vectors and matrices - springer link

A

Vectors and Matrices

Vector notation provides a compact representation of the equationsdealing with motion in multiple dimensions. In simple linear motion,an object can be characterized by its position x in some reference sys-tem. For objects that are constrained to move in a plane, the object ischaracterized by two numbers x and y that specify its position in theplane. By convention, we can write the position as (x,y), where the orderis important. In general, the point (y,x) is not the same as the point (x,y).For objects that move through three dimensions, we now require threenumbers to specify location (x,y,z). If we had more dimensions, we couldcontinue analogously.

So, in some basic sense, a vector is an ordered list of numbers. Onemight imagine a fruit vector, in which each component of the vector rep-resents the number of apples, bananas, grapefruit, etc., that one pos-sesses. In the pharmaceutical industry, chemists construct quantitativestructure/activity relationship (QSAR) vectors for small molecules thatcapture their chemical properties like acidity and polarity and may havemillions of components. In these examples, there is no particular relationbetween the components of the vector: one obviously cannot mix applesand bananas. In physics applications, however, we make a further restric-tion on the vectors that we utilize: the components must have the samedimensionality. This means that if x has the units of a length, then so willy and z.

In this text, and most printed matter, we utilize the notation that boldfacecharacters represent vectors. Hence, we will write r = (x,y,z), which hasthe obvious advantage that we can replace several characters with justone. On the blackboard, or in homework submissions, it is more commonto utilize an arrow atop the character: �r = (x,y,z). This style is not nearlyas aesthetically pleasing as the use of boldface but is a practical solution tothe problem that it is quite difficult and time consuming to draw boldfacecharacters by hand.

© Mark A. Cunningham 2015M.A. Cunningham, Neoclassical Physics, Undergraduate LectureNotes in Physics, DOI 10.1007/978-3-319-10647-2

353

354 Vectors and Matrices

One can perform algebraic manipulations on vectors. The scalar productof a number and a vector is defined as

αx = (αx1,αx2, . . . ,αxn),

where the vector is assumed to have n components. One can add vectors:

x+ y = (x1 + y1,x2 + y2, . . . ,xn + yn),

provided that the vectors have the same dimensionality. There is also anadditive inverse −x:

x+ (−x) = 0,where the inverse is obtained by negating all of the components of x.

There are two means of multiplying vectors. The first is the inner productand results in a scalar:

x · y = x1y1 + x2y2 + · · ·+ xnyn

The inner product of a vector with itself produces the square of the mag-nitude of the vector:

x · x ≡ |x|2 = x2,

where we utilize the common notation that the magnitude of a vector iswritten in italics. In places in the text where this notation may be confus-ing, we utilize the first form.

The second vector product is the outer product, which results in a vec-tor. Restricting ourselves to three dimensions for the moment, the outerproduct is also known as the cross product:

x× y =((x2y3 − x3y2), (x3y1 − x1y3), (x1y2 − x2y1)

).

The cross product does not commute. In fact, x× y = −y× x. The vectors xand y are orthogonal to their cross product, by which we mean that:

x · (x× y) = 0 and y · (x× y) = 0.Higher order products can generally be reduced via one of several vectoridentities:

a · (b× c) = b · (c× a) = c · (a×b)a× (b× c) = (a · c)b− (a ·b)c

(a×b) · (c×d) = (a · c)(b ·d)− (a ·d)(b · c)

Students are undoubtedly familiar with the Cartesian representation ofvectors, which is predicated on some choice of coordinate system. As weshall see, the choice of coordinate system will not affect the physical in-terpretation of our equations, so any choice of coordinate system will suf-fice. As a practical matter, though, it will often prove highly useful to

§A Vectors and Matrices 355

choose coordinate systems to minimize the algebraic effort in solving theresulting equations.

To define a coordinate system, we will most often define a set of normal-ized, orthogonal basis vectors. In this text, we shall define these as follows:

x = (1,0,0) y = (0,1,0) z = (0,0,1).

For an arbitrary vector v = (vx,vy,vz), the inner product of the basis vectorswith the vector produces the components of v:

v · x = vx v · y = vy v · z = vz.

The basis vectors serve as projection operators.

It is not necessary for the basis vectors to be orthogonal to still serve asa basis. In fact, it is common in crystallography to choose a set of basisvectors that are aligned with the crystal lattice. In this case, the basisvectors ei are not orthogonal and are often not normalized to have unitmagnitude. To obtain the components of a vector in such a basis set, it isnecessary to define dual vectors ei :

(A.1) e1 =e2 × e3

e1 · (e2 × e3)e2 =

e3 × e1e1 · (e2 × e3)

e3 =e1 × e2

e1 · (e2 × e3).

The components of an arbitrary vector v are obtained by projections withthe dual vectors:

vi = v · ei .In the text, we shall want to differentiate and integrate vector quantities.We shall interpret the derivative of a vector as the derivative of its com-ponents:

d

dtx =

(dx1dt

,dx2dt

, . . . ,dxndt

).

The derivative is an operator that acts on the components of the vector.Similarly, we will treat integration as an operator acting on the compo-nents of a vector:

∫dtx =

(∫dt x1,

∫dt x2, . . . ,

∫dt xn

).

Vector calculus admits additional forms of integration. For example, onemay integrate a vector function along a directed path or over a surface.This gives rise to integrals of the following forms:

∫ds ·F and

∫dA ·B,


where F and B are vector functions. Both of these integrals result in scalarvalues. In Cartesian coordinates the infinitesimal path element ds can beresolved intocoordinates (See figure A.1):

ds = xdx + ydy + zdz.

Figure A.1. Consider a volume in space (light gray object) and a por-tion of the surface of that volume S. The direction of the surfaceis routinely taken to be directed outward and a path around theboundary ∂S of the surface is positively directed in a right-handedsense

Differential surface elements are constructed from the cross products ofthe unit vectors:

dA = xdx × ydy + ydy × zdz+ zdz× xdx= xdy dz+ ydxdz+ zdxdy.

Integrals over the volume take the following form:∫

d3rF,

where the volume element is composed by the triple product:

d3r = xdx · (ydy × zdz) = dxdy dz.

The result of this integration will be a vector function. Note that the tripleproduct is an invariant; one constructs the same invariant from permuta-tions of x, y and z.

For problems that have cylindrical symmetry, we shall frequently utilizea cylindrical coordinate system. In cylindrical coordinates, the z-axis re-mains the same as in the Cartesian system but we utilize polar coordinates


Figure A.2. A cylindrical coordinate system utilizes the same zvector as the Cartesian system but uses polar coordinates in the x-yplane

in the x-y plane. In physics, the most common notation for this is to de-note the azimuthal angle by θ and the radial distance by r. We shall fre-quently also utilize spherical coordinates, where θ is used by physicists tomean the polar angle and ϕ denotes the azimuthal angle. The distance rin spherical coordinates reflects the distance to the origin, not the distanceto the z-axis. To avoid these issues, we shall use the (somewhat nonstan-dard) symbolϕ to denote the azimuthal angle as measured from the x-axisand the (completely nonstandard) symbol ζ to denote the radial distanceto the z-axis, as illustrated in figure A.2.

It is somewhat more common to utilize the symbol ρ to denote the radialdistance to the z-axis but, in physics, we shall often use ρ to mean a den-sity. This results in the particularly awkward equation for computing thetotal charge from a charge distribution:

Q =∫

d3rρ(r) =∫ rb

ra

dρ

∫ 2π

0dϕ

∫ zb

za

dzρρ(ρ,ϕ,z),

where we intend to integrate the function ρ(ρ,ϕ,z) over the range of theradial coordinate ρ. We note though that our choice of ζ to replace rand ρ (which is the Greek form of the Latin r) is not completely unjus-tified. We are using the string of Latin characters x, y, z to denote posi-tions. The equivalent Greek characters are χ, υ and ζ. It it quite likelythat handwriting-challenged faculty will have difficulty differentiating χfrom x on the blackboard but it seems completely impossible for even themost diligent faculty to differentiate the Greek υ from the Latin v. It isimperative that we differentiate the radial position υ from the velocity v.


So, weighing these three choices, we are left with ζ as the radial coordinatein a cylindrical system. Another possibility would be the Greek symbolξ , which has the advantage that it is dramatically different than any Latincharacter and not used commonly. In the author’s mind, ζ is sometimes apseudonym for the direction z, not orthogonal to it, but the Greek ξ willprove overly taxing to the aforementioned handwriting challenged to useroutinely. So, we’ll stick with ζ as the radial coordinate in cylindrical sys-tems. Again, the choice of notation is partly an aesthetic choice. Studentsare ultimately free to make their own choices.

The same point r1 = (x1, y1, z1) can be expressed in both Cartesian andcylindrical forms. To avoid confusion, we shall only use the parenthesisform of the vector to mean Cartesian coordinates. That is,

r1 = (x1, y1, z1) = xx1 + yy1 + zz1

are different representations of the same point in space. In cylindricalcoordinates, we have the following:

r1 = (ζ1 cosϕ1,ζ1 sinϕ1, z1)

where ζ1 = (x21+y21 )1/2 and ϕ1 = tan−1(y1/x1). This latter definition implies

that the following relations hold:

cosϕ1 =x1

(x21 + y21 )1/2and sinϕ1 =

y1(x21 + y21 )1/2

.

The unit vectors in a cylindrical coordinate system are given by the fol-lowing:

ζ = (cosϕ,sinϕ,0) ϕ = (−sinϕ,cosϕ,0) z = (0,0,1).

These are related to the Cartesian unit vectors by the following:

x = ζ cosϕ − ϕ sinϕ ζ = x cosϕ + y sinϕ

y = ζ sinϕ + ϕ cosϕ ϕ = −x sinϕ + y cosϕ

The differential elements in cylindrical coordinates are given by the fol-lowing:

ds = ζ dζ + ϕζ dϕ + zdz

dA = ζ ζ dϕdz+ ϕdζ dz+ zζ dζ dϕ

d3r = ζ dζ dϕdz.

We will also make use of spherical coordinates, as depicted in figure A.3,in situations where we can exploit the symmetry. A point in space r1 ischaracterized by the distance to the origin r1 = (x21 + y21 + z21)

1/2, and twoangles. The azimuthal angle is measured from the x-axis and we denote it


Figure A.3. In a spherical coordinate system, points are representedby the radial distance to the origin, the polar angle θ and the az-imuthal angle ϕ

by ϕ1 = tan−1(y1/x1). The polar angle is measured from the z-axis and wedenote it by θ1 = tan−1(

√x21 + y21 /z1).

Unit vectors in the spherical coordinate system are given by the following:

r = (sinθ1 cosϕ1,sinθ1 sinϕ1,cosθ1)

θ = (cosθ1 cosϕ1,cosθ1 sinϕ1,−sinθ1)ϕ = (−sinϕ1,cosϕ1,0).

These are related to the Cartesian vectors by the following expressions:

x = r sinθ1 cosϕ1 + θ cosθ1 cosϕ1 − ϕ sinϕ1

y = r sinθ1 sinϕ1 + θ cosθ1 sinϕ1 + ϕ cosϕ1

z = r cosθ1 − θ sinθ1.

Differential elements are listed below:

ds = rdr + θ r dθ + ϕr sinθdϕ

dA = rr2 sinθdθdϕ + θ r sinθdr dϕ + ϕr dr dθ

d3r = r2 sinθdr dθ dϕ.

Just as vector calculus includes new forms of integration, there are addi-tional derivative operators that can be defined. We can define the symbol∇ to mean the following:

∇ ≡(∂

∂x,∂

∂y,∂

∂z

).


This mathematical quantity is not a vector in the sense that we have usedbefore but it does have ordered components. (Mathematicians will callthis entity a 1-form.) We can construct a vector from the gradient of ascalar function f :

∇f ≡(∂f

∂x,∂f

∂y,∂f

∂z

).

We can also define the (scalar) divergence of a vector function:

∇ ·F =∂Fx∂x

+∂Fy∂y

+∂Fz∂z

.

Finally, we can define the curl of a vector function:

∇×F =(∂Fz∂y−∂Fy∂z

,∂Fx∂z−∂Fz∂x

,∂Fy∂x−∂Fx∂y

).

The vector derivatives can be used to derive an alternative representa-tion of our various equations of motion. This is done primarily because,while it is exceptionally difficult to solve the resulting partial differen-tial equations, it is even more difficult to solve integral equations. Stu-dents may recognize that transforming between integral and differentialforms of equations amounts to the mathematical equivalent of pushingone’s peas about the plate in the hopes that one’s parents do not noticethat they have been eaten by the dog. The author cannot provide a cogentargument against such a proposal.

We have introduced matrix notation in the text primarily as a visual con-venience. If we define a vector x to have N components xj and a matrix Ato have M ·N components Aij , then the product y of the multiplication ofAx is defined as follows:

(A.2) yi =N∑

j=1

Aij xj .

In a more graphical display, we can write this in the following form:

(A.3)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y1y2y3...

yM

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A11 A12 A13 · · · A1NA21 A22 A23 · · · A2NA31 A32 A33 · · · A3N...

......

. . ....

AM1 AM2 AM3 · · · AMN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1x2x3...xN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

by which we mean that, for example,

y2 = A21x1 +A22x2 +A23x3 + · · ·+A2NxN .

Thus, Equations A.2 and A.3 are simply different representations of thesame mathematical quantity.


The matrix representation of a system of linear equations provides aconvenient and compact means for representing those equations. Thereare a number of courses on linear algebra that provide far more detail onmatrices than we shall attempt to cover here. We note that one can alsodefine multiplication of two matrices A and B in a systematic mannerwhere the components of AB are obtained by taking the inner productsof the columns of the right-hand matrix B with the rows of the left-handmatrix A, similar to the form displayed in Equation A.2. For this to makesense, the number of columns of the right-hand matrix B must be thesame as the number of rows of the left-hand matrix A. Matrix algebradiffers from the algebra of the real numbers in that the commutativeproperty is generally lost, i.e., AB � BA.

In the graphical representation of matrices, for Ax to make sense, the vec-tor x needs to have as many rows as the matrix A has columns. Conse-quently, x is represented as a column vector. The Mathematica syntaxdefines vectors as ordered lists and matrices as lists of lists. Perform-ing a matrix-vector multiplication implicitly uses the rule stated in Equa-tion A.2. For example the Mathematica script v={1,2,3} creates a listof three numbers that will be interpreted in the usual sense as a vector.Thus the script yhat={0,1,0}; yhat.v will produce the result 2, i.e., thesecond component of the vector v.

As a consequence, there is no need to define column vectors, although ina Mathematica script, we could define a single-column matrix as follows:

p = {{E/c}, {px}, {py}, {pz}}.The dual vector would then be the row vector pd={E/c,-px,-py,-pz}.When using the MatrixForm function, these two entities would appear inthe expected form. The dual vector can be constructed from the transposeof the product of the vector and the metric tensor. For the four-vector x,we have the following:

x =[ct −x −y −z

]=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ctxyz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

.

From which we can recover the invariant interval:

x · x =[ct −x −y −z

]⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ctxyz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦= (ct)2 − x2 − y2 − z2.

Note that the graphical representation of row vectors and column vectorsis not required if we simply utilize the definition of Equation A.2. Both


vectors and their duals are ordered lists of n elements. Their inner prod-uct can be defined without resort to a graphical display. Indeed, the astutestudent will recognize that we have refrained from illustrating vector ad-dition in the text with pictures of arrows linked head to tail. We haveused the precise definition that vector addition is defined by adding thecomponents of the vectors. This activity can be performed exactly wheresketching arrows cannot.

We referred to the metric tensor for the theory of Special Relativity, whichwe can write as the matrix with (1,−1,−1,−1) on the diagonal and zeroelsewhere. The metric tensor is a mathematical extension of the Euclideandistance between two points A = (x1, y2) and B = (x2, y2) on the plane:d = [(x2 − x1)2 + (y2 − y1)2]1/2. The Euclidean metric tensor is the identitymatrix, i.e., a matrix with ones on the diagonal and zero elsewhere. In thisinitial course, the mathematical structure underlying metric spaces is notnecessary to provide adequate descriptions of physical systems. It was,of course, essential for the development of Einstein’s General Theory ofRelativity.

We have made no effort in this text to distinguish four-vectors from three-vectors with different notation. This may be something of a controversialchoice but texts that make this distinction generally jump into notationalchoices that are difficult to understand initially. We made a consciouschoice in this text to use notation that is not compact, so that studentswill learn to appreciate the value of compact notation as they progress.Eventually, use of the Einstein summation convention will prove usefulbut not for introductory students. It cannot be expected that such stu-dents will grasp the subtlety that by pμ, with a Greek subscript, we meanthe four-momentum of something, whereas by pi , with a Latin subscript,we mean the ith component of the three-momentum of something. Weshall leave such intricacies to the instructors of subsequent courses.

B

Noether’s Theorem

We would now like to make the connection between symmetries of theequations of motion and conservation laws. To do so, we will first needto revisit a result first established by the mathematician Joseph-Louis La-grange.1 Lagrange suggested that one should consider the quantity T −U ,where T is the kinetic energy and U is the potential energy that we havedefined previously. Recall that T is a function only of the velocities andU is only a function of the positions. For concreteness, let us use the en-ergies defined for two masses interacting through the gravitational force;these were defined by Equations 2.26 and 2.27.

The Lagrangian function L = T − U is a function of six quantities: thethree components of the vector r2(t) − r1(t) and the three components ofthe vector v2(t) − v1(t). Consider taking the derivative of L with respectto one of the components of r2(t)− r1(t). As T does not depend upon theposition, we have, for the x-component:

∂Ld(r2(t)− r1(t))x

= − ∂Ud(r2(t)− r1(t))x

= GM1M2

|r2(t)− r1(t)|3(r2(t)− r1(t))x,(B.1)

where by the ()x notation we mean the x-component of the vector. Simi-larly, if we take the derivative of L with respect to the x-component of thevector v2(t)− v1(t), we would obtain the following result:

∂Ld(v2(t)− v1(t))x

=∂T

d(v2(t)− v1(t))x=

M1M2M1 +M2

(v2(t)− v1(t))x.(B.2)

1Lagrange was actually born in Turin, Italy and baptized as Guiseppe Lodovico Lagrangia in1736. He spent much of his working life in Berlin before moving to Paris in 1787, where heremained for the remainder of his life. Lagrange considered himself French at heart, signinghis name with the French spelling even in his youth.


363

364 Noether’s Theorem

What Lagrange observed is that if we take the time derivative of Equa-tion B.2 and add Equation B.1 we obtain the following:

(B.3)M1M2M1 +M2

d

dt(v2(t)− v1(t))x +G

M1M2|r2(t)− r1(t)|3

(r2(t)− r1(t))x

This is just the x-component of the equations of motion that we wrotedown in Equation 2.6 and we know that it vanishes!

Lagrange’s approach to mechanics seems somewhat contrived here but,in fact, offers a very general strategy for deriving the equations of mo-tion of physical systems, which is completely equivalent to the Newtonianmethodology that we have investigated previously. In using the Lagrangestrategy, one must define the kinetic energy T and the potential energy Uof the system in terms of whichever variables are convenient and then theequations of motion can be systematically constructed by taking deriva-tives of the Lagrangian L. Suppose that we have a system defined by a setof variables {xi} = x1, . . . ,xn. The equations of motion for the xi are thenobtained from the following:

(B.4)∂L∂xi− d

dt

∂L

∂

[∂xidt

] = 0,

where there will be n total equations.

Exercise B.1. We just stated the result in Equation B.1. Considerusing our concise notation: r2(t) − r1(t) = r21 = (x21, y21, z21). Here,we would write U = −GM1M2/r21, where r21 = |r21|. Show that youdo indeed obtain the result found in Equation B.1

Exercise B.2. Likewise, we just stated the result in Equation B.2.Consider using our concise notation: v2(t)− v1(t) = v21 = (vx,vy,vz).Here, we would write T = M1M2v

221/2(M1 +M2), where v21 = |v21|.

Show that you obtain the result found in Equation B.2

Our introduction of Lagrange’s methodology allows us now to discussNoether’s ideas on conservation laws. Consider that the parameters thatdefine Lagrange’s function L are subjected to some continuous transfor-mation, like our Lorentz transformation. Then, the parameters can bethought of as themselves functions of another parameter like ζ: xi = xi(ζ).That is, if L is a function of x and its time derivative dx/dt, we would havethe following:

(B.5)dLdζ

=∂L∂x

∂x

∂ζ+

∂L∂dx/dt

∂dx/dt

∂ζ.

§B Noether’s Theorem 365

Now this quantity vanishes if Lagrange’s function L and, consequently,the equations of motion do not depend explicitly on ζ.

Noether defined the following quantity:

(B.6) η =∂L

∂(dx/dt)∂x

∂ζ.

Noether’s theorem is that the time derivative of η vanishes; η is a con-served quantity. We find explicitly that:

dη

dt=[d

dt

∂L∂dx/dt

]∂x

∂ζ+

∂L∂dx/dt

[d

dt

∂x

∂ζ

]

=[d

dt

∂L∂dx/dt

− ∂L∂x

]∂x

∂ζ= 0,(B.7)

where we have used the result that Equation B.5 vanishes to replace thesecond term on the right-hand side of the first line. Note that the quantityin the square brackets in Equation B.7 is just the equation of motion forthe parameter x. This vanishes by definition, completing the proof thatthe quantity η does not depend upon time and is thereby conserved. It iscommonly referred to as the Noether current.

For a free particle of mass m and velocity v, there is no potential energy,just kinetic energy. In this case, we find the Lagrangian is given by thefollowing:

L =mv2x + v2y + v2z

2,

where the vj are the components of velocity in a Cartesian coordinate sys-tem. The Lagrangian has no explicit dependence upon x, so here ζ = xand the corresponding Noether current is just

ηx =∂L∂vx

∂x

∂x=mvx.

This is the x-component of momentum and we have just demonstratedthat it is an invariant of the system, through the use of Noether’s theorem.

As a second example, consider the theory of Newtonian gravitation, wherethe Lagrangian has the following form:

L =M1M2M1 +M2

|v2 − v1|2 +GM1M2|r2 − r1|

.

Suppose that we rotate the coordinate system by an angle ζ around thez-axis. In the new (primed) coordinate system, we have the following:

r′2 − r′1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

cosζ(r2 − r1)x − sinζ(r2 − r1)ysinζ(r2 − r1)x + cosζ(r2 − r1)y

(r2 − r3)z

⎞⎟⎟⎟⎟⎟⎟⎟⎠

366 Noether’s Theorem

and we have a similar expression for the velocity vector:

v′2 − v′1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

cosζ(v2 − v1)x − sinζ(v2 − v1)ysinζ(v2 − v1)x + cosζ(v2 − v1)y

(v2 − v3)z

⎞⎟⎟⎟⎟⎟⎟⎟⎠

The x-component of the Noether current is given by the following:

ηx =∂L

∂(v′2 − v′1)x∂(x′2 − x′1)x

∂ζ

= − M1M2M1 +M2

(v′2 − v′2)x(r′2 − r′1)y.

Similarly, the y-component of the Noether current is given by thefollowing:

ηy =∂L

∂(v′2 − v′1)y∂(x′2 − x′1)y

∂ζ

=M1M2M1 +M2

(v′2 − v′2)y(r′2 − r′1)x.

Each of these two components is individually conserved, so their sum willbe conserved as well. Thus the following quantity is conserved:

ηx + ηy =M1M2M1 +M2

((r′2 − r′1)x(v′2 − v′2)y − (r′2 − r′1)y(v′2 − v′2)x

)

=M1M2M1 +M2

((r′2 − r′1)× (v′2 − v′2)

)z

= Lz

This is just the z-component of the angular momentum vector definedin Equation 2.24. Hence, invariance to rotations about any axis leads toconservation of angular momentum about that axis.

It is a bit beyond most student’s present mathematical abilities but onecan define a Lagrangian function from which Maxwell’s equations can bederived. The application of Noether’s theorem results in a conservationlaw for electric charge:∫

Vd3rρ(r) +

∫

∂VdA · J(r) = 0.

We recognize this result as Equation 9.3. While electric charge conserva-tion is an experimental observation, it is also the result of an importantsymmetry that is reflected in the Maxwell equations.

Index

abstraction, 1, 4, 164acceleration, 22, 24, 134, 161, 177, 184,221, 227

definition, 14relation to force, 17

accuracy, 94definition, 90

Jean-Baptiste le Rond d’Alembertwave equation, 102, 132

algebramatrix, 128, 360, 361vector, 6, 354

α particle, 64, 239scattering, 66, 74–85

alphabetGreek, 7, 64, 244, 358Latin, 7

André-Marie Ampèremagnetic field, 251, 267

Ampère-Maxwell equation, 267amplifier, 316Anders Jonas Ångströmsolar spectrum, 157

angular momentumdefinition, 36

anode, 315antimatter, 243approximation, 2, 13, 253Dominique François Jean Arago, 251Argonne National Laboratoryneutrino event, 249

asteroid, 214Francis Astonisotopes, 230Nobel Prize (1922), 230

astronomical unit (AU), 43asymptote, 79asymptotic behavior, 61, 63, 123, 208,234

atomgraphical representation, 157nuclear structure, 86, 239size, 153

axis, see also coordinate systemazimuthal angle, 80definition, 357, 359

backspin, 183baryon, 119, 244baseball, 173–184pitch, 183–184

basketball, 8–11battery, 218, 253beamα particle, 82canal rays, 226cathode rays, 222

beats, 138Daniel Bernoullivibrating string, 134

Jacob Bernoullistatistics, 320

Claude Louis Bertholletreversibility of chemical reactions, 352

William Bertozzirelativistic electrons, 127

β particle, see also electronJean-Baptiste Biotmagnetic field, 251

Biot-Savart equation, 257, 260, 267, 272Ludwig Eduard Boltzmannkinetic theory of gases, 333

boost, see also Lorentz transformRobert Boylegas law, 340

William Henry BraggNobel Prize (1915), 148x-ray diffraction experiments, 148

William Lawrence BraggNobel Prize (1915), 148x-ray diffraction theory, 148

Bertram N. Brockhouseneutron spectroscopy, 158Nobel Prize (1994), 158

Robert Brownrandom motion, 324


367

368 Index

Brownian motion, 285, 323–330bubble chamberinterpretation, 244–250operation, 241

calculus, 40, 52, 254fundamental theorem, 28, 32integral representation, 273vector, 355, 359

caloric fluid, 331canal rays, 226–232capacitancedefinition, 302

capacitor, 302, 303bypass, 314

Giovanni Domenico Cassini, 88cathode, 315cathode rays, see also electroncausality, 113, 290center of mass, 178coordinate origin, 34definition, 33motion, 33, 178, 189, 279

central limit theorem, see also statisticsCERN, 119–121James Chadwickneutron, 233–238Nobel Prize (1935), 238

changeup, see also baseballchannel rays, see also canal rayschaos, 215chargeatomic, 65conservation, 254, 297, 317current, 253definition, 55density, 253, 272, 317electric field, 66, 252electric force, 56, 217, 240electron, 223–225, 271magnetic, 252magnetic force, 219mass ratio, 228, 232storage, 301thermionic emission, 314transport velocity, 256

Jacques Charlesgas law, 340

chemical reaction, 347chemistry, 347reaction coordinate, 348transition state, 348

Stephen ChuNobel Prize (1997), 309optical molasses, 309

circuit, 257, 292, 296LR, 299LRC, 305multiply-connected, 297RC, 303

circular orbit, 29Benoît Paul Émile Clapeyronideal gas law, 340

CODATA, 239Claude Cohen-TannoudjiNobel Prize (1997), 309optical molasses, 309

Peter Collinson, 303Arthur Holly ComptonNobel Prize (1927), 236photon scattering, 236–238

conductivity, 253, 257conductordefinition, 255

conic section, 35, 58definition, 29

conservation, 11angular momentum, 26, 31, 36, 42,57, 178, 219

definition, 10eccentricity, 31energy, 171, 210momentum, 34, 197, 233, 241Noether current, 365

coordinate systemcenter of mass, 33definition, 6invariance, 9polar, 40

coordinatesCartesian, 6, 161, 356chemical reaction, 348cylindrical, 49, 260, 329, 357spherical, 168, 273, 358

Charles Augustin de Coulomb, 55electric force, 85, 86, 286inverse square law, 56torsion balance, 56

cross section, see also scatteringcrystal lattice, 148, 149, 342Andreas CuneausLeyden jar, 301

§I Index 369

current, 252–256induced, 294

curve fitting, 3, 46least squares, 95

curveball, see also baseball

Clinton Davissonelectron diffraction, 157Nobel Prize (1937), 158

Humphrey Davy, 293Louis Victor Pierre Raymond duc de

Broglieelectron wavelength, 158Nobel Prize (1929), 158

Lee de ForestAudion, 315

René Antoine Ferchault de Réaumur,301

Peter Debyeheat capacity, 346

deflectionα particle in an electric field, 76moving charge in magnetic field, 219of a spinning baseball, 180

density, 174diffraction, 144–154diffusion, 327–330dimensional analysis, 3, 22, 164, 174,256, 265

diode, 312Paul Diracclassical electrodynamics, 290delta function, 258

distributioncharge, 73electrical power, 295Gaussian probability, 91, 326mass, 49Maxwell-Boltzmann, 339probability, 320random, 82

Christian Dopplerfrequency shift of a moving source,119

Doppler shift, 119, 122, 309Paul Karl Ludwig Drudeelectrons in metals, 255, 296

Pierre-Louis Dulongheat capacity of metals, 343

eccentricity, 29Thomas Edison, 296

electrical meter patent, 315effusion, 337Felix Ehrenhaftsubelectron, 287

Albert Einstein, 10Brownian motion, 324Eigenzeit, 121general theory of relativity, 129heat capacity, 343relativistic mechanics, 124–128respect for mathematics, 44special theory of relativity, 114, 116temperature, 344

electric fieldcharged sphere, 278parallel plates, 224point charge, 66

electrodynamics, 289electron, 64, 65, 74, 93, 156, 217, 222,239, 253

cloud, 79, 86, 158diffraction, 158finite size, 74lepton, 119

energy, 60conservation, 32, 37definition, 32free, 350kinetic, 32, 37, 60potential, 32, 37, 60relativistic, 127

entropy, 332ephemeris table, 88equations of motion, 16, 25, 180electron in metal, 255

equilibrium, 350equipartition theorem, 336, 342error function, 93definition, 328

Leonhard EulerLagrange points, 195wave equation, 132

Paul Peter Ewaldgraphical solution to diffraction, 152

experimentβ decay, 240μ lifetime (CERN), 121π0 decay/constant light velocity

(CERN), 120

370 Index

Bertozzi relativistic electrons, 127Davisson-Germer electron diffraction,157

existence of the ν, 249flag waving, 2G. P. Thomson electron diffraction,157

Geiger-Marsden, 66, 81, 130Miller-Kusch, 337Millikan oil drop, 285Perrin, 324statistical analysis, 90Stern-Gerlach, 281Thomson cathode ray, 229

exponentialmass dependence of rockets, 199

Fagnanoelliptical arc length, 262

Michael Faradaymagnetic field lines, 252, 266magnetic induction, 292–296

fastball, see also baseballEnrico Fermineutrino, 241

Adolf Fickdiffusion, 330

fieldelectric, 66gravitational, 48magnetic, 257

Hyppolite Fizeauvelocity of light, 101

John Ambrose Flemingthermionic valve, 315

fluorescence, 66forceaerodynamic, 177attractive, 56central, 20, 56, 78conservative, 169Coulombic, 217definition, 16dissipative, 159, 244, 307, 309electromagnetic, 55gravity, 21inverse square, 24magnetic, 219repulsive, 56resistive, 164–167, 256strong nuclear, 85weak nuclear, 239

Léon Foucaultvelocity of light, 101

Jean Baptiste Joseph Fouriersine and cosine series, 135, 312

Benjamin Franklinelectrical novelties, 302

free energy, 350free-body diagram, 160frequencydefinition, 119

friction, 185–188Walter Friedrichx-ray diffraction, 147

functioncontinuous, 253of a continuous variable, 4piecewise continuous, 18vector, 356

Galileo Galilei, 23, 53, 88Jovian moons, 87

γ radiation, 64Carl Friedrich Gauss, 67Law of electric flux, 67mathematical proof, 133

Gauss’s law, 67–72, 315, 316Joseph Louis Gay-Lussacgas law, 340

geosynchronous orbit, 201Walther Gerlachmagnetic moment of the electron, 281

Lester Germerelectron diffraction, 157

Josiah Willard Gibbs, 319chemical potential, 349free energy, 350statistical mechanics, 333

Donald Glaserbubble chamber, 241Nobel Prize (1960), 241

glibness, 132, 133gold, 77Eugen Goldsteincanal rays, 226

golf, 159gravitationrestricted three-body problem,190–195

gravitational field, 52Henry Graysonruling engine, 155

James Gregory

§I Index 371

diffraction grating, 144grid electrode, see also triodeFrancesco Maria Grimaldidiffraction, 144

Marcel Grossmann, 44differential geometry, 129

Frederick Guthriethermal electron emission, 314

hadron, 244Edmund Halley, 189William Rowan Hamiltondynamics, 189

Serge HarocheNobel Prize (2012), 121

heat, 331Joseph Henrymagnetic induction, 292

David Hilbert, 10Walter Hohmanntransfer orbit, 201

Christiaan Huygens, 54wavelets, 145

idealization, 13, 27, 48, 75, 129, 135,162, 177, 186, 192, 200, 203, 214,254, 296

impact parameter, see also scatteringimpedancedefinition, 313

inductance, 300inductor, 300Jan Ingenhouszcoal dust particles, 324

insulatordefinition, 255

interference, 137, 140–144interpolation, 46, 62invariance, 10coordinate, 21definition, 10scale, 325

invariant interval, 106, 110, 125isotope, 230, 239

JupiterGalilean moons, 87

Kanalstrahlen, see also canal raysJohannes KeplerFirst law of planetary motion, 24, 30Second law of planetary motion, 42Third law of planetary motion, 42

kinematicequation, 15, 165, 326relativistic theory, 124–128, 235

Gustav Kirchhoffcircuit equations, 296–299, 313spectra of alkalis and earth-alkalis,155

Daniel Kirkwoodorbital resonance, 212

Felix Klein, 10Ewald Georg von KleistLeyden jar, 301

Paul Knippingx-ray diffraction, 147

knuckleball, see also baseballPolykarp KuschMaxwell-Boltzmann distribution, 337Nobel Prize (1955), 337

Joseph-Louis Lagrangeco-orbital points, 196dynamics, 189, 363wave equation, 134

Willis LambNobel Prize (1955), 337

Pierre-Simon marquis de Laplacefinite velocity of gravity, 290

large number limit, 83Joseph Larmorradiation by accelerated charges, 86

latitude, 87Max von LaueNobel Prize (1914), 148x-ray diffraction, 147

Lawrence Berkelery Laboratorypair production, 242

Lawrence Berkeley Laboratorystrange matter, 245

Leon LedermanΥ discovery, 96Nobel Prize (1988), 96

Adrien-Marie Legendreelliptic integrals, 262

Heinrich Friedrich Emil Lenzinductance, 299

lepton, 119, 243lightning, 55Gabriel LippmannNobel Prize (1908), 65

lodestone, 218, 269longitude, 87Hendrik Antoon Lorentz

372 Index

electromagnetic force, 219, 279Nobel Prize (1902), 105wave equation invariance, 105

Lorentz force, 219, 244, 318Lorentz transform, 105–114physical interpretation, 122

magic, 126magnetic fieldcurrent density, 268current loop, 260–267, 277equivalence, 269line segment, 258–260spinning electron, 272–277spinning sphere, 277torque, 280

magnetic permeability, 257Heinrich Magnusforce on spinning sphere, 178

Benoît Mandelbrotfractal geometry, 325

many body problem, 52, 333Simon MariusJovian moons, 87

Mars, 8mass, 55definition, 17, 22

mass spectrometry, 230, 232mathematicsdifferential geometry, 129group theory, 128need for more, 1, 44, 54, 62representation, 1, 135, 275, 294, 361

Cato Maximillianlaw of mass action, 352

James Clerk Maxwellelectromagnetic theory, 101, 218, 267

mean, see also statisticsmechanicsstatistical, 319

meson, 119, 244meteorChelyabinsk, 163

R. C. MillerMaxwell-Boltzmann distribution, 337

William Hallowes Millercrystallographic indices, 150

Robert Millikancharge quantization, 287electron charge, 285Nobel Prize (1923), 285

Michael Minovitch

gravity assist, 207modulation, 139, 310molecular biology, 117momentumdefinition, 17electromagnetic field, 277photon, 235relativistic, 126

motor, 295Andrew Motte, 120muon, 121Pieter van MusschenbroekLeyden jar, 301

Napoleon, 218NASA, 11, 53, 122, 198GOES satellite, 36

Franz Ernst Neumann, 297neutrino, 249neutronlifetime, 93

Isaac Newton, 29Philosphiæ Naturalis Principia

Mathematica, 17, 120calculus, see also calculuscorpuscular theory of light, 146First law of motion, 85, 185flight of tennis balls, 178light through a prism, 154mechanics, 124, 128, 159Opticks, 100Second law of motion, 17Third law of motion, 20three-body problem, 189Universal gravitation, 19, 24, 37, 52

NIST, 342Emmy Noetherconservation theorem, 10, 103, 171,365–366

normal distribution, see alsodistribution, Gaussian

notation, 7, 15, 21, 24, 25, 34, 40, 77,353, 358

numerical solutionbaseball trajectory, 176–185electron trajectory, 244precession of magnetic moment, 280three-body problem, 192

Hans Christian Ørstedmagnetic field, 251

Georg Simon Ohm

§I Index 373

electrical circuits, 253, 296optical molasses, 308orbit, see also trajectory

partial derivativedefinition, 101

partial differential equation, 102, 128particle, 158, 242partition function, 334Wolfgang Paulineutrino, 241

pendulum, 307perfect differential, 28, 50period, 117, 142periodic motion, 42Jean Baptiste PerrinBrownian motion, 324Nobel Prize (1926), 324

Alexis-Thérèse Petitheat capacity of metals, 343

William D. PhillipsNobel Prize (1997), 309optical molasses, 309

Thomas Phippsmagnetic moment of hydrogen, 282

physical insight, 19, 59, 78, 116, 122physicsas an experimental science, 3definition, 1process, 1, 2terminology, 2vocabulary, 12

pion, 119Max Planckenergy-frequency relationship, 156,343

Nobel Prize (1918), 343Platomagnetism in ancient Greece, 217

plumbing, 291polar angle, 80definition, 359

polonium, 77, 237potentialelectric, 296

potential energy surface, 190Henry Powergas law, 339

power, 295precession, 192, 280precision, 2definition, 90

product, 347propagationof light, 118

proper time, see also Einstein, EigenzeitPtolemygeocentric model of the universe, 88

QSAR, 353quantizationcharge, 285energy, 156spin, 285

quantumdefinition, 1theory, 11, 86, 156, 217

radio receiver, 310radium, 66radon, 66randomsample, 320walk, 323

John William Strutt, third Baron ofRayleigh of Terling Place, Witham

Nobel Prize (1904), 174velocity-dependent force, 174

reactant, 347rectifier, 312, 315refraction, 100, 154relativity, 114representationatomic, 85, 157, 239

resistor, 296resonance, 212guitar string, 306LRC circuit, 306radio receiver, 310skyscraper, 306

rings of Saturn, 54rocket equation, 197–203Wilhelm Conrad RöntgenNobel Prize (1901), 156x-ray tube, 156

Ole Rømer, 88eclipses of Io, 88–89, 96–99, 115

Ernest Rutherford, 64α, β radiation, 64finite nuclear size, 85Nobel Prize (1908), 65nuclear model of the atom, 79, 270

satellite, 190

374 Index

Saturn, 53Félix Savartmagnetic field, 251

scatteringcross section, 80, 84elastic, 234impact parameter, 80

Melvin SchwartzNobel Prize (1988), 96

scientific priority, 87, 196semiconductor, 312sequencetime ordered, 114

Clifford Shullneutron diffraction, 158Nobel Prize (1994), 158

Le Système International d’Unités (SI),76

significance, 93, 98singular, 61, 63, 67, 275skew, see also statisticsMarian SmoluchowskiBrownian motion, 324

solenoid, 267solid angle, 80Arnold Sommerfeld, 44spacetime, 124event, 106

specific heat capacity, 341spectroscopy, 217spectrumdefinition, 154visible, 153

square root of time, 326standard deviation, see also statisticsstatistics, 90central limit theorem, 92inference, 94mean, 91skew, 91standard deviation, 91, 97variance, 91

Jack SteinbergerNobel Prize (1988), 96

Otto Sternmagnetic moment of the electron, 281Nobel Prize (1943), 281

stoichiometry, 351George Stokesresistive force on droplets, 165–167,286

Reynolds number, 173Strategic Defense Initiative (Star Wars),102

Gerald Sussmannchaotic solar system, 215

tankage factor, 199Brook Taylorvibrating string, 134

John Taylormagnetic moment of hydrogen, 282

Taylor, Brookseries expansion, 12

telecommunications, 310Nikolai Tesla, 296thermodynamics, 331first law, 331second law, 332

Benjamin Thompson, Reichsgraf vonRumford

work/energy equivalence, 331George Paget Thomsonelectron diffraction, 157Nobel Prize (1937), 158

Joseph John Thomson, 64cathode rays, 222electron, 65Nobel Prize (1906), 223plum pudding model, 65, 74, 225

Richard Townleygas law, 339

trajectory, 8, 12, 18, 20, 232, 247baseball, 174–185bats, 5bounded, 192definition, 5elliptical, 30, 38–48gravity assist, 208–211helical, 221hyperbolic, 30, 58–64, 78parabolic, 30random walk, 325time on orbit, 43–48

transducer, 311transformcoordinate, 34, 103, 273coordinate rotation, 110rotation, 196

transformer, 295transmutation, 64trend, see also statisticstriode, 316

§I Index 375

uncertainty, 2, 90, 91uncorrelated, 91unit systemBritish, 177conversion between, 198le Système International d‘unités(SI), 3

universe, 11, 20, 88, 243unphysical solution, 58

vacuum tubediode, 315triode, 316

variance, see also statisticsvectoralgebra, 6, 354calculus, 13, 289components, 6cross product, 25definition, 6, 353dual, 149, 355identity, 29, 38, 354light-like, 107normal to a surface, 68space-like, 107time-like, 107

velocityasymptotic, 208characteristic, 101definition, 13exhaust, 198gravity, 290relativistic, 125relativistic addition, 123sound, 115terminal, 166

Venus, 8, 21vibrationdamping, 307

viscosity, 174

Alessandro Voltabattery, 218, 253

voltaic pile, see also batteryRobert von Liebentriode, 316

Peter Waagelaw of mass action, 352

waveintensity, 142

wave equationone-dimensional, 101–106, 107point source, 135–137three-dimensional, 111–113two-dimensional, 108–111

wave vector, 150, 153wavelength, 150definition, 118

Victor Weisskopf, 247George Westinghouse, 296Charles Thomson Rees WilsonNobel Prize (1927), 236

wind, 184as a vector field, 67

David J. WinelandNobel Prize (2012), 121

Jack Wisdomchaotic solar system, 215

Ernest Wollandneutron diffraction, 158

workdefinition, 168energy equivalence, 169

Pieter ZeemanNobel Prize (1902), 105

Zeus, 87Zustandsumme, see also partition

function

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