This page intentionally left blank


ADVANCED MECHANICS AND GENERAL RELATIVITY
Aimed at advanced undergraduates with background knowledge of classical
mechanics and electricity and magnetism, this textbook presents both the particle dynamics relevant to general relativity, and the field dynamics necessary to
understand the theory.
Focusing on action extremization, the book develops the structure and predictions of general relativity by analogy with familiar physical systems. Topics ranging
from classical field theory to minimal surfaces and relativistic strings are covered in
a consistent manner. Nearly 150 exercises and numerous examples throughout the
textbook enable students to test their understanding of the material covered. A tensor manipulation package to help students overcome the computational challenge
associated with general relativity is available on a site hosted by the author. A link to
this and to a solutions manual can be found at www.cambridge.org/9780521762458.
joel franklin is an Assistant Professor in the physics department of Reed
College. His work spans a variety of fields, including stochastic Hamiltonian
systems (both numerical and mathematical), modifications of general relativity,
and their observational implications.



ADVANCED MECHANICS AND
GENERAL RELATIVITY
JOEL FRANKLIN
Reed College


CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521762458
© J. Franklin 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13

978-0-511-77654-0

eBook (NetLibrary)

ISBN-13

978-0-521-76245-8

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.


For Lancaster, Lewis, and Oliver




Contents

Preface
Acknowledgments
1 Newtonian gravity
1.1 Classical mechanics
1.2 The classical Lagrangian
1.2.1 Lagrangian and equations of motion
1.2.2 Examples
1.3 Lagrangian for U (r)
1.3.1 The metric
1.3.2 Lagrangian redux
1.4 Classical orbital motion
1.5 Basic tensor definitions
1.6 Hamiltonian definition
1.6.1 Legendre transformation
1.6.2 Hamiltonian equations of motion
1.6.3 Canonical transformations
1.6.4 Generating functions
1.7 Hamiltonian and transformation
1.7.1 Canonical infinitesimal transformations
1.7.2 Rewriting H
1.7.3 A special type of transformation
1.8 Hamiltonian solution for Newtonian orbits
1.8.1 Interpreting the Killing vector
1.8.2 Temporal evolution
2 Relativistic mechanics
2.1 Minkowski metric


vii

page xiii
xviii
1
1
2
2
5
7
9
12
16
21
25
26
30
33
34
37
37
39
43
46
46
52
56
57



viii

Contents

2.2 Lagrangian
2.2.1 Euclidean length extremization
2.2.2 Relativistic length extremization
2.3 Lorentz transformations
2.3.1 Infinitesimal transformations
2.4 Relativistic Hamiltonian
2.5 Relativistic solutions
2.5.1 Free particle motion
2.5.2 Motion under a constant force
2.5.3 Motion under the spring potential
2.5.4 The twin paradox with springs
2.5.5 Electromagnetic infall
2.5.6 Electromagnetic circular orbits
2.5.7 General speed limit
2.5.8 From whence, the force?
2.6 Newtonian gravity and special relativity
2.6.1 Newtonian gravity
2.6.2 Lines of mass
2.6.3 Electromagnetic salvation
2.6.4 Conclusion
2.7 What’s next
3 Tensors
3.1 Introduction in two dimensions
3.1.1 Rotation
3.1.2 Scalar

3.1.3 Vector (contravariant) and bases
3.1.4 Non-orthogonal axes
3.1.5 Covariant tensor transformation
3.2 Derivatives
3.2.1 What is a tensorial derivative?
3.3 Derivative along a curve
3.3.1 Parallel transport
3.3.2 Geodesics
4 Curved space
4.1 Extremal lengths
4.2 Cross derivative (in)equality
4.2.1 Scalar fields
4.2.2 Vector fields
4.3 Interpretation
4.3.1 Flat spaces

61
61
63
67
67
71
74
75
76
78
80
82
83
85

86
87
87
89
92
93
94
98
99
99
101
102
104
106
111
112
117
117
122
127
128
129
130
130
132
134


Contents


4.4 Curves and surfaces
4.4.1 Curves
4.4.2 Higher dimension
4.5 Taking stock
4.5.1 Properties of the Riemann tensor
4.5.2 Normal coordinates
4.5.3 Summary
4.6 Equivalence principles
4.6.1 Newtonian gravity
4.6.2 Equivalence
4.7 The field equations
4.7.1 Equations of motion
4.7.2 Newtonian deviation
4.7.3 Geodesic deviation in a general space-time
4.8 Einstein’s equation
5 Scalar field theory
5.1 Lagrangians for fields
5.1.1 The continuum limit for equations of motion
5.1.2 The continuum limit for the Lagrangian
5.2 Multidimensional action principle
5.2.1 One-dimensional variation
5.2.2 Two-dimensional variation
5.3 Vacuum fields
5.4 Integral transformations and the action
5.4.1 Coordinate transformation
5.4.2 Final form of field action
5.5 Transformation of the densities
5.5.1 Tensor density derivatives
5.6 Continuity equations
5.6.1 Coordinates and conservation

5.7 The stress–energy tensor
5.7.1 The tensor T µν for fields
5.7.2 Conservation of angular momentum
5.8 Stress tensors, particles, and fields
5.8.1 Energy–momentum tensor for particles
5.8.2 Energy and momenta for fields
5.9 First-order action
6 Tensor field theory
6.1 Vector action
6.1.1 The field strength tensor

ix

137
137
140
144
145
146
150
152
152
153
157
157
158
159
162
167
167

170
173
174
175
177
178
181
181
183
184
185
188
189
192
192
195
197
197
199
200
205
206
210


x

Contents

6.2 Energy–momentum tensor for E&M

6.2.1 Units
6.3 E&M and sources
6.3.1 Introducing sources
6.3.2 Kernel of variation
6.4 Scalar fields and gauge
6.4.1 Two scalar fields
6.4.2 Current and coupling
6.4.3 Local gauge invariance
6.5 Construction of field theories
6.5.1 Available terms for scalar fields
6.5.2 Available terms for vector fields
6.5.3 One last ingredient
6.6 Second-rank tensor field theory
6.6.1 General, symmetric free field equations
6.7 Second-rank consistency
6.7.1 Modified action variation
6.7.2 Stress tensor for H µν
6.7.3 Matter coupling
6.8 Source-free solutions
7 Schwarzschild space-time
7.1 The Schwarzschild solution
7.1.1 The Weyl method and E&M
7.1.2 The Weyl method and GR
7.2 Linearized gravity
7.2.1 Return to linearized field equations
7.3 Orbital motion in Schwarzschild geometry
7.3.1 Newtonian recap
7.3.2 Massive test particles in Schwarzschild geometry
7.3.3 Exact solutions
7.4 Bending of light in Schwarzschild geometry

7.5 Radial infall in Schwarzschild geometry
7.5.1 Massive particles
7.5.2 Light
7.6 Light-like infall and coordinates
7.6.1 Transformation
7.7 Final physics of Schwarzschild
7.7.1 Black holes
7.7.2 Gravitational redshift in Schwarzschild geometry
7.7.3 Real material infall
7.8 Cosmological constant

211
212
213
214
216
218
218
220
222
225
226
226
229
231
231
236
237
239
243

245
251
253
254
255
260
260
264
265
265
271
272
277
278
280
281
281
285
285
286
287
293


Contents

8 Gravitational radiation
8.1 Gauge choice in E&M
8.2 Gauge choice in linearized GR
8.3 Source-free linearized gravity

8.4 Sources of radiation
8.4.1 Source approximations and manipulation
8.4.2 Example – circular orbits
8.4.3 Energy carried away in the field
9 Additional topics
9.1 Linearized Kerr
9.1.1 Spinning charged spheres
9.1.2 Static analogy
9.1.3 General relativity and test particles
9.1.4 Kerr and the weak field
9.1.5 Physics of Kerr
9.2 Kerr geodesics
9.2.1 Geodesic Lagrangian
9.2.2 Runge–Kutta
9.2.3 Equatorial geodesic example
9.3 Area minimization
9.3.1 Surfaces
9.3.2 Surface variation
9.3.3 Relativistic string
9.4 A relativistic string solution
9.4.1 Nambu–Goto variation
9.4.2 Temporal string parametrization
9.4.3 A σ string parametrization (arc-length)
9.4.4 Equations of motion
9.4.5 A rotating string
9.4.6 Arc length parametrization for the rotating string
9.4.7 Classical correspondence
Bibliography
Index


xi

296
297
299
302
305
311
315
318
327
328
328
329
330
331
333
334
334
337
339
340
340
342
345
347
348
350
350
352

354
355
357
361
363



Preface

Classical mechanics, as a subject, is broadly defined. The ultimate goal of mechanics is a complete description of the motion of particles and rigid bodies. To find
x(t) (the position of a particle, say, as a function of time), we use Newton’s laws,
or an updated (special) relativistic form that relates changes in momenta to forces.
Of course, for most interesting problems, it is not possible to solve the resulting
second-order differential equations for x(t). So the content of classical mechanics
is a variety of techniques for describing the motion of particles and systems of
particles in the absence of an explicit solution. We encounter, in a course on classical mechanics, whatever set of tools an author or teacher has determined are most
useful for a partial description of motion. Because of the wide variety of such tools,
and the constraints of time and space, the particular set that is presented depends
highly on the type of research, and even personality of the presenter.
This book, then, represents a point of view just as much as it contains information and techniques appropriate to further study in classical mechanics. It is the
culmination of a set of courses I taught at Reed College, starting in 2005, that
were all meant to provide a second semester of classical mechanics, generally to
physics seniors. One version of the course has the catalog title “Classical Mechanics II”, the other “Classical Field Theory”. I decided, in both instantiations of the
course, to focus on general relativity as a target. The classical mechanical tools,
when turned to focus on problems like geodesic motion, can take a student pretty
far down the road toward motion in arbitrary space-times. There, the Lagrangian
and Hamiltonian are used to expose various constants of the motion, and applying
these to more general space-times can be done easily. In addition, most students
are familiar with the ideas of coordinate transformation and (Cartesian) tensors,

so much of the discussion found in a first semester of classical mechanics can be
modified to introduce the geometric notions of metric and connection, even in flat
space and space-time.

xiii


xiv

Preface

So my first goal was to exploit students’ familiarity with classical mechanics
to provide an introduction to the geometric properties of motion that we find in
general relativity. We begin, in the first chapter, by reviewing Newtonian gravity,
and simultaneously, the role of the Lagrangian and Hamiltonian points of view,
and the variational principles that connect the two. Any topic that benefits from
both approaches would be a fine vehicle for this first chapter, but given the ultimate
goal, Newtonian gravity serves as a nice starting point. Because students have seen
Newtonian gravity many times, this is a comfortable place to begin the shift from
L = 12 m v 2 − U to an understanding of the Lagrangian as a geometric object. The
metric and its derivatives are introduced in order to make the “length-minimizing”
role of the free Lagrangian clear, and to see how the metric dependence on coordinates can show up in the equations of motion (also a familiar idea).
Once we have the classical, classical mechanics reworked in a geometric fashion,
we are in position to study the simplest modification to the underlying geometry –
moving the study of dynamics from Euclidean flat space (in curvilinear coordinates)
to Minkowski space-time. In the second chapter, we review relativistic dynamics,
and its Lagrange and Hamiltonian formulation, including issues of parametrization
and interpretation that will show up later on. Because of the focus on the role of
forces in determining the dynamical properties of relativistic particles, an advertisement of the “problem” with the Newtonian gravitational force is included in this
chapter. That problem can be seen by analogy with electrodynamics – Newtonian

gravity is not in accord with special relativity, with deficiency similar in character
to Maxwell’s equations with no magnetic field component. So we learn that relativistic dynamics requires relativistic forces, and note that Newtonian gravity is
not an example of such a force.
Going from Euclidean space in curvilinear coordinates to Minkowski spacetime (in curvilinear coordinates, generally) represents a shift in geometry. In the
third chapter, we return to tensors in the context of these flat spaces, introducing
definitions and examples meant to motivate the covariant derivative and associated
Christoffel connection. These exist in flat space(-time), so there is an opportunity
to form a connection between tensor ideas and more familiar versions found in
vector calculus. To understand general relativity, we need to be able to characterize
space-times that are not flat. So, finally, in the fourth chapter, we leave the physical
arena of most of introductory physics and discuss the idea of curvature, and the
manner in which we will quantify it. This gives us our first introduction to the
Riemann tensor and a bit of Riemannian geometry, just enough, I hope, to keep
you interested, and provide a framework for understanding Einstein’s equation.
At the end of the chapter, we see the usual motivation of Einstein’s equation,
as an attempt to modify Newton’s second law, together with Newtonian gravity,
under the influence of the weak equivalence principle – we are asking: “under


Preface

xv

what conditions can the motion of classical bodies that interact gravitationally, be
viewed as length-minimizing paths in a curved space-time?” This is Einstein’s idea,
if everything undergoes the same motion (meaning acceleration, classically), then
perhaps that motion is a feature of space-time, rather than forces.
At this point in the book, an abrupt shift is made. What happened is that I
was asked to teach “Classical Field Theory”, a different type of second semester
of classical mechanics geared toward senior physics majors. In the back of most

classical mechanics texts, there is a section on field theory, generally focused on
fluid dynamics as its end goal. I again chose general relativity as a target – if
geodesics and geometry can provide an introduction to the motion side of GR in
the context of advanced mechanics, why not use the techniques of classical field
theory to present the field-theoretic (meaning Einstein’s equation again) end of
the same subject? This is done by many authors, notably Thirring and Landau
and Lifschitz. I decided to focus on the idea that, as a point of physical modelbuilding, if you start off with a second-rank, symmetric tensor field on a Minkowski
background, and require that the resulting theory be self-consistent, you end up,
almost uniquely, with general relativity. I learned this wonderful idea (along with
most of the rest of GR) directly from Stanley Deser, one of its originators and early
proponents. My attempt was to build up enough field theory to make sense of the
statement for upper-level undergraduates with a strong background in E&M and
quantum mechanics.
So there is an interlude, from one point of view, amplification, from another,
that covers an alternate development of Einstein’s equation. The next two chapters detail the logic of constructing relativistic field theories for scalars (massive
Klein–Gordon), vectors (Maxwell and Proca), and second-rank symmetric tensors
(Einstein’s equation). I pay particular attention to the vector case – there, if we
look for a relativistic, linear, vector field equation, we get E&M almost uniquely
(modulo mass term). The coupling of E&M to other field theories also shares similarities with the coupling of field theories to GR, and we review that aspect of
model-building as well. As we move, in Chapter 6, to general relativity, I make
heavy use of E&M as a theory with much in common with GR, another favorite
technique of Professor Deser. At the end of the chapter, we recover Einstein’s
equation, and indeed, the geometric interpretation of our second-rank, symmetric,
relativistic field as a metric field. The digression, focused on fields, allows us to
view general relativity, and its interpretation, in another light.
Once we have seen these two developments of the same theory, it is time (late
in the game, from a book point of view) to look at the physical implications of
solutions. In Chapter 7, we use the Weyl method to develop the Schwarzschild
solution, appropriate to the exterior of spherically symmetric static sources, to
Einstein’s equation. This is the GR analogue of the Coulomb field from E&M,



xvi

Preface

and shares some structural similarity with that solution (as it must, in the end,
since far away from sources, we have to recover Newtonian gravity), and we
look at the motion of test particles moving along geodesics in this space-time. In
that setting, we recover perihelion precession (massive test particles), the bending
of light (massless test particles), and gravitational redshift. This first solution also
provides a venue for discussing the role of coordinates in a theory that is coordinateinvariant, so we look at the various coordinate systems in which the Schwarzschild
space-time can be written and its physical implications uncovered.
Given the role of gravitational waves in current experiments (like LIGO), I
choose radiation as a way of looking at additional solutions to Einstein’s equation
in vacuum. Here, the linearized form of the equations is used, and contact is again
made with radiation in E&M. There are any number of possible topics that could
have gone here – cosmology would be an obvious one, as it allows us to explore
non-vacuum solutions. But, given the field theory section of the book, and the
view that Maxwell’s equations can be used to inform our understanding of GR,
gravitational waves are a natural choice.
I have taken two routes through the material found in this book, and it is the
combination of these two that informs its structure. For students who are interested
in classical mechanical techniques and ideas, I cover the first four chapters, and
then move to the last three – so we see the development of Einstein’s equation,
its role in determining the physical space-time outside a spherically symmetric
massive body, and the implications for particles and light. If the class is focused
on field theory, I take the final six chapters to develop content. Of course, strict
adherence to the chapters will not allow full coverage – for a field theory class,
one must discuss geodesic and geometric notions for the punchline of Chapter 7 to

make sense. Similarly, if one is thinking primarily about classical mechanics, some
work on the Einstein–Hilbert action must be introduced so that the Weyl method
in Chapter 8 can be exploited.
Finally, the controversial ninth chapter – here I take some relevant ideas from
the program of “advanced mechanics” and present them quickly, just enough to
whet the appetite. The Kerr solution for the space-time outside a spinning massive sphere can be understood, qualitatively and only up to a point, by analogy
with a spinning charged sphere from E&M. The motion of test bodies can be
qualitatively understood from this analogy. In order to think about more exotic
motion, we spend some time discussing numerical solution to ODEs, with an eye
toward the geodesic equation of motion in Kerr space-time. Then, from our work
understanding metrics, and relativistic dynamics, combined with the heavy use of
variational ideas throughout the book, a brief description of the physics of relativistic strings is a natural topic. We work from area-minimization in Euclidean spaces to


Preface

xvii

area-minimization in Minkowski space-times, and end up with the standard equations of motion for strings.
I have made available, and refer to, a minimal Mathematica package that
is meant to ease some of the computational issues associated with forming the
fundamental tensors of Riemannian geometry. While I do believe students should
compute, by hand, on a large piece of paper, the components of a nontrivial
Riemann tensor, I do not want to let such computations obscure the utility of the
Riemann tensor in geometry or its role for physics. So, when teaching this material,
I typically introduce the package (with supporting examples, many drawn from the
longer homework calculations) midway through the course. Nevertheless, I hope
it proves useful for students learning geometry, and that they do not hesitate to use
the package whenever appropriate.
A note on the problems in this book. There are the usual set of practice problems,

exercises to help learn and work with definitions. But, in addition, I have left
some relatively large areas of study in the problems themselves. For example,
students develop the Weyl metric, appropriate to axially symmetric space-times,
in a problem. The rationale is that the Weyl metric is an interesting solution to
Einstein’s equation in vacuum, and yet, few astrophysical sources exhibit this axial
symmetry. It is an important solution, but exploring the detailed physics of the
solution is, to a certain extent, an aside. In the end, I feel that students learn best
when they develop interesting (if known) ideas on their own. That is certainly the
case for research, and I think problems can provide an introduction to that process.
In addition to practicing the techniques discussed in the text, working out long,
involved, and physically interesting problems gives students a sense of ownership,
and aids retention. Another example is the verification that the Kerr solution to
Einstein’s equation is in fact a vacuum solution. Here, too, a full derivation of
Kerr is beyond the techniques introduced within the book, so I do not consider
the derivation to be a primary goal – verification, however, is a must, and can be
done relatively quickly with the tools provided. I have marked these more involved
problems with a ∗ to indicate that they are important, but may require additional
tools or time.
As appears to be current practice, I am proud to say that there are no new ideas
in this book. General relativity is, by now, almost a century old, and the classical
mechanical techniques brought to its study, much older. I make a blanket citation to
all of the components of the Bibliography (found at the end), and will point readers
to specific works as relevant within the text.


Acknowledgments

I would like to thank my teachers, from undergraduate to postdoctoral: Nicholas
Wheeler, Stanley Deser, Sebastian Doniach, and Scott Hughes, for their thoughtful advice, gentle criticism, not-so-gentle criticism, and general effectiveness in
teaching me something (not always what they intended).

I have benefitted greatly from student input,1 and have relied almost entirely
on students to read and comment on the text as it was written. In this context, I
would like to thank Tom Chartrand, Zach Schultz, and Andrew Rhines. Special
thanks goes to Michael Flashman who worked on the solution manual with me, and
provided a careful, critical reading of the text as it was prepared for publication.
The Reed College physics department has been a wonderful place to carry out
this work – my colleagues have been helpful and enthusiastic as I attempted to
first teach, and then write about, general relativity. I would like to thank Johnny
Powell and John Essick for their support and advice. Also within the department,
Professor David Griffiths read an early draft of this book, and his comments and
scathing criticism have been addressed in part – his help along the way has been
indispensable.
Finally, Professor Deser introduced me to general relativity, and I thank him for
sharing his ideas, and commentary on the subject in general, and for this book in
particular. Much of the presentation has been informed by my contact with him –
he has been a wonderful mentor and teacher, and working with him is always a
learning experience, that is to say, a great pleasure.
1

The Oxford English Dictionary defines a student to be “A person who is engaged in or addicted to study” – from
that point of view, we are all students, so here I am referring to “younger” students, and specifically, younger
students at Reed College.

xviii


1
Newtonian gravity

The first job, in studying general relativity, must be to establish the predictions

and limitations of Newtonian gravity, its immediate predecessor. So we begin
by studying the usual mechanical formulations of central potentials associated
with spherically symmetric central bodies. We shall study the orbital motion
of “test particles”, and in general, review the language and results of classical
gravity.
1.1 Classical mechanics
The program of classical mechanics is to determine the trajectory of a particle
or system of particles moving under the influence of some force. The connection
between force and motion is provided by Newton’s second law:
m x¨ (t) = F,

(1.1)

supplemented by appropriate initial or boundary conditions. The forces are provided (F might be made up of a number of different forces) and we solve the above
for x(t), from which any measurable quantity can be predicted.
As an approach to solving problems, Newton’s second law can be difficult to
work with. Given a generic force and multiple particles, a direct and complete
solution proceeding from (1.1) is often unattainable. So we content ourselves with
supplying less information than the full x(t) (sometimes, for example, we can
easily find x˙ (t), but cannot proceed to x(t)), or, we work on special classes of
forces for which alternate formulations of the second law are tractable. It is with
the latter that we will begin our work on Newtonian gravity – following a short
review of the Lagrangian formulation of the equations of motion given a force
derivable from a potential, we will see how the Lagrange approach can be used
to simplify and solve for the trajectories associated with the Newtonian central

1


2


Newtonian gravity

potential. From there, we will move on to the Hamiltonian formulation of the same
problem.

1.2 The classical Lagrangian
Here we will define the Lagrangian formulation of the fundamental problem of
classical mechanics: “Given a potential, how do particles move?” This section
serves as a short review of Lagrangians in general, and the next section will specialize to focus on Keplerian orbits in the classical setting – if we are to understand
the changes to the motion of particles in general relativity (GR), it behooves us to
recall the motion in the “normal” case. Our ultimate goal is to shift from the specific
sorts of notations used in introductory cases (for example, spherical coordinates),
to a more abstract notation appropriate to the study of particle motion in general
relativity.
As we go, we will introduce some basic tensor operations, but there will be more
of this to come in Chapter 3. We just need to become comfortable with summation
notation for now.

1.2.1 Lagrangian and equations of motion
A Lagrangian is the integrand of an action – while this is not the usual definition,
it is, upon definition of action, more broadly applicable than the usual “kinetic
minus potential” form. In classical mechanics, the Lagrangian leading to Newton’s
second law reads, in Cartesian coordinates:1
L=

1
m (v(t) · v(t)) −U (x(t)),
2


(1.2)

≡T

where we view x, y and z as functions of a parameter t which we normally
interpret as “time”. The first term is the kinetic energy (denoted T ), the second
is the potential energy. Remember, the ultimate goal of classical mechanics is to
find the trajectory of a particle under the influence of a force. Physically, we control
the description of the system by specifying the particle mass, form of the force or
potential, and boundary conditions (particle starts from rest, particle moves from
point a to point b, etc.). Mathematically, we use the equations of motion derived
from the Lagrangian, together with the boundary conditions, to determine the curve
x(t) = x(t) xˆ + y(t) yˆ + z(t) zˆ through three-dimensional space.
1

I will refer to the “vector” (more appropriately, the coordinate differential is the vector) of coordinates as
x = x xˆ + y yˆ + z zˆ , and its time-derivative (velocity) as v = dx
dt .


1.2 The classical Lagrangian

3

Extremization of an action
The Euler–Lagrange equations come from the extremization, in the variational
calculus sense, of the action:
S[x(t)] =

L(x(t), v(t)) dt.


(1.3)

We imagine a path connecting two points x(0) and x(T ), say. Then we define the
dynamical trajectory to be the unique path that extremizes S. Suppose we have an
arbitrary x(t) with the correct endpoints, and we perturb it slightly via
x(t) −→ x(t) + η(t). In order to leave the physical observation of the endpoints
unchanged, we require η(0) = η(T ) = 0. The action responds to this change:
T

S[x(t) + η(t)] =

˙
L(x(t) + η(t), x˙ (t) + η(t))
dt

0

(1.4)

T

≈

∂L
∂L
·η+
· η˙
L(x(t), x˙ (t)) +
∂x

∂ x˙

0

dt,

where the second line is just the Taylor expansion of the integrand to first order in η.
The unperturbed value S[x(t)] is recognizable from the leading term in the integrand,
all the rest is the change:
T

S = S[x(t) + η(t)] − S[x(t)] =

∂L
∂L
·η+
· η˙
∂x
∂ x˙

0

dt.

(1.5)

The small change η(t) is arbitrary, but once chosen, its time-derivative is fixed. We
would like to write the integrand of (1.5) entirely in terms of the arbitrary trajectory
perturbation, η(t), rather than quantities derived from this. We can use integration by
parts on the second term to “flip” the t-derivative onto the L-derivative:

T
0

∂L
· η˙ dt =
∂ x˙

T
0

d
dt

∂L
d
·η −η·
∂ x˙
dt

∂L
∂ x˙

dt.

(1.6)

The first term, as a total time-derivative, gets evaluated at t = 0 and t = T where
η vanishes. We can use (1.6) to make the replacement under the integral in (1.5):
d
∂L

· η˙ −→ −η ·
∂ x˙
dt
and this leaves us with

∂L
,
∂ x˙

(1.7)

S that depends on η(t) only:
T

S=
0

d
∂L
−
∂x
dt

∂L
∂ x˙

· η dt.

(1.8)



4

Newtonian gravity
Now extremization means S = 0, and the arbitrary value of η(t) allows us to set the
term in parentheses equal to zero by itself (that’s the only way to get S = 0 for
arbitrary η(t)).
As a point of notation, we use the variational derivative symbol δ to indicate that
we have performed all appropriate integration by parts, so you will typically see (1.8)
written as:
T

δS =
0

d
∂L
−
∂x
dt

∂L
∂ x˙

· δx dt,

(1.9)

where δx replaces η – this tells us that it is the variation with respect to x that is
inducing the change in S. For actions that depend on more than one variable that can

be varied, the notation makes it clear which one is being varied. In addition to this δS,
δx shift, we will also refer to the Euler–Lagrange equations from variation with
respect to x as:
d
∂L
δS
=
−
δx
∂x
dt

∂L
∂ x˙

,

(1.10)

the “variational derivative” of S with respect to x. Extremization is expressed by
= 0.
δS = 0, or equivalently in this case, δS
δx

Variation provides the ordinary differential equation (ODE) structure of interest,
a set of three second-order differential equations, the Euler–Lagrange equations of
motion:
d ∂L ∂L
−
= 0.

dt ∂v
∂x

(1.11)

In Cartesian coordinates, with the Lagrangian from (1.2), the Euler–Lagrange
equations reproduce Newton’s second law given a potential U :
m x¨ (t) = −∇ U.

(1.12)

The advantage of the action approach, and the Lagrangian in particular, is that
the equations of motion can be obtained for any coordinate representation of the
kinetic energy and potential. Although it is easy to define and verify the correctness
of the Euler–Lagrange equations in Cartesian coordinates, they are not necessary
to the formulation of valid equations of motion for systems in which Cartesian
coordinates are less physically and mathematically useful.
The Euler–Lagrange equations, in the form (1.11), hold regardless of our association of x with Cartesian coordinates. Suppose we move to cylindrical coordinates
{s, φ, z}, defined by
x = s cos φ

y = s sin φ

z = z,

(1.13)


1.2 The classical Lagrangian


5

then the Lagrangian in Cartesian coordinates can be transformed to cylindrical
coordinates by making the replacement for {x, y, z} in terms of {s, φ, z} (and
associated substitutions for the Cartesian velocities):
L(s, φ, z) = L(x(s, φ, z)) =

1
m (˙s 2 + s 2 φ˙ 2 + z˙ 2 ) − U (s, φ, z).
2

(1.14)

But, the Euler–Lagrange equations require no modification, the variational procedure that gave us (1.11) can be applied in the cylindrical coordinates, giving three
equations of motion:
0=

d ∂L ∂L
−
dt ∂ s˙
∂s

0=

d ∂L ∂L
−
dt ∂ φ˙
∂φ

0=


d ∂L ∂L
−
.
dt ∂ z˙
∂z

(1.15)

The advantage is clear: coordinate transformation occurs once and only once, in
the Lagrangian. If we were to start with Newton’s second law, we would have three
equations with acceleration and coordinates coupled together. The decoupling of
these would, in the end, return (1.15).

1.2.2 Examples
In one dimension, we can consider the Lagrangian L = 12 m x˙ 2 − 12 k (x − a)2 ,
appropriate to a spring potential with spring constant k and equilibrium spacing a.
Then the Euler–Lagrange equations give:
d ∂L ∂L
−
= m x¨ + k (x − a) = 0.
dt ∂ x˙
∂x

(1.16)

Notice that for a real physical problem, the above equation of motion is not
enough – we also need to specify two boundary conditions. We can phrase this
choice in terms of boundaries in time at t = t0 and t = tf (particle starts at 1 m
from the origin at t = 0 and ends at 2 m from the origin at t = 10 s), or as an

initial position and velocity (particle starts at equilibrium position with speed
5 m/s) – there are other choices as well, depending on our particular experimental
setup.
In two dimensions, we can express a radial spring potential as:
L=

1
1
m (x˙ 2 + y˙ 2 ) − k ( x 2 + y 2 − a)2 ,
2
2

(1.17)