**Chapter 6
**

**Some Methods in
the Calculus of Variations
**

__Euler’s
Equation__

Consider
the two-dimensional plane shown in Figure 1. Our initial position is specified by (*x*_{1},*y*_{1}) and the final position is specified by (*x*_{2},*y*_{2}).

Figure 1. Path *y*(*x*)
that is used to move from position 1 to position 2.

Consider that we are asked to
minimize the path integral of a function *f* between position 1 and position 2. Suppose the path integral is minimized when the use path *y*(*x*). If we change the path slightly by
adding a second function *n*(*x*) then we expect that the path integral is expected
to increase. Consider the
following path:

The function *h*(*x*) is a
arbitrary function of *x* and is
used to make small changes to the path. The only requirements of *h*(*x*) are that *h*(*x*) has a continuous first derivative and that *h*(*x*) vanishes at the end points of the path, that is *h*(*x*_{1}) = *h*(*x*_{2}) = 0.

Since
the path integral is minimized when we follow the path, the path integral must
have an extreme value when *a* =
0. This requires that

The left hand side of this equation
can be rewritten by differentiating the argument of the integral with respect
to *a*:

Using our definition of *y*(*a*, *x*) we can easily show that

and

The requirement that the path integral has an extreme is equivalent to requiring that

The second term in the integrant can be rewritten as

This path integral is an extreme if

Since *h*(*x*) is an arbitrary function, this equation can only
satisfied if

This equation is known as **Euler’s
equation**.

__Example:
Problem 6.4__

Show that the geodesic on the surface of a right circular cylinder is a segment of a helix.

The
element of distance along the surface of a cylinder is

(6.4.1)

In cylindrical coordinates (*x*, *y*, *z*) are related to (*r*, *f*, *z*) by

(6.4.2)

Since
we consider motion on the surface of a cylinder, the radius *r* is constant. The expression for *x*, *y*, and *z* can be used to express *dx*, *dy*, and *dz* in cylindrical coordinates:

(6.4.3)

Substituting (6.4.3) into
(6.4.1) and integrating along the entire path, we find

(6.4.4)

If *S* is to be a minimum,
must satisfy the
Euler equation:

(6.4.5)

Since
, the Euler equation becomes

(6.4.6)

This condition will be
satisfied if

= constant º C (6.4.7)

or,

(6.4.8)

Since *r* is constant, (6.4.8) implies that

and for any point along the path, *z* and *f* change at the same
rate. The curve described by this
condition is a *helix*.

__Example:
Problem 6.7__

Consider
light passing from one medium with index of refraction *n*1 into another medium with index of refraction *n*2 (see Figure x). Using Fermat’s principle to minimize time, and derive the
law of refraction: *n*_{1} sin *q*_{1} = *n*_{2} sin *q*_{2}.

Figure x. Problem 6.7

The
time to travel the path shown in Figure x is

(6.7.1)

The velocity *v* = *v*_{1} when *y* > 0 and *v* = *v*_{2} when *y* < 0. The velocity is thus a function of *y*, *v* = *v*(*y*),
and *dv*/*dy *= 0 for all values of *y*, except for *y* = 0. The function *f* is given by

(6.7.2)

The Euler equation tells us

(6.7.3)

Now use *v* = *c*/*n* and *y*¢ = -tan*q* to obtain

(6.7.4)

This proves the assertion.

** Second
Form of Euler’s Equation
**

In
some applications, the function *f* may
not depend explicitly on *x*: *¶**f/**¶**x* = 0. In order to benefit from this
constraint, it would be good to try to rewrite Euler’s equation with a term *¶**f/**¶**x* instead of a
term *¶**f/**¶**y*.

The
first step in this process is to examine *df*/*dx*. The
general expression for *df*/*dx* is

Note that we have not assume that *¶**f/**¶**x* = 0 in order to derive this expression for *df*/*dx*. This equation can be rewritten as

We also know that

Applying Euler’s theorem to the term in the parenthesis on the right hand side, we can rewrite this equation as

or

This equation is called **the second form of Euler’s
equation**. If *f* does not depend explicitly on *x* we conclude that

__Example:
Problem 6.4 – Part II__

Show that the geodesic on the surface of a right circular cylinder is a segment of a helix.

We
have already solved this problem using the “normal” form of Euler’s
equation. However, looking back at
the solution we realize that the expression for *f, *
, does not depend explicitly on *f*. We thus should be able to use the second form of Euler’s
equation to solve this problem.

This equation requires that

Since *r* is constant, this equation implies that

and for any point along the path, *z* and *f* change at the same
rate. The curve described by this
condition is a *helix*.

__Euler’s
Equation with Several Dependent Variables__

Consider
a situation where the function *f* depends
on several dependent variables *y*_{1}, *y*_{2}, *y*_{3}, … etc., each of which depends on the
independent variable *x*. Each dependent variable *y*_{i}(*a**,
x)* is related to the solution *y*_{i}(0*, x)* in the following manner:

If the independent functions *y*_{1}, *y*_{2}, *y*_{3}, etc. minimize the
path integral of *f*, they must
satisfy the following condition:

for *i* = 1, 2, 3,
…. The procedure to find the
optimum paths is similar to the procedures we have discussed already, except
that we need to solve the Euler equation for each dependent variable *y*_{i}.

__Euler’s
Equation with Boundary Conditions__

In
many cases, the dependent variable *y* must satisfy certain boundary conditions. For example, in problem 6.4 the function *y* must be located on the surface of the cylinder. In this case, any point on *y* must satisfy the following condition:

In general, we can specify the constraint on the path(s) by
using one or more functions *g* and
requiring that *g*{*y*_{i}; *x*} = 0.

Let’s
start with the case where we have two dependent variables *y* and *z*. In this case, we can write the function *f* as

In this case, we can write the differential of *J* with respect to *a* as

Using our definition of *y*_{i}(*a**,
x*) we can rewrite this relation as

For our choice of constraint we can immediately see that the
derivative of *g* must be zero:

Using our definition of *y* and *z* we can rewrite
this equation as

This equation shows us a general relation between the
functions *h*_{x} and *h*_{y}:

Using this relation we can rewrite our expression for the
differential of *J*:

Since the function *h*_{x} is an arbitrary function, the equation can only evaluate to 0 if the term in
the brackets is equal to 0:

This equation can be rewritten as

This equation can only be correct if both sides are equal to
a function that depends only on *x*:

or

Note that in this case, where we have one auxiliary
condition, *g*{*y*_{, }*z*; *x*} = 0, we end up with
one Lagrange undetermined multiplier *l*(*x*). Since
we have three equations and three unknown, *y*, *z*,
and *l*, we can determine
the unknown.

In
certain problems the constraint can only be written in integral form. For example, the constraint for
problems dealing with ropes will be that the total length of the path is equal
to the length of the rope *L*:

The curve *y* then must satisfy the following differential
equation:

__Problem
6.12
__

Repeat example 6.4, finding the shortest path between any two points on the surface of a sphere, but use the method of the Euler equation with an auxiliary condition imposed.

The path length is given by

(6.12.1)

and our equation of constraint is

(6.12.2)

The Euler equations with undetermined multipliers (6.69) tell us that

(6.12.3)

with a similar equation for *z*. Eliminating the factor *l*, we obtain

(6.12.4)

This simplifies to

(6.12.5)

(6.12.6)

and using the derivative of (2),

(6.12.7)

This looks to be in the simplest form we can make it, but is it a plane? Take the equation of a plane passing through the origin:

(8)

and make it a differential equation by taking derivatives
(giving *A* + *By*¢ = *z*¢ and *By*² = *z*²)
and eliminating the constants. The
substitution yields (7) exactly. This confirms that the path must be the intersection of the sphere with
a plane passing through the origin, as required.

__
__

__
__

__Example:
Problem 6.4 – Part III__

Show that the geodesic on the surface of a right circular cylinder is a segment of a helix.

We
have already solved this problem using the “normal” form of Euler’s
equation. However, this problem is
a good example of how to approach problems with constraints. **Note: doing it in this way is NOT
easier then the approaches we have used previously.
**

** **Let us consider two points on the surface of the
cylinder:

and

Consider an arbitrary path connecting the initial and final position. The length of a tiny segment of this path is

The integral we want to minimize is

We immediately see that *z* is our independent variable and that *x*, and *y* are our dependent variables. The
function *f* is thus given by

The solution *y* is constrained to be on the surface of the cylinder
and therefore, the following equation of constraint needs to be applied:

For this equation of constraint we know that

and

Note: if we had picked our function of constraint to be

we would get more complicated
partial derivatives of *g*.

To solve the current problem we thus need to solve the following Euler equations:

These two equations can be rewritten as

Eliminating *l* we find that

This equation can be rewritten as

After simplifying this equation we obtain

Is this equation describing a helix? Yes it is! How do you see that? Let's look at the definition of a helix:

We find that

and

Taking these relations and substituting them in the solution we obtained we find:

__The __**d**__ notation__

It
is common to use the *d* notation
in the calculus of variations. In
order to use the *d* notation we
use the following definitions:

In terms of these variables we find

Since *d**y* is an arbitrary function, the requirement that *d**J* = 0 requires that the term in the parenthesis is 0. This of course is the Euler equation we have encountered
before!

It
is important to not that there is a significant difference between *d**y* and *dy*. Based on the definition of *d**y* we see that *d**y* tells us how *y* varies when we
change *a* while keeping
all other variables fixed (including for example the time *t*).

In this Chapter we focus on an important method of solving certain problems in Classical Mechanics.

In many problems we need to determine how a system evolves between an initial state and a final state. For example, consider two locations on a two-dimensional plane. Consider an object moving from one location to the other, and assume the object experiences a friction force when it moves over the surface. One can ask questions such as “What is the path between the initial and the final condition that minimizes the work done by the friction force?” that can be most easily answered using the calculus of variations. The evolution that can be studied using the calculus of variations is not limited to evolutions in real space. For example, one can consider the evolution of a gas in a pV diagram and ask “What is the path between the initial and final state that maximizes the work by the gas?”