Introduction to classical mechanics

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Introduction to
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Lakshmi - 11750 - Introduction to Classical Mechanics.indd 1

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January 28, 2020

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BC: 11750 - Introduction to Classical Mechanics

For Kay

v

Mechintroroot

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Preface

The author recently published a book entitled Introduction to Electricity
and Magnetism [Walecka (2018)]. It is based on an introductory course
taught several years ago at Stanford, with over 400 students enrolled. The
only requirements were an elementary knowledge of calculus and familiarity with vectors and Newton’s laws; the development was otherwise
self-contained. The lectures, although relatively concise, take one from
Coulomb’s law to Maxwell’s equations and special relativity in a lucid and
logical fashion. The book has an extensive set of accessible problems that
enhances and extends the coverage. As an aid to teaching and learning, the
solutions to those problems were subsequently published in a separate text
[Walecka (2019)].
Although never presented in an actual course, it occurred to the author
that it would be fun to compose an equivalent set of lectures, aimed at the
very best students, that would serve as a prequel to that Electricity and
Magnetism text. These lectures would assume a good, concurrent, course
in calculus and familiarity with basic concepts in physics (say, from a good
high-school course); they would otherwise, again, be self-contained. For my
own amusement, I did just that.
The lectures start with a review of the necessary mathematics and a
review of vectors. The idea of an inertial frame is then introduced, and
Newton’s laws are stated, with several applications included. The concepts
of energy and angular momentum are introduced, and the analysis is then
extended to many-particle systems.
The notions of generalized coordinates and Lagrange’s equations are
first introduced on the basis that they reproduce Newton’s laws in the
chosen examples. After a lecture introducing the calculus of variations, Lagrange’s equations are derived from what then serves as the basic principle
vii

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BC: 11750 - Introduction to Classical Mechanics

Mechintroroot

Introduction to Classical Mechanics

of classical mechanics —Hamilton’s principle of stationary action. Several
more examples are given of lagrangian mechanics.
Hamilton’s equations are similarly first introduced on the basis that
they reproduce Lagrange’s equations and Newton’s laws for the chosen
examples, and they are then subsequently similarly derived from Hamilton’s
principle of stationary action. Several examples are included of hamiltonian
mechanics and phase space.
A lecture then discusses the transition from the mechanics of discrete
particle systems to that of continuous media. Lagrange’s equations for continuous systems are exhibited and then derived from Hamilton’s principle.
The wave motion of a string under tension serves as the paradigm for continuum mechanics, and the analysis extends up through the construction
of the energy-momentum tensor and the reflection and radiation of those
waves.
Irrotational, isentropic fluid flow, where the velocity field is derived from
a potential and there is no internal (reversible) heat flow, serves as the final
example of lagrangian continuum mechanics. The lagrangian density is
constructed. Bernoulli’s equation and the continuity equation for the mass

(number) density are then derived from Lagrange’s equations, and they
are related back to Newton’s laws for fluid mechanics. The energy density
and energy flux are constructed, and the analysis is then applied to sound
waves, where reflection and radiation are again examined.
The goal of this text is to provide a clear and concise set of lectures
that take one from the introduction and application of Newton’s laws up to
Hamilton’s principle and the lagrangian mechanics of continuous systems.
This, indeed, provides the point of departure from classical mechanics to
modern quantum field theory.1 An extensive set of accessible problems
again enhances and extends the coverage.
Now readers may feel that this is an overly ambitious goal for a set of
introductory lectures on classical mechanics, and it is hard to argue with
that. I did not feel the goals were too ambitious in the case of the Electricity
and Magnetism text. It may be that the current lectures are only relevant
to a more advanced honors course. Nevertheless, after completing this text
and reading it over several times, I am convinced that the whole thing fits
together well, and the book serves as a useful text for good students. I
do also believe that the current book provides a good introduction to the
more advanced mechanics texts, such as [Fetter and Walecka (2003)]. In
1 See,

for example, [Walecka (2010)] .

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Preface

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ix

addition, I feel the book serves as a good introduction and companion to
other standard mechanics texts such as [Kleppner and Kolenkow (2013);
Morin (2008); Thornton and Marion (2012); Kibble and Berkshire (2004);
Taylor (2004); Landau and Lifshitz (1976); Goldstein et al. (2011)], etc. I
am therefore submitting the present manuscript for publication. It is my
hope that students and teachers alike will share some of the pleasure I took
in writing this book.
I would like to once again thank my editor, Ms. Lakshmi Narayanan,
for her help and support on this project.
Williamsburg, Virginia
November 4, 2019

John Dirk Walecka
Governor’s Distinguished CEBAF
Professor of Physics, emeritus
College of William and Mary

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Contents

Preface

vii

1. Introduction
1.1
1.2
1.3
1.4

Physics
Calculus
Example

Units . .

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2. Vectors
2.1

2.2
2.3
2.4
2.5

Vector . . . . . . . . . . .
Scalar Product . . . . . .
Vector Product . . . . . .
Length and Direction . . .
Gradient . . . . . . . . . .
→
2.5.1 Example − ∇ V (r)

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3. Inertial Coordinate Systems

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4. Newton’s Laws

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5. Examples

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5.1 Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . .
6. Energy

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21

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Introduction to Classical Mechanics

6.1 Spring . . . . . . . . . . .
6.2 Projectile Motion . . . . .
6.3 Two-Body Problem . . . .
6.3.1 Conservative Force
6.3.2 Two-Body System

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7. Angular Momentum

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7.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . .
7.2 Uniform Circular Motion . . . . . . . . . . . . . . . . . . .
7.2.1 Gravitational Orbits . . . . . . . . . . . . . . . . . .
8. System of Particles
8.1
8.2
8.3
8.4

C-M Motion . . . . .
Energy . . . . . . . .
Angular Momentum
Rigid-Body Motion .

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9. Generalized Coordinates
9.1
9.2
9.3
9.4

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Pendulum . . . . . . . . . .
Particle on Table Connected
Bead on a Rotating Hoop .
Coupled Oscillators . . . . .

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to Hanging Mass
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10. Hamilton’s Principle

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10.1 Calculus of Variations . . . . . . . . . . . . . . . . . . . . .
10.2 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . .
10.3 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . .
11. Lagrangian Dynamics
11.1 Examples . . . . . . . . . . . . . . . . . .
11.1.1 Another Bead on a Rotating Hoop
11.1.2 Cylinder Rolling on Incline Plane .
11.2 Canonical Momentum . . . . . . . . . . .
11.3 Hamiltonian . . . . . . . . . . . . . . . . .
11.4 Previous Examples . . . . . . . . . . . . .

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12. Hamiltonian Dynamics
12.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . .

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xiii

Contents

12.2 Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . .
13. Continuum Mechanics

77

13.1 Oscillations of Particles Connected by Springs
13.2 Continuum Limit . . . . . . . . . . . . . . . .
13.3 Wave Equation for String . . . . . . . . . . .
13.3.1 Normal Modes . . . . . . . . . . . . .
13.3.2 Boundary Conditions . . . . . . . . . .
13.3.3 General Solution . . . . . . . . . . . .

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14. Waves

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14.1 One-Dimensional Wave Equation

14.2 Superposition . . . . . . . . . . .
14.3 Traveling Waves . . . . . . . . .
14.3.1 Snapshot at Fixed t . . . .
14.3.2 Disturbance at a Fixed x .
14.4 Standing Waves . . . . . . . . . .
14.5 Amplitude Modulation . . . . . .

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15. Continuum Mechanics of String

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15.1 Energy . . . . . . . . . . . . . . . . .
15.2 Lagrangian . . . . . . . . . . . . . .
15.3 Lagrange’s Equation . . . . . . . . .
15.4 Hamilton’s Principle . . . . . . . . .
15.5 Lagrangian Dynamics . . . . . . . .
15.5.1 Canonical Momentum Density
15.5.2 Hamiltonian Density . . . . .
15.5.3 Continuity Equation . . . . .
15.5.4 Momentum Density . . . . . .
15.6 Energy-Momentum Tensor . . . . . .

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. 91
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. 100
. 101

16. Mechanics of Fluids
16.1 Assumptions . . . . . . . .
16.2 Lagrangian Density . . . . .
16.3 Lagrange’s Equations . . . .
16.3.1 Continuity Equation
16.3.2 Energy Flux . . . . .
16.3.3 Bernoulli’s Theorem

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105
107
108
109
110
111

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Introduction to Classical Mechanics

16.4 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.5 Sound Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 117
17. Problems

119

Appendix A Numerical Methods

157

Appendix B

159

Significant Names in Classical Mechanics

Bibliography

161

Index

163

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Chapter 1

Introduction

The author recently published a book entitled Introduction to Electricity
and Magnetism based on a one-quarter, calculus-based course he taught at
Stanford some years ago [Walecka (2018)].1 The purpose of the present
book, written just for fun, is to design a one-quarter series of lectures on
Introduction to Classical Mechanics that could serve as a prequel to the
E&M text.2
It is assumed that the reader has taken a good high-school physics course
and is familiar with the basics concepts of units, measurements, vectors, etc.
It is also assumed that he or she has taken, or is taking, a good course on
calculus.

1.1

Physics

We again start with some comments on physics. Physics provides a way
of looking at the world. We describe physical phenomena in mathematical
terms with the goal of
• Correlating phenomena
• Predicting new phenomena

The description is tested with experiment. Physics is an experimental science. The payoff is that
• The description is either correct or incorrect
• The correct results are universal

1 See

also [Walecka (2019)] .
mechanics course also serves as a nice introduction to the graduate text [Fetter
and Walecka (2003)] .
2 This

1

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Introduction to Classical Mechanics

The goal is to develop a physical law, presented as a mathematical relation, usually (but not always) a statement on the instantaneous development of a system, and then derive, and test, the mathematical consequences
of that law.

The two towering geniuses of physics are Newton and Einstein — Newton, who invented calculus to implement his second law, and Einstein, who
realized our concepts of space and time depend on how fast one is moving
and on any nearby mass.3
We start our discussion with homage to Newton, and give a brief review
of the elements of calculus that we will need for our Introduction to Classical
Mechanics.

1.2

Calculus

Consider the curve described by the function f (x). For every smooth curve,
there will be a straight-line tangent to that curve at the point x (Fig. 1.1).
Now move to a neighboring point x+∆x where ∆x is a very small increment.
The function will change to f (x) + ∆f (x).

tangent
Δx

Δ f(x)

θ

f(x)

x
Fig. 1.1 Tangent to the curve f (x) at the point x, and increment ∆f (x) in the function
when x increases by ∆x.

The angle that the tangent to the curve makes with the x-axis (the slope)

3 Maxwell

and Schră
odinger, with their equations, are not far behind.

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3

Introduction

is given by
tan θ =

∆f (x)
∆x

; Limit ∆x → 0

(1.1)

where this expression is exact in the limit that the displacement ∆x becomes
vanishingly small. This quantity is called the derivative of f (x)
∆f (x)
df (x)
≡ Lim ∆x→0
dx
∆x

; derivative

(1.2)

Given the value of f (x1 ), one can obain the value f (x) by stepping along
the curve (Fig. 1.2)
N

(∆f )i

f (x) = f (x1 ) +

; ∆x ≡

i=1

(x − x1 )
N

(∆f )i = f (x1 + i∆x) − f [x1 + (i − 1)∆x]

f(x)

f(x1 )

x1
Fig. 1.2

(1.3)

x

Δx

From f (x1 ) to f (x) and area under the curve.

This is rewritten as
N

f (x) = f (x1 ) +
i=1

(∆f )i
∆x
∆x

(1.4)

The limit N → ∞, which implies ∆x → 0, serves to define the integral

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BC: 11750 - Introduction to Classical Mechanics

4

MechIntroroot

Introduction to Classical Mechanics

I(x, x1 )
N

f (x) − f (x1 ) = Lim ∆x→0

(∆f )i
∆x
∆x

i=1

≡ I(x, x1 )

(1.5)

Since
f (x1 + i∆x) − f [x1 + (i − 1)∆x]

∆x

Lim ∆x→0

=

df (xi )
dx

(1.6)

and as ∆x → 0 we call it the differential dx, one has
x

df (u)
du
du

I(x, x1 ) =
x1

; integral

(1.7)

where we have simply re-labeled the dummy integration variable.
Stepping past x with a finite interval ∆x gives
I(x + ∆x, x1 ) = I(x, x1 ) +

∆f (x)

∆x
∆x

(1.8)

which is rewritten as
∆f (x)
I(x + ∆x, x1 ) − I(x, x1 )
=
∆x
∆x

(1.9)

The limit ∆x → 0 then gives
dI(x, x1 )
df (x)
=
dx
dx

(1.10)

The derivative of the integral with respect to its upper limit is the integrand
evaluated at that upper limit.
It is evident from Fig. 1.2 that as the width of each rectangle decreases,
and the number of the rectangles increases, one calculates the area under
the curve f (x)
x

f (u)du

A(x, x1 ) =

; area

(1.11)

x1

Exactly as above, the derivative of this area with respect to x is just the
function f (x) itself
dA(x, x1 )
= f (x)
dx

(1.12)

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5

Introduction

In summary, calculus relates the infinitesimal change in a function,
the slope, or derivative df (x)/dx, to the function itself f (x) − f (x1 ) =
x
x
[df (u)/du]du, and then to the area under the curve x1 f (u)du, both
x1
of the latter being obtained by integration, which involves summing the
infinitesimal elements.
1.3

Example

Let us give a simple example of how this works. It is observed empirically
that in the absence of air resistance, the velocity of an object that is dropped
from rest at the surface of the earth increases linearly with time.4 If z is
the distance the object falls, and dz/dt is the instantaneous velocity, then
dz
= gt
dt

; observation

(1.13)

where g, the “acceleration of gravity at the surface of the earth”, is a
constant given by
g = 9.8 m/sec2

; acceleration of gravity

(1.14)

The total distance the object falls after a time t is then given by integrating
this differential equation
t

z=g

1 2
gt
2

t dt =
0

; distance fallen

(1.15)

We can go back one step and write the rate of change of velocity, the
acceleration, as
dvz
=g
dt

; acceleration

(1.16)

Integration of this relation reproduces Eq. (1.13)
vz =

dz
=g
dt

t

dt = gt

(1.17)

0

Now note that Eq. (1.16) reads
d
dt

dz
dt

≡

d2 z
=g
dt2

; Newton

(1.18)

4 The original experiment is due to Galileo, who dropped different objects from the
Leaning Tower of Pisa.

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BC: 11750 - Introduction to Classical Mechanics

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Introduction to Classical Mechanics

Where we have introduced the second derivative—the derivative of the
derivative. This is the simplest form of Newton’s second law for the motion
of a body dropped at the surface of the earth. Note the interesting fact
that, as first observed by Galileo, the body’s mass does not enter into this
relation.
We can further manipulate these results. Consider
1 2
v = gz ≡ −U (z)

2 z

; gravitational potential

(1.19)

where U (z) = −gz is the gravitational potential per unit mass (remember that z is measured down). This gives rise to the statement of energy
conservation for an object released from rest at z = 0
1 2
v + U (z) = 0
2 z
1.4

; energy conservation

(1.20)

Units

In this book we use Standard International (SI) units, essentially m.k.s.
Masses are measured in kilograms (kg), distances in meters (m), and forces
in newtons, where
1 newton ≡ 1 N =

1 kg-m
1 sec2

(1.21)

Energy is measured in joules, where

1 kg-m2
(1.22)
1 sec2
To write Newton’s laws in three dimensions, we must employ vectors,
to which we next turn our attention.
1 joule ≡ 1 J = 1 N-m =

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Chapter 2

Vectors

We summarize the essential elements of vectors that we will need in our
development.

2.1

Vector

A vector can be characterized by its cartesian components (Fig. 2.1)

v : (vx , vy , vz )

(2.1)

z

vz

v
vy

vx

y

x
Fig. 2.1

A vector v in a cartesian coordinate system, with components (vx , vy , vz ).

• Vectors can be added by adding their components
a + b = (ax + bx , ay + by , az + bz )

(2.2)

• Vectors can be multiplied by a scalar through multiplication of their
7

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Introduction to Classical Mechanics

components
λa = (λax , λay , λaz )
2.2

(2.3)

Scalar Product

The scalar product (“dot product”) of two vectors is given by
a · b ≡ ax b x + ay b y + az b z

; scalar product

(2.4)

; (length)2

(2.5)

• The square of the length of a vector is
|v |2 ≡ v · v = vx2 + vy2 + vz2

• It follows from the above that (see Fig. 2.2)
(a − b )2 = a 2 + b 2 − 2a · b

b

(2.6)

a-b

a
Fig. 2.2

The quantity a − b.

From the law of cosines
(a − b )2 = a 2 + b 2 − 2|a ||b | cos θ

(2.7)

a · b = |a ||b | cos θ

(2.8)

Hence

We can alternatively express the vector v as a linear combination of
orthonormal unit vectors1
v = vx x
ˆ + vy yˆ + vz zˆ
x
ˆ2 = yˆ2 = zˆ2 = 1
1 In

xˆ · yˆ = x
ˆ · zˆ = yˆ · zˆ = 0

this work, a hat consistently denotes a unit vector.

(2.9)

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9

Vectors

2.3

Vector Product

The vector product (“cross product”) of two vectors is given by
a × b ≡ (ay bz − az by ) x
ˆ + (az bx − ax bz ) yˆ + (ax by − ay bx ) zˆ
; vector product

(2.10)

This is most conveniently written as a determinant
x
ˆ yˆ zˆ
a × b ≡ det ax ay az
bx by bz
= det

a a
a a
ay az
x
ˆ − det x z yˆ + det x y zˆ
bx by
bx bz
by bz

(2.11)

The last line is obtained through an expansion in minors.

The cross product has the following properties:
• The dot product of a × b with either of its constituents vanishes
a · (a × b) = b · (a × b) = 0

(2.12)

This is obtained either from the definition in Eq. (2.10), or from
the fact that these expressions lead to a determinant with two
identical rows in Eq. (2.11). As a consequence, the vector a × b is
perpendicular to both a and b (see Fig. 2.3).

a b
b

a

Fig. 2.3

The cross product a × b.

• The direction of a × b is given by the right-hand rule
Put the fingers of your right hand along a, curl them into
b, then your thumb points along a × b.

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