Analytical Mechanics for Relativity and Quantum Mechanics

634 Pages·2011·86.97 MB·English

by  Oliver Johns

ParIt
IntrodTuhcTetr iaodni:t ional
Theory
1
Analytical Mechanics for Relativity and
Quantum Mechanics
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AnalytMieccahla nics
for
Relativity and Quantum Mechanics
SecEodnidt ion
Oliver Davis Johns
San Francisco State University
OXFORD
UNIVERSITY PRESS
OXFORD
\TNIVBllSrrY PUSS
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First published 2005
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Second Edition published 2011
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1
BASIC DYNAMICS OF POINT PARTICLES AND COLLECTIONS
Modemme chabneigcwisin thths e p ubliicn1a 6ti8o7nI soaNfae cw tPorni'nsca ipni a,
extensoifto hwneo rokfh ipsr edecneostsaGobarllsyia, l nDedeo s cathratatel sl,o ws
hitmoe xplmaaithne matiwchaahltecl aytl lh"seS ysotfthe eWm o rltdhm"eo: ti ons
planmeotosnc,so ,m etitdseT,sh .the r ee 'i'\oxrLi aowomsfMs o,ti oinnt" h Per ionf cipia
(Newton,
1729)
are:
LaIw: in
Every boey perseveres its state of rest, or of uniform motion in a right
is forces
line, unless it compelled to change that state by impressed thereon.
LaIwI : is im­
The alteration of motion ever proportional to the motive force
is is
pressed; and made in the direction of the right line in which that force
impressed.
LawI ll: is
To every Action there always opposed an equal Reaction: or the
mutual actions of two bodies upon each other are always equal, and directed
to contrary parts.
Theasxei ormesft eothr eg enebreahla ovfai" obro dIy.it"cs l efraormN ewteoxn­'s
ampl(epsr ojaet cotppil,la encseo,tm sea,st tso,in nteh )se a msee ctitohhnae int t ends
thesbeo dtiobe esm acrosocrodpioinbcaj,r eyc ts.
1
Buetl sewNheewrtero enf teotr hs"e p articloefbs o diienws a"yt sh astu ggest
ana tothmeiocir nwy h itchhpe r imietilveem,eo nbtjaearcryste ms a ilnld,e structible,
"somlaisdhs,ay, r idm,p enemtroavbalpbeal,re t iThcelsees w.h"aw tewill catlhle
ofN ewtonian phsyasoyifstc hset.aremh a'Nt'e,thwp etrimsoient Piavre­
point particles
ticlesS obleaiirdninesgc , o mpahraarbthdlaeynra npyo roBuosdci oemsp ounded
theemv;es nov ehrya radsn evteowr e aorrb reianpk i enceoos r;d inarbyeo fi Pnogw er
abtlode i vwihdaGeto hdi mselofn iemnt a hfiderse Ct r eation."
Thper escheanpttw eirbl elg wiintht hases umpttihoNanet w tthorne'axieso ms
reffuenrd ametnott ahleplsoyeip natr tiAftcelred se.r ithveli anwgos fm omentaunm­,
gulmaorm entaunwmdo, r- ekneforrgp yo ipnatrti cwleew sis,lh lot wh agti,vc eenr tain
plauasniudbn liev earcscaelapldtyde idt aixoinoeamslss ,e nthteis aalmlleay wc sa n
bep rovteoad p ptlomya croscboopdiicceo sn,s iadsce orlelde ocftti hoeenl se mentary
poipnatr ticles.
1.1 Newton's Space and Time
Befodries cuthsels aiwonsfmg o tioonfp oint wmeam susscetos n,st ihdseep raa cned
timien w hithcahmt o tiotna kesF opNrle awcteso.pn a,wc aels o giacnapdlh lyys ically
distfrionmthc etm asstheamsti gohctc uiptSy.p apcreo vais dteadat bisco,al nudt e,
independent rreesfeptreowec hntic acelhp l aw rititchl e apnmodos tiiotinosn s were
1See query 31, page 400 of Newton (1730).
3
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4 BASIC DYNAMICS OF POINT PARTICLES AND COll.ECTIONS
to be measured. Space could be perceived by looking at the fixed stars which were
presumed to be at rest relative to it. Newton also emphasized the ubiquity of space,
2
comparing it to the sensorium of God.
Newton thought of time geometrically, comparing it to a mathematical point moving
steadily along a straight line. As witb space, the even flow of time was absolute and
independent of objects. He writes in the Principia, "Absolute, true and mathematical
time, of itself, and from its own nature, flows equably without relation to anything
3
extemal."
In postulating an absolute space, Newton was breaking with Descartes, who held
that the proper definition of motion was motion witb respect to nearby objects. In the
Principia, Newton uses the ex.ample of a spinning bucket filled with water to argue for
absolute motion. If tbe bucket is suspended by a rope from a tree limb and then twisted,
upon release the bucket will initially spin rapidly but the water will remain at rest. One
observes that the surface of the water remains flat. Later, when tbe water has begun
to rotate witb the bucket, the surface of the water will now be concave, in response
to the forces required to maintain its accelerated circular motion. If motion were to
be measured witb respect to proximate objects, one would expect the opposite obser­
vations. Initially, there is a large relative motion between the water and the proximate
bucket, and later the two have nearly zero relative motion. So the Cartesian view would
predict inertial effects initially, with the water surface becoming flat later, contrary to
observation.
Newton realized that, as a practical matter, motion would often be measured by
reference to objects rather than to absolute space directly. As we discuss in Section
16.1, the Galilean relativity principle states that Newton's laws hold when position is
measuted with respect to inertial systems that are either at rest, or moving with constant
velocity, relative to absolute space. But Newton considered these relative standards to
be secondary, merely stand-ins for space.
Nearly the opposite view was held by Newton's great opponent, Leibniz, who held
that space is a "mere seeming thing" and that the only reality is the relation of objects.
Their debate took the form of an exchange of letters, later published, between Leibniz
4
and Clarke, Newton's surrogate. Every student is urged to read them. The main diffi­
culty for the modem reader is the abundance of theological arguments, mixed almost
inextricably with the physical ones. One can appreciate the enormous progress that
has been made since the seventeenth century in freeing physics from the constraints
of theology. In the century after Newton and Leibniz, their two philosophical traditions
continued to compete. But the success of the Newton5 ian method in explaining experi­
ments and phenomena led to its gradual ascendency.
2Seventeenth century physiology held that the information from human sense organs is collected in a
"'sensorium" which the soul then views.
3Newton's ideas about time were possibly influenced by those of his predecessor at Cambridge, Isaac
Barrow. See Chapter 9 ofWhitrow (1989).
4The correspondence is reprinted, with portions of Newton's writings, in Alexander (1956).
5Lefunizian ideas continued to be influential, however. The great eighteenth centruy mathematician Euler,
to whom our subject owes so much., pubJished in 1768 a widely read book, Letters Addressed to a Gernum
Princess, in which he explained the science of his day to the lay person (Euler, 1823). He felt it necessary to
SINGLE POINT PARTICLE 5
Newton's space and time were challenged by Mach in the late nineteenth century.
Mach argued, like Leibniz, that absolute space and time are illusory and that the only
6
reality is the relation of objects. Macb also proposed that the inertia of a particle is
related to the existence of other particles and presumably would vanish without them,
Mach's Principle.
an idea that Einstein referred to as
Einstein's special relativity unifies space and time. And in his general relativity the
metric of the combined spacetime becomes dynamic rather than static and absolute.
General relativity is Machian in the sense that the masses of the universe affect the
local curvature of spacetime, but Newtonian in the sense that spacetime itself (now
represented by the dynamic metric field) is something all pervasive that has definite
properties even at points containing no masses.
For the remainder of Part I of the book, we will adopt the traditional Newtonian
definition of space and time. In Part II, we will consider the modifications of Lagrangian
and Hamiltonian mechanics that are needed to accommodate special relativity, in whicb
space and time are combined and time becomes a transformable coordinate.
1.2 Single Point Particle
In this section, we assume the applicability of Newton's laws to point particles, and
introduce the basic derived quantities: momentum, angular momentum, work, kioetic
energy, and their relations.
An uncharged point particle is characterized completely by its mass m and its po­
sition r relative to the origin of some inertial system of coordinates. The velocity v =
dr/dt and acceleration a= dv/dt are derived by successive differentiation. !ts momen­
tum (which is what Newton called "motion" in his second law) is defined as
p=mv (1.1)
Newton's second law then can be expressed as the law of momentum for point particles,
dp
f= (1.2)
dt
Since the mass of a point panicle is unchanging, this is equivalent to the more familiar
f=ma. The requirement that the change of momentum is "in the direction of the right
line" of the impressed force f is guaranteed in modem notation by the use of vector
quantities in the equations.
For the point particles, Newton's first law follows directly from eqn (1.2). When
f = 0, the time derivative of p is zero and so p is a constant vector. Nate that eqn
x
(1.2) is a vector relation. If, for example, the x-component of force f is zero, then the
x
corresponding momentum component P will be constant regardless of what the other
components may do.
devote some thhty pages of that book to refute Wolff, the chief proponent of Leibniz's philosophy. See also
the detailed defense of Newton's ideas in Euler. L. (17-48) "Reflexions sur l'Espace et le Terns [sic]" Me'moires
de l�imie des Sciences de Berlin,. reprinted in Series III, Volume 2 of Euler (1911).
6See Mach (1907). Discussions of Mach's ideas are found in Rindler (1977, 2001) and Misner, Thome
and Wheeler (1973). A review of the history of spacetime theories from a Machi.an perspective is found in
Barbour (1989, 2001). See also Barbour and Pl'i<!er (1995).
6 BASIC DYNAMICS OF POINT PARTICLES AND COll.ECTIONS
The angular momentum j of a point particle and the torque -r acting on it are de­
fined, respectively, as
(1.3)
j =rxp T=rxf
It follows that the law of angular momentum for point particles is
(1.4)
= -d----j--
T
dt
since
dj dr dp (1.5)
- = - xp+rx -=vxmv+rxf=O+ T
dt dt dt
In dr v
a time dt the particle moves a vector distance = dt. The work dW done by
force f in this time is defined as
(1.6)
dW=f·dr 2
This work is equal to the increment of the quantity (1/2)mv since
(4�v) 2
m (1mv
dW= f·v dt= dt) ·V= (dv) ·V= d ) (1.7)
Taking a particle at rest to have zero kinetic energy, we define the kinetic energy T as
T= 1 mv 2
(1.8)
2
with the result that a work-energy theorem for point particles may be expressed as
dW = dT or
(1.9)
f·V=­dT
dt
If the force f is either zero or constantly perpendicular to v (as is the case for purely
magnetic forces on a charged particle, for example) then the left side of eqn (1.9) will
vanish and the kinetic energy T will be constant.
1.3 Collective Variables
1,
Now imagine a collection of N point particles labeled by index n, with masses m
and positions
m2, ... , mN r1,r2, ... , rN.
The other quantities defined in Section 1.2 will be indexed similarly, with Pn =
m.v ., for example, referring to the momentum of the nth particle and f. denoting the
force acting on The total mass, momentum, force, angular momentum, torque, and
it.
kinetic energy of this collection may be defined by
N N N N N N
Lm• LP• Lf• 'l ETn T=LT•
M = 11=1 p = n=l F = n=l J = nI.:=l j. = 11=1 n=l
(1.10)
vector
Note that, in the cases of P, F, J, and T, these are sums. If a particular collection
consisted of two identical particles moving at equal speeds in opposite directions, for
example, Pw ould be zero.
TIIE LAW OF MOMEN'IUM FOR COLl.ECTIONS
7
-----
.,"' .;'.,,. •m2
,I/ . e m1 ... • ' ' '\ I
I
I I
I\ ,
I
I e,z
I
I
' I
,,
FIG. 1.1. A collection of point masses.
In the following sections, we derive the equations of motion for these collective
variables. All of the equations of Section 1.2 are assumed to hold individually for each
particle in the collection, with the obvious addition of subscripts n nto each quantnity to
label the particular particle being considered. For example, = dr /dt, an = dv /dt,
n n n n n n n
P = m V , f = dp /dt, f = ffl iln, etc. Vn
1.4 The Law of Momentum for Collections
We begin with the law of momentum. Differentiation of the sum for Pin eqn (1.10),
n n
using eqn (1.2) in the indexed form dp /dt = f , gives
N N n N
dP d n dp n
- = -LP = L- = Lf =F
dt dt dt
n=l n=l n=l
(1.11)
The time rate of change of the total momentum is thus the total force.
n
But the force f on the nth particle may be examined inext more detail. Suppose that
it can be written as the vector sum of an external force r,. ) coming from influences
operating on the collection from outside it, and an internal force t<,.mt) consisting of all
forces that cannot be identified as external, such as forces on particle n coming from
collision or other interaction with other particles in the collection. For example, if the
collection were a globular cluster of stars (idealized here as point particles!) orbiting a
galactic center, the external force on star n would be the gravitational attraction from
the galaxy, and the internal force would be the gravitational attraction of the other stars
in the cluster. Thus
ext
fn = fnext ) + fnin t) and, correspondingly, F = p( ) + p(int)
where (1.12)
I:N �=> N
ext
p< > = and p(int) --�" fnin t) (1.13)
n=l n=l