =
d
dt
∂ L
∂ ˙
y
−
∂ L
∂ y
=
d
dt
m ˙
y − [−mg + λ] = m ¨
y + mg − λ
(3.5)
0 =
∂ L
∂λ
= y − x tan α .
The derivative with respect to λ recovers the constraint and the two first equations
have the constraint built-in with the aid of the Lagrange multiplier λ. In the next
step we eliminate λ from the equations by multiplying the first equation by cos α
and the second by sin α and then adding the two equations. This leads us to 0 =
¨
x cos α + ¨
y sin α + g sin α. Now we use the constraint to eliminate ¨
y = ¨
x tan α and
arrive at
0 =
¨
x
cos α
+ g sin α
or
0 = ¨
s + g sin α
(3.6)
where s = x/ cos α is the distance measured along the inclined plane, which is the
result one can expect from more elementary considerations.
Note that we simply patch the constraint onto the Lagrange function with the
help of a Lagrange multiplier λ and then treat the latter as an additional dynamical
parameter. This “recipe” then incorporates the constraint into the equation of motion
and, at the same time, the Euler-Lagrange equation with respect to λ reproduces the
constraint.
After this refresher to illustrate the use of Lagrange multipliers for treating constraints in variational problems we address the optimization of portfolios.
17
We discuss the Lagrange multipliers by considering the simple mechanical
model of a falling rock and then constraining the rock’s motion to an inclined
plane. Figure 3.1 illustrates the geometry. We describe the unconstrained motion
of the rock in two dimensions x and y by the Lagrangian L = T − U, where
T = (m/2)( ˙
x
2
+ ˙
y
2
) is the kinetic energy and U = mgy is the potential energy. Constraining the “falling” motion to the inclined plane, given by the equation y = x tan α,
results in the adapted Lagrange function
L = T − U + λ(y − x tan α)
=
1
2
m( ˙
x
2
+ ˙
y
2
) − mgy + λ(y − x tan α)
(3.4)
with the Lagrange multiplier λ, which we treat as an additional dynamic variable.
There is no term with ˙
λ, such that the Euler-Lagrange equations of motion are given
by
0 =
d
dt
∂ L
∂ ˙
x
−
∂ L
∂ x
=
d
dt
m ˙
x − [−λ tan α] = m ¨
x + λ tan α
0 =
d
dt
∂ L
∂ ˙
y
−
∂ L
∂ y
=
d
dt
m ˙
y − [−mg + λ] = m ¨
y + mg − λ
(3.5)
0 =
∂ L
∂λ
= y − x tan α .
The derivative with respect to λ recovers the constraint and the two first equations
have the constraint built-in with the aid of the Lagrange multiplier λ. In the next
step we eliminate λ from the equations by multiplying the first equation by cos α
and the second by sin α and then adding the two equations. This leads us to 0 =
¨
x cos α + ¨
y sin α + g sin α. Now we use the constraint to eliminate ¨
y = ¨
x tan α and
arrive at
0 =
¨
x
cos α
+ g sin α
or
0 = ¨
s + g sin α
(3.6)
where s = x/ cos α is the distance measured along the inclined plane, which is the
result one can expect from more elementary considerations.
Note that we simply patch the constraint onto the Lagrange function with the
help of a Lagrange multiplier λ and then treat the latter as an additional dynamical
parameter. This “recipe” then incorporates the constraint into the equation of motion
and, at the same time, the Euler-Lagrange equation with respect to λ reproduces the
constraint.
After this refresher to illustrate the use of Lagrange multipliers for treating constraints in variational problems we address the optimization of portfolios.