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Lecture 21
Spherical polar coordinates (cont’d)
(Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 15.8)
In the previous lecture, we introduced the spherical polar coordinate system. We must now
determine the infinitesimal volume element, dV , generated by infinitesimal increments of r, θ, φ at a
point (r, θ, φ) in R3 .
Consider the tiny volume ∆V
obtained by revolving the shaded
∆V
area ∆A about the z-axis by angle
r sin φ
∆θ.
∆θ
∆A
r
∆φ
Recall, from polar coordinates,
that
∆A ≃ r ∆r ∆φ
∆A is rotated about the z-axis with an effective radius of r sin φ, NOT simply r, so that
∆V
= ∆A(r sin φ) ∆θ
= r ∆r ∆φ r sin φ ∆θ
= r 2 sin φ ∆r ∆θ ∆φ.
Therefore, in the infinitesimal limit
dV = r 2 sin φ dr dθ dφ.
(1)
Suppose that we wished to integrate a function f over a sphere of radius R > 0. The limits of
integration would then be
0≤r≤R
0 ≤ θ ≤ 2π
0≤φ≤π
156
The final integral would then have the form,
Z Z Z
Z
f dV =
D
Z
π
0
2π
0
Z
R
f (r, θ, φ) r 2 sin φ dr dθ dφ
0
or
Z
0
R Z 2π
0
Z
π
f (r, θ, φ) r 2 sin φ dφ dθ dr
0
or
etc...
Special Case:
Z Z Z
Let f (r, θ, φ) = 1, so that
dV = V (D), the volume of the sphere. Then
D
V =
Z Z Z
dV =
D
Z
Z
π
2π
R
r 2 sin φ dr dθ dφ.
(2)
0
0
0
Z
Because the integrand is a simple product of r and φ-dependent functions, and the integration limits
are constants, this integral is separable:
V
=
Z
Z
π
sin φ dφ
= (2)(2π)
4 3
πR ,
3
dθ
0
0
=
2π
1 3
R
3
"Z
R
2
r dr
0
#
(3)
the correct result for the volume of the sphere.
That being said, let’s evaluate the above integral in terms of nested integrals. It’s a little more
tedious, but safer:
Inner integral:
Z
R
r 2 sin φ dr = sin φ
Z
R
r 2 dr =
0
0
1 3
R sin φ
3
Middle integral:
Z
0
2π
1
1 3
R sin φ dθ = R3 sin φ
3
3
Z
2π
dθ =
0
2π 3
R
3
Outer integral:
2π 3
R
3
Z
π
sin φ dφ =
0
4π 3
R .
3
Once again, we have obtained the correct result. In retrospect, we have seen that variables not being
integrated are simply placed in front of an integration. This is essentially equivalent to “separation.”
157
Let us now perform the volume integration in a different order, namely,
Z
0
R Z 2π
0
Z
π
r 2 sin φ dφ dθ dr
0
Inner:
Z
π
r 2 sin φ dφ = 2r 2 .
0
Middle:
Z
2π
2r 2 dφ = 4πr 2
0
Outer:
Z
R
4
4πr 2 dr = πR3
3
0
(volume of sphere)
In this example, we integrated over angles θ and φ first to produce
4πr 2 (this is the area of a spherical surface of radius r).
In the final integral,
Z
R
4πr 2 dr
0
4πr 2 dr
is the volume dV of a thin spherical shell with radius r,
thickness dr.
This method of integration is commonly employed in physics.
Example: Given a hemisphere with radius R and constant density ρ0 , find the center of mass.
158
Let O be located at the center of the circular base.
By symmetry (if you don’t believe it, calculate it!),
x̄ = ȳ = 0.
We still need to calculate z̄:
RRR
z dm
z̄ = R R RD
.
D
(4)
dm
The denominator is the mass of the hemisphere:
1
2
M=
2
4 3
πR ρ0 = πR3 ρ0
3
3
. We now need to compute
Z Z Z
z dm = ρ0
D
Z Z Z
z dV.
(5)
D
To scan all the points in D (hemisphere), we must integrate over the spherical polar coordinates
0 ≤ φ ≤ π2 , 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π:
Z Z Z
z dV =
D
Z
0
π/2 Z 2π
0
Z
R
0
(r cos φ) r 2 sin φ dr dθ dφ.
|
{z
(6)
}
z
Note that the integrand z has been translated into spherical coordinates.
This integral is separable:
Z Z Z
z dV =
D
"Z
# Z
π/2
0
φ=π/2
1
sin φ cos φ dφ =
sin2 φ
2
"Z
R
0
0
The first integral is
π
2
dθ
sin φ cos φ dφ
0
Z
2π
3
#
r dr .
(7)
1
= /
2
(8)
1 4
πR4
1
R =
.
(2π)
2
4
4
(9)
φ=0
The second and third integrals are straightforward:
Z Z Z
z dV =
D
As a result, we have
z̄ =
ρ0 π4 R4
3
= R.
2
3
8
3 πR ρ0
159
(10)
Another application of triple integration - Kinetic energy of a rotating solid
Consider a solid body that is rotating about an axis with angular speed ω. For convenience we shall
assume that the axis lies along the z axis. And for the moment, we assume that the mass density
function ρ(r) is not necessarily constant. The question: What is the kinetic energy of this rotating
body? In particular, we shall examine the rotation of a sphere of radius R about an axis that passes
through its centre but, for the moment, we’ll keep the discussion general.
In the “spirit of calculus,” we consider a mass element dm centered at (x, y, z). What is the kinetic
energy of dm?
The kinetic energy of this mass element dm is
dK =
1
1
(dm)v 2 = (ρdV )v 2 .
2
2
(11)
But what is the speed v of the revolving mass element dm?
It is v = sω where s is the distance from dm to the
z-axis, s =
The kinetic energy of dm is therefore
p
x2 + y 2 – see diagram below.
1
dK = ρ(x, y, z)(x2 + y 2 )ω 2 dV.
2
160
(12)
Top View:
The total kinetic energy of the rotating body is
obtained by integrating over D:
K =
Z Z Z
dK
D
=
1 2
ω
2
Z Z Z
(x2 + y 2 )ρ(x, y, z) dV. (13)
D
It is convenient to express the kinetic energy of the rotating body as follows,
1
T = Iω 2 ,
2
(14)
where
I=
Z Z Z
(x2 + y 2 )ρ(x, y, z) dV
(15)
D
is the moment of inertia of the body with respect to the z-axis.
Special case - homogeneous body: In this case, ρ(x, y, z) = ρ, a constant, so that
I =ρ
Z Z Z
(x2 + y 2 ) dV.
(16)
D
Example: In the case of a homogeneous sphere of radius R, the moment of inertia is given by
(Exercise):
I=
8
ρπR5 .
15
(17)
(Hint: Use polar coordinates and the fact that x2 + y 2 = r 2 sin2 φ.) Note that this moment of inertia
can be rewritten as follows:
8
π R5
15
2 4
3
π R ρ R2
5 3
2
M R2 ,
5
I = ρ0
=
=
where M is the total mass of the sphere. You may have seen this expression for I in a first-year physics
course.
161
Lecture 22
Triple integrals - finale
Cylindrical coordinates
(Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 15.7)
As the name suggests, cylindrical coordinates are useful in the treatment of physical systems
that exhibit cylindrical symmetry, e.g., a straight conducting wire with nonnegligible cross-sectional
area, fluid flow through a pipe. This coordinate system is simply constructed by using planar polar
coordinates in the xy-plane and adding the z coordinate. In other words, the cylindrical coordinates
of a point r ∈ R3 are given by (r, θ, z), where r ≥ 0, 0 ≤ θ ≤ 2π, and z ∈ R. The relation betwee
cylindrical and Cartesian cooordinates is straightforward – the z coordinate is the same, and x and y
are related to the planar polar coordinates in the familiar way:
x = r cos θ,
y = r sin θ,
z.
(18)
The element of volume dV in cylindrical coordinates will be given by
dV = r dr dθ dz.
(19)
Geometrically, this may be seen by taking the two-dimensional area element dA = r dr dθ and
translating it upward/downward by dz, so that dV = dA dz.
162
Appendix: Integration over a solid region to find net potential
This material was not covered in any lecture but is provided for the benefit of
the interested reader. The ideas sketched below are used in many applications of
physics.
Here is a typical problem from electrostatics. Suppose that a charged body occupies a region D ⊂ R3
and that the charge density at a point r ∈ D is given by ρ(r). Recall how the charge density function
is defined:
∆q
,
∆V →0 ∆V
ρ(r) = lim
(20)
where ∆q is the amount of charge contained in a cube or sphere of volume ∆V centered at r.
Now consider a point P with position vector r that lies outside region D. The object is to find
the electrostatic potential U (r) at P due to the charge distribution on D. The situation is sketched
below.
D
Q
r − r′
dq
P
r′
r
O
Proceeding in the “Spirit of Calculus”, we consider the potential dU at r that is due to the
infinitesimal element of charge dq at a point r′ = (x′ , y ′ , z ′ ) ∈ D and then integrate over all point
r′ ∈ D. Note that we shall now use primed coordinates to designate points in D, which is common
practice. Let us proceed in simple steps to arrive at the desired result.
1. Step 1: Recall that the electrostatic potential at a point P situated at r = (x, y, z) due to the
presence of a point charge Q at the origin O is given by
U (r) =
1
1
1
1
=
.
4πǫ0 k r k
4πǫ0 [x2 + y 2 + z 2 ]1/2
163
(21)
2. Step 2: Now keep the point P where it is and move the charge Q to a point Q with coordinates
r0 = (x0 , y0 , z0 ) The electrostatic potential at P is now determined by the distance between Q
and P and will be given by
U (r) =
1
1
1
1
.
=
4πǫ0 k r − r0 k
4πǫ0 [(x − x0 )2 + (y − y0 )2 + (z − z0 )2 ]1/2
(22)
Step 3: From Step 2, the potential dU (r′ ) at r that is due to the infinitesimal element of charge
dq at a point r′ = (x′ , y ′ , z ′ ) in the charged region D is given by
dU (r′ ) =
1
1
1
1
=
.
′
′
2
4πǫ0 k r − r k
4πǫ0 [(x − x ) + (y − y ′ )2 + (z − z ′ )2 ]1/2
(23)
Note that we have treated the infinitesimal element of charge dq as a point charge.
3. Step 4: We now sum up the contributions in Step 3 over all points r′ ∈ D by integrating over
D:
U (r) =
Z Z Z
dU (r)
(24)
D
=
=
Z Z Z
1
1
dV ′
′
4πǫ0
D kr−r k
Z Z Z
1
1
dx′ dy ′ dz ′ .
′
2
′ 2
′ 2 1/2
4πǫ0
D [(x − x ) + (y − y ) + (z − z ) ]
Of course, it may be more advantageous to use systems other than Cartesian coordinates.
The same idea may be used to compute gravitational potential at a point due to a mass distribution. In an upcoming assignment, you will be asked to perform such a calculation where region D
represents a spherical, homogeneous earth.
164
Vector Calculus
We now return to a discussion of vector fields and vector calculus – for the remainder of the term!
Gradient fields/conservative forces
(Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 16.1, p. 1031-1032.
That being said, the treatement of gradient fields/conservative forces in this textbook is extremely
brief. These lecture notes should be used as the primary source.)
Let us first consider the problem of motion in one dimension under the action of a force, which you
saw in first-year Calculus and Physics. Suppose that a mass m is acted upon by a force F(x) = f (x)i
and that the motion of the mass, restricted to the x-axis, is determined by Newton’s Law, i.e., F = ma.
Equating components, we have
f = ma,
(25)
Here, a(t) = x′′ (t), where x(t) is the position of the mass.
Eq. (25) is a differential equation for the position function x(t). In fact, it is a second-order
ordinary differential equation (ODE) in x(t), since the second derivative x′′ (t) is involved. For f (x)
sufficiently “nice,” which covers many common applications, there are mathematical theorems that
guarantee the existence of solutions to this ODE. And as you will see in your course on differential
equations (MATH 228), since Eq. (25) is a second-order ODE, if we impose two initial conditions on
the solution, i.e., (i) x(0) = x0 (initial position) and (ii) x′ (0) = v0 (initial velocity), we can extract a
unique solution x(t) that satisfies these conditions. For some special cases of f (x), including the two
examined below, the solutions to Eq. (25) may be obtained in closed form, i.e., in terms of simple
functions. In other cases, one may have to rely on series expansions or numerical methods to generate
approximate solutions.
Here, however, we are not interested in solving for x(t) but rather in establishing a fundamental
feature of the motion of the mass. First of all, we may define the potential energy associated with the
165
force F as follows,
V (x) = −
Z
x
f (s)ds,
(26)
x0
where x0 is a reference point. (Typically, we choose x0 such that V (x0 ) = 0, which often – but not
always! – implies that x0 = 0.) The physical interpretation of this definition is that the potential
energy V (x) is the work done against the force f (x) in moving the mass m from x0 to x.
Note, from the Fundamental Theorem of Calculus II, that
V ′ (x) = −f (x), or equivalently, f (x) = −V ′ (x).
(27)
A force for which this relation holds, you may recall, is called a conservative force.
Here we insert an important note: Eq. (42) is to be considered as the definition of a
conservative force, i.e., one that may be written as the negative derivative of a potential
energy function V (x). The conservation of energy, which you know to be associated with
a conservative force, is a consequence of Eq. (25), as we’ll show after these examples.
In one dimension, all forces that are functions only of position are conservative. (A frictional force
which, for a moving mass, is typically dependent upon the particle’s velocity, is not conservative.) The
total mechanical energy of the mass during its motion is given by the sum of its kinetic and potential
energies, i.e.,
1
E(t) = mv(t)2 + V (x(t)).
2
(28)
We emphasize the dependence of the position x(t) and velocity v(t) on time t – the kinetic and
potential energies need not be constant during the motion of the mass. (However, the energy E(t)
will be, but more on this later.)
Example 1: Near the surface of the earth, the gravitational force on a small object of mass m is well
approximated by F(x) = −mgi, where x denotes the height of the object above the surface of the
earth and the unit vector i points directly upward. Here, f (x) = −mg. For convenience, we let x = 0
be the surface of the earth so that the potential energy of the mass becomes
V (x) = −
Z
x
−mg ds = mgx.
(29)
0
Then for an object moving under the influence of gravity near the surface of the earth (for example,
a ball that has been thrown upward, or dropped from a height h > 0), its total mechanical energy is
166
given by
E(t) =
1
mv(t)2 + mgx(t).
2
(30)
Example 2: Recall Hooke’s “Law” (it is NOT a law but an approximation) for the force exerted by
a spring on a mass that is allowed to move only in one dimension: F(x) = −kxi, where k is the spring
constant and x is the displacement from the equilibrium position x = 0. We also assume that there
is no frictional force exerted on the mass. Here, f (x) = −kx. Once again we let x = 0 be reference
point so that the potential energy function is defined as
V (x) = −
Z
0
x
1
−ks ds = kx2 .
2
(31)
The total mechanical energy of the mass moving under the influence of the spring (oscillatory motion
about x = 0) is given by
1
1
E(t) = mv(t)2 + kx2 .
2
2
(32)
Conservation of energy
We now come to the following fundamental result, which accounts for the term “conservative” in
“conservative forces.” The motion of the mass m, when acted upon by a conservative force, is such
that the total mechanical energy E(t) is conserved. The total mechanical energy is the sum of the
kinetic energy (KE) and the potential energy (PE) of the mass. Along the trajectory x(t), then, the
energy of the mass is given by
1
E(t) = mv(t)2 + V (x(t)).
2
(33)
We now prove that E(t) is constant along the trajectory x(t) by showing that E ′ (t) = 0 for all
time t. Let us differentiate (33) with respect to time t, using the Chain Rule (yes, the one of the most
important methods in Calculus!):
E ′ (t) = mv(t)v ′ (t) + V ′ (x(t))x′ (t)
= mv(t)a(t) − f (x(t))v(t)
= v(t)[ma(t) − f (x(t))]
= v(t) · 0
= 0,
167
(34)
where we have used the result that f = ma at all points/times on the trajectory x(t). Since E ′ (t) = 0,
it follows that E(t) is constant in time. This constant value of the energy could be determined, for
example, from the initial conditions, i.e., x(0) and x′ (0) = v(0).
We now wish to extend this result to motion in higher dimensions, i.e. Rn for n = 2, 3, · · ·. The
question is, “What is a conservative force F in R2 or R3 ?” Let us go back to the definition of a
conservative force in one dimension, Eq. (27), and rewrite it in vector form as
F = f (x)i,
F=−
∂V
i.
∂x
(35)
The expression on the right hand side is the one-dimensional form of a gradient vector. If we can
generalize this result to Rn – and we can – then the definition of a conservative force F in Rn is one
for which there exists a scalar-valued function V : Rn → R such that
~
F = −∇V.
(36)
Final notes: Using conservation of energy to solve simple physical problems
In addition to being a a fundamental principle of classical mechanics, conservation of energy can often
be an effective tool to analyze physical systems. In many cases, it is simpler, perhaps much simpler,
to use conservation of energy in such analyses. You have probably encountered this idea in a first-year
physics course, but it is important to recall this idea.
Example 1 revisited: Motion of a projectile near the surface of the earth
To illustrate, let us return to Example 1 above, specifically, to analyze motion under the influence of
gravity near the surface of the earth. We consider the following well-known problem,
A ball is thrown directly upwards with speed v0 . How high does it travel from the point
at which it was released?
There are two ways to solve this problem:
1. Method No. 1 – Solving for the particle’s trajectory: Given that the force f (x) = −mg,
the equation of motion arising from Newton’s law f = ma is
mx′′ (t) = m
d2 x
= −mg
dt2
168
→
d2 x
= −g.
dt2
(37)
This is a very simple differential equation for x(t) which can be integrated twice to obtain x(t).
1
x(t) = − gt2 + c1 t + c2 .
2
(38)
Imposing the initial conditions x(0) = 0 (reference point - the position from which the ball was
launched) and x′ (0) = v0 (initial velocity), we obtain the well-known result,
1
x(t) = v0 t − gt2 .
2
(39)
From this equation, it follows that the velocity v(t) = x′ (t) is given by
v(t) = v0 − gt,
(40)
which we would have obtained by integrating Eq. (37) only once. At the highest point of the
v0
ball’s trajectory, the velocity is zero. From (40), we see that velocity v(t) = 0 at time t∗ = .
g
Substitution into Eq. (39) yields that that
x(t∗ ) =
v02
.
2g
(41)
This is the maximum height of the ball’s trajectory.
2. Method No. 2 – Using conservation of energy: The total mechanical energy of the ball,
given by Eq. (30), is constant in time. At t = 0, when the ball is launched, its mechanical energy
1
is E(0) = mv02 , i.e., all kinetic energy and zero potential energy. At the highest point of the
2
ball’s trajectory, which we shall denote as x = h, it has zero kinetic energy and potential energy
mgh. By conservation of energy,
1
mgh = mv02 ,
2
from which it follows that h =
(42)
v02
, in agreement with the result of Method No. 1.
2g
You can see that Method No. 2 is much simpler – we didn’t have to solve for the ball’s trajectory, then
solve for the time at which it attains its maximum height, etc.. In more complicated problems, e.g.,
the motion of the earth around the sun (the so-called “Kepler problem” of motion under the influence
of gravity, not near the surface of the earth!), it is quite tedious to solve for the position of the object
of concern. Conservation of energy allows us to extract much information much more easily. Indeed,
in problems involving more than two objects, e.g., sun-earth-moon, the trajectories of the masses may
not be obtained in closed form.
169
Example 2 and its generalization: One-dimensional periodic motion in a “potential well”
We return to the mass-spring system of Example 2 above. Suppose that we place the mass m at a
point x0 > 0 – therefore stretching the spring – and then release it from rest. We expect the resulting
motion of the mass to be oscillatory: The mass will move leftward, past the equilibrium point x = 0
and continue to move leftward, but slowing down until it stops and begins travelling to the right. It
then moves past the equilibrium point x = 0 and then continues to its original starting point, x = x0 .
Then it again begins its leftward journey, etc.. You are probably not surprised by the fact that the
particle returned to its original starting point, since the total energy of the mass is conserved. You
probably also know that when the mass is going through the equilibrium point x = 0, it possesses its
maximum kinetic energy.
In fact, as we established above, the total mechanical energy of the mass, E(t), is constant for
all time t ≥ 0, t = 0 being the time at which we released the mass. Recall that we determined the
1
potential energy of this system (i.e., the potential energy stored in the spring) to be V (x) = kx2 .
2
1 2
Therefore, the initial mechanical energy of the system is E(0) = kx0 . At any time t ≥ 0, the
2
mechanical energy of the system is
1
1
1
E(t) = mv(t)2 + kx(t)2 = kx20 ,
2
2
2
(43)
where v(t) = x′ (t) is the velocity of the mass.
Note that if we set v(t) = 0, we obtain x(t) = x0 , i.e., the original starting point. But another
possibility is x(t) = −x0 . In other words, the mass executes oscillatory motion between the points
x = x0 and x = −x0 . In this case, the amplitude of the oscillation is clearly A = x0 .
This oscillatory motion may be represented graphically in a plot of potential energy vs. position, as
1
shown in the figure below. In this figure, we have plotted the potential energy function V (x) = kx2 .
2
On the x-axis, we have also identified the points ±x0 representing the extremities of the trajectory
x(t) of the mass. These points are also called the turning points of the oscillatory trajectory – they
are the points at which the mass “turns around” and reverses its motion. They are the points at which
the instantaneous velocity of the mass is zero, implying that its potential energy is equal to its total
energy E(0).
In fact, we can solve Newton’s equation of motion f = ma for this case to obtain the trajectory
x(t) explicitly as a function of time. The result, which you may see in your course on differential
170
Energy
V (x)
1
2
2
kx0
x
x0
−x0
oscillatory motion in potential well
between turning points
equations, is
x(t) = x0 cos(ωt),
ω=
s
k
.
m
(44)
Here, ω is the (angular) frequency of oscillation.
For more general one-dimensional potentials, however, we may not be able to solve for the trajectory x(t) of a mass oscillating in a potential well. However, if we know the (constant) mechanical
energy E of the mass, we may be able to solve for its turning points x1 and x2 , shown schematically
in the figure below. They are the points at which V (x1 ) = V (x2 ) = E. Moreover, the period of the
oscillation may also be expressed as an integral involving the potential function V (x), the total energy
E and the turning points x1 and x2 .
Energy
V (x)
E
x1
x2
x
oscillatory motion in potential well
between turning points
171
Lecture 23
Gradient fields/conservative forces in physics (cont’d)
Definition: In physics, a force field F : Rn → Rn is conservative if there exists a scalar-valued
potential function U : Rn → Rn such that
~
F = −∇U.
(45)
The above definition is quite specific to Physics because of the appearance of the minus sign on the
LHS. (As you have seen earlier, this allows the total mechanical energy to be a sum of the kinetic
energy and the potential energy U , not −U .) Mathematicians tend not to talk about conservative
fields but rather gradient vector fields:
Definition: A vector field F : Rn → Rn is a gradient vector field if there exists a scalar-valued
function f : Rn → Rn such that
~
F = ∇f.
(46)
Of course, f and U are related by U = −f . We shall be using both definitions throughout this course.
But when definite applications to Physics are considered, we shall be using the definition of conservative forces.
Important note: Unfortunately, the textbook (Stewart, p. 1032) uses the above definition of a
gradient field to define a conservative force. In other words, f in (46) is referred to as the potential
function.
Another note: The above definition includes the one-dimensional case studied in the previous lecture
as a special case. Recall that in this case, we considered a force F(x) = f (x)i that was dependent
only upon the position variable x. By defining the associated potential function U (x) as follows,
U (x) =
Z
x
f (s) dx,
(47)
x0
we have, by the Fundamental Theorem of Calculus II that
f (x) = −U ′ (x) = −
172
dU
.
dx
(48)
Since we are working only in one dimension, this relation implies the equality of the following two
vectors,
f (x) i = −
dU
i.
dx
(49)
The left-hand side is F(x). The right-hand side is the negative one-dimensional gradient of U (x), so
that we have
~
F = −∇U,
(50)
confirming that F is conservative.
Conservative forces and conservation of energy
The following represents one of the most important results of this course.
Suppose that a mass m is moving in Rn under the influence of a conservative force F : Rn → Rn .
The trajectory x(t) of the mass is determined by Newton’s Law, F = ma. Then the total mechanical
energy E of the mass is constant over the trajectory. Here, E is defined as the usual sum of kinetic
and potential energies of the mass, i.e.,
1
E(t) = mkv(t)k2 + U (x(t)).
2
(51)
It is important to note that the energy E(t) is evaluated at points x(t) of the trajectory.
To prove this result, it will be useful to rewrite the total energy function as follows:
1
E(t) = mv(t) · v(t) + U (x1 (t), x2 (t), · · · , xn (t)).
2
(52)
We now differentiate both sides with respect to time t:
E(t) =
1 d
d
m [v(t) · v(t)] + U (x1 , x2 (t), · · · , xn (t)).
2 dt
dt
(53)
To determine the first derivative, we use the result:
d
a(t) · b(t) = a′ (t) · b(t) + a(t) · b′ (t).
dt
(54)
(This is easily proven from the definition of the dot product:
a(t) · b(t) = a1 (t)b1 (t) + a2 (t)b2 (t) + · · · + an (t)bn (t).
173
(55)
Simply use the product rule for differentiation and rearrange the results.)
Then
d
v(t) · v(t) = v′ (t) · v(t) + v(t) · v′ (t)
dt
= 2v′ (t) · v(t)
(56)
= 2a(t) · v(t).
The derivative involving U is computed using the Chain rule:
d
∂U dx1
∂U dxn
U (x1 (t), · · · , xn (t)) =
+ ···
dt
∂x1 dt
∂xn dt
~
= ∇U · v.
(57)
Putting these two results together gives
~
E ′ (t) = v · [ma(t)] + v(t) · ∇U
(58)
~ ]
= v(t) · [ma(t) + ∇U
= v(t) · [ma(t) − F(x(t))]
(since F is conservative)
= v(t) · 0 (trajectory determined by Newton’s Law)
= 0.
Therefore the energy E(t) is constant along the trajectory of the mass.
Some examples of conservative forces in higher dimensions
Example 1: Consider the two-dimensional mass-spring system sketched below. (We are assuming
that the mass is lying on a frictionless floor.)
For small displacements from the equilibrium point (0, 0), the force exerted by the springs on the
mass m is well-approximated as follows,
F(x, y) = −k1 xi − k2 yj.
(59)
(We are ignoring higher-order corrections. The exact result will involve square roots of sums involving
squares of the coordinates x and y.) We now ask whether this force is conservative or not. In other
~ ? In component form,
words, does there exist a scalar-valued function U (x, y) such that F = −∇U
this means that
−k1 xi − k2 yj = −
174
∂U
∂U
i−
j.
∂x
∂y
(60)
y
m
k1
x
O
equilibrium point
k2
By equating components, we are looking for a function U (x, y) such that
∂U
= k2 y.
∂y
∂U
= k1 x,
∂x
(61)
This problem is rather easy. By “inspection” or “educated guessing”, we can come up with a suitable
candidate:
U (x, y) =
1
1
k1 x2 + k2 y 2 ,
2
2
(62)
which you can easily verify by differentiation. (Very shortly, we shall describe a more systematic
procedure to determine U , provided that such a U exists.) Note that U (x, y) is simply the sum of
the “Hookean” potential energies of the two springs. Thus the force F is conservative. This implies
that the total energy of the mass is constant during any motion (or even if it is lying at rest at the
equilibrium point!). Its total energy will be given by
E(t) =
1
mv(t)2 + U (x(t), y(t)),
2
(63)
where v(t) =k v(t) k is the speed of the mass at time t. If we drop the reference to time t and let
v = (v1 , v2 ), then the total mechanical energy is given by
1
1
1
1
mv12 + mv22 + k1 x2 + k2 x2
2
2
2
2
1
1
1
1
2
2
2
2
=
mv + k1 x + mv2 + k2 x .
2 1 2
2
2
E =
(64)
(65)
In other words, the total energy can be viewed as the the sum of the mechanical energies in the x and
y directions where each spring acts independently along a specific direction.
Before moving on, let us return to the potential energy function U (x, y) for this problem, as given
in Eq. (62). A sample graph of this function is sketched in the figure below. This is a two-dimensional
175
“potential well”, as opposed to the one-dimensional potential well sketched in the previous lecture.
And the minimum of this potential well, (0, 0), is the equilibrium point of the system – the point at
which the net force on the mass is zero.
z
Energy
z = V (x, y)
y
x
If the mass has a total mechanical energy E0 ≥ 0 at some time t, then its total energy E(t) = E0
for all time t. The “classical turning points” of its oscillation, that is, the points at which it will
change its direction of motion, consist of all points (x, y) such that U (x, y) = E0 . Recall that for
one-dimensional motion, there will be only two such turning points at which the mass reverses its
direction. But in two dimensions, the turning points form a curve. In this case, the curve is the ellipse
in the xy-plane given by
U (x, y) =
1
1
k1 x2 + k2 y 2 = E0 .
2
2
(66)
At these points, the mass has zero kinetic energy – all of its energy has been converted into potential
energy.
Example 2: Let us return to the following result in R3 derived earlier in the course (you also worked
with it in an earlier assignment):
~ 1 = − 1 r,
∇
r
r3
where r =
p
r = xi + yj + zk,
(67)
x2 + y 2 + z 2 . Note that for any constant K,
~ K
∇
r
=−
K
r.
r3
(68)
As we said earlier, the term on the right side resembles the form of force vector fields associated with
either a charge or mass situated at the origin.
176
1. Case 1: Electrostatic fields
Recall that the electrostatic field (that is, the force per unit charge) E(r) that is generated by
the presence of a charge Q at the origin is given by
E(r) =
Q
r.
4πǫ0 r 3
(69)
Q
,
4πǫ0
(70)
Note that if we set
K =−
then Eq. (68) becomes
~ − Q
∇
4πǫ0 r
=
Q
r = E(r).
4πǫ0 r 3
(71)
This implies that the electrostatic field E is conservative, i.e.
~ = ∇(−Φ),
~
E = −∇Φ
where Φ(r) =
Q
.
4πǫ0 r
(72)
Φ(r) is the electrostatic potential associated with the electrostatic field E(r). In Cartesian coordinates, it is given by
1
Q
p
.
4πǫ0 x2 + y 2 + z 2
Φ(x, y, z) =
(73)
The level sets of Φ are spheres in R3 that are centered at the origin (0, 0, 0). The value of the
function Φ(r) decreases as we move away from the origin. Once again, this is reflected in the
fact that its gradient vectors point inward. And since E is the negative of these gradient vectors,
E points outward.
You may be more familiar with the electrostatic force F(r) exerted by a charge Q at (0, 0, 0) on
a test charge q that is situated at r. It will be given by
F(r) =
Qq
r = qE(r).
4πǫ0 r 3
(74)
The potential energy of the charge q at r is then
U (r) =
Qq
= qΦ(r).
4πǫ0 r
(75)
These quantities are related, respectively, to the field quantitites E and Φ by the test charge
constant q so that
~ (r).
F(r) = −∇U
In other words, the electrostatic force field F is conservative.
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2. Case 2: Gravitational fields
The above method is also easily applied to the gravitational field G(r) due to a mass M situated
at the origin (0, 0, 0). Here,
G(r) = −
GM
r.
r3
(77)
In this case, we choose
K = GM
so that Eq. (68) becomes
~ GM
∇
r
=−
(78)
GM
r = G(r).
r3
(79)
This implies that
~ = ∇(−Φ),
~
G = −∇Φ
where Φ(r) = −
GM
.
r
(80)
Φ(r) is now the gravitational potential associated with the gravitational field G(r).
Once again, you are more familiar with the gravitational force F(r) exerted by a mass M at
(0, 0, 0) on a mass m that is situated at r. It will be given by
F(r) = −
GM m
r = mG(r).
r3
(81)
The potential energy of the mass m at r is then
U (r) = −
GM m
= mΦ(r).
r
(82)
These quantities are related, respectively, to the field quantitites E and Φ by the mass m so that
~ (r).
F(r) = −∇U
In other words, the gravitational force field F is conservative.
178
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