Classical applications of the Klein–Gordon equation a
Pierre Gravel
Collège Militaire Royal du Canada, Département de Mathématiques et d’Informatique, Kingston, Ontario, Canada K7K 5L0 b
Claude Gauthier
Département de Mathématiques et de Statistique, Université de Moncton, Moncton, Nouveau-Brunswick, Canada E1A 3E9
Received 9 September 2008; accepted 4 February 2011 The quantum mechanical origin of the Klein–Gordon equation hides its capability to model many classi cla ssical cal sys system tems. s. We con consid sider er thr three ee exa exampl mples es of vib vibrat rating ing sys system temss who whose se mat mathem hemati atical cal descriptions lead to the Klein–Gordon equation. These examples are adapted to applications such as thee mo th moti tion on of su susp spen ende ded d ca cabl bles es an and d In Inca ca ro rope pe su susp spen ensi sion on br brid idge ges. s. We al also so di disc scus usss th thee correspondence between the classical and quantum settings of this equation as a way to provide an explanation of the concept of mass. © 2011 American Association of Physics Teachers. 10.1119/1.3559500 19/1.3559500 DOI: 10.11 2 t u
I. INTRODUCTION
It often happens that a model put forward to describe a given giv en phe phenom nomeno enon n is rep replac laced ed by a bet better ter model. model. It als also o happens that the former model can be used to solve problems that are sometimes very different from the one for which it was introduced. Such a situation is the case for the Klein– Gordon equation, which first appeared in the notebooks of Schröd Sch röding inger er in lat latee 192 1925 5 as an att attemp emptt to des descri cribe be t he de 1 Brogli Bro gliee wav waves es of the ele electr ctron on in the hyd hydrog rogen en ato atom. m. Because this equation does not take into account the electron’s spin, it led to predictions which did not match th thee experi2 mental men tal results results and was thu thuss put aside by Pau Pauli. li. Between April and September 1926, the Klein–Gordon equation independently pende ntly appeared appeared in seven papers to descr describe ibe rela relativi tivistic stic 3 massive particles without spin. But, in these early days of quantum mechanics, there arose insurmountable difficulties related to the interpretation of the function representing the probability density. We now know that these difficulties disappear in the quantum theory of fields. The aim of the present paper is to show how the Klein– Gordon equation is useful in the description of some vibrating systems in classical mechanics. Also, we shall relate the classical and quantum settings of this equation to explain the concept of mass. Because of their simplicity, we shall focus our pre presen sentat tation ion on vib vibrat rating ing sys system temss ma made de of flex flexibl iblee strings. Reca Re call ll th thee us usua uall as assu sump mpti tion onss fo forr or ordi dina nary ry vi vibr brat atin ing g strings. Each string is assumed to be of constant linear mass density and to be under uniform tension T . The strings are assumed to be perfectly flexible and free from any internal or external friction. The amplitude of the vibrations is assumed to be sufficiently small for the tension in the strings to stay constant in time. To describe the vibrations of a string, we use a Cartesian coordinate system whose x-axis is collinear with the position of the string at rest. The equation describing the vibrations in a plane can be deduced from Hamilton’s principle or Newton’s second law applied to an infinitesimal section of the string. If u x , t denotes the displacement from rest position of the point x at time t , it is easily shown that these the se vib vibrat ration ionss are loc locall ally y des descri cribed bed by the lin linear ear wav wavee equation 447
Am. J. Phys. 79 5, May 20 201 11
htttp: ht p:// //aa aapt pt.o .orrg/aj ajp p
= c2 x2u ,
1
where the positive constant c = T / represents the speed at which traveling waves move along the string.
II. THE BRACED STRING
Our first exam example ple of a class classical ical mechanics mechanics phenome phenom enon 4 thatt lea tha leads ds to the Kle Klein– in–Gor Gordon don equ equati ation on is not new new.. We discuss it briefly because it is easy to visualize many features of all phenomena described by this equation. We assume that the plane in which the string vibrates offerss res fer resist istanc ancee to the str string ing’’s mot motion ion.. Thi Thiss set settin ting g can be realized by embedding the string in a piece of elastic material, such as a thin flat sheet of rubber. In addition to the restoring force due to the string tension, there is then a restorin sto ring g for force ce due to the ela elasti sticc mat materi erial al sur surrou roundi nding ng the stri st ring ng.. Th Thee ne new w re rest stor orin ing g fo forc rcee is gi give ven n by − Yu x , t dx , where Y is the Young modulus of the material surrounding the string. The equation describing the vibrations of this elastically braced string is 2 t u
= c2 x2u − 2u ,
2
where c has the same meaning as in Eq. 1 and = Y / . Equation 2 is the Klei Klein–Gor n–Gordon don equat equation, ion, orig original inally ly used to describe a particle of spin zero and mass m = / c, where c is th thee sp spee eed d of li ligh ghtt an and d is Planc Planck’ k’ss cons constant tant divided by 2 . In this context the Klein–Gordon equation is not deduced from a mechanical model as is done here for the strings. Instead it is obtained by considering the relativistically invariant energy equation, E 2 = c2 p p2 + m2c4 ,
3
and applying the usual operator substitutions p → −i and E → i / t t , where p is the momentum of the particle. The resulting resulti ng differential operator equation acts on a wave func5 tion . If the string vibrates between two rigid supports separated by a distance L, we can show that the allowed frequencies are
© 20 201 11 Am Amer eric ican an Ass sso oci ciat atiion of Ph Phy ysi sics cs Tea each cher erss
447
n
=
nc L
2 2
+
n = 1,2, .. . .
4
These frequencies are larger than the corresponding ones for an ordinary string for which =0. The waves traveling along a semi-infinite elastically braced string driven from its finite end by a transverse alternating force are also different from those of an ordinary string. If is the driving frequency, then the phase velocity of the waves along the braced string is larger than c by the factor 1 / 1 − / 2. The additional elastic force thus increases the speed of the traveling waves along the string. However, the energy of these waves is transported according to their group velocity, which is smaller than c by the factor 1 − / 2. Because these velocities depend on the frequency, a wave shape composed of waves of different frequencies will change from its original shape. Conversely, this dependence of the velocities on the frequencies means that non-localized disturbances along the string may temporarily form localized packets. These properties apply to all solutions of the Klein–Gordon equation with 0. To illustrate this case for a simple model, consider a cable under tension linked at regular intervals to a beam by suspension rods of length a and negligible mass. Each element of the cable behaves as the bob of a pendulum attached to the beam by a suspension rod. The cable corresponds to the string of our model and the gravitational force plays the role of the elastic material which braces the string. In addition to the tension in the cable, there is a restoring force on the cable elements due to the pendulum configuration. If is the angle of the lateral motion of a suspension rod with respect to its equilibrium position and 1, this force is proportional to g sin g , where g is the acceleration of gravity at Earth’s surface. It is easy to show that the horizontal motion of the suspended cable obeys the Klein–Gordon equation with = g / a.
III. TWO PARALLEL STRINGS
Consider the coupling of transverse waves propagating along two parallel taut strings when the motion is restricted to the plane defined by the strings at rest. Examples corresponding to this model are the dividing nets used in the games of badminton and table tennis. In these examples the waves are vertical. For simplicity, we shall assume the coupling between the strings to be through a massless elastic material with Young modulus Y . The equations describing the motions of the coupled strings are derived in Appendix A. If u0 and u1 denote the displacements from their rest positions of the points on strings with string number 0 and 1, respectively, then 2 t u 0
= c2 x2u0 − 2u0 − u1
5a
2 t u 1
= c2 x2u1 − 2− u0 + u1 ,
5b
where c and have the same meanings as in Eq. 2. The functions u0 and u1 appear in Eqs. 5a and 5b. These equations decouple by introducing the variables v0 = u0 − u1 and v 1 = u 0 + u 1: 2 t v0
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6a
2 t v1
= c 2 x2v1 .
6b
The variable v0 represents the crosswise stretching of the elastic strip between the strings and describes vibrations resulting from forces without a net transverse component such that in pure modes, the center of mass at x does not move. In contrast, v1 describes the transverse motion of the elastic strip as a whole or, if preferred, the motion of the curve made up of the points always at mid-distance between the two vibrating strings. Its vibrations result from forces with a nonzero net transverse component. These vibrations correspond to two kinds of transverse waves propagating along the twostring structure, with only those of the second kind do so at the speed equal to that of waves along each of the two free strings. IV. THREE PARALLEL STRINGS
This example is the most complex of the three discussed in this paper. Its use is illustrated by rope bridges, the classical example of which are the three-rope Inca suspension bridges. These bridges were a part of the Inca road syste m 6 and a good example of Inca innovations in engineering. Such a rope bridge is till in use today near Cuzco, Peru, and crosses the 67 m of the Apurimac Canyon and is 36 m above the river. To build a three-rope bridge, a stone anchor is needed on each side of the canyon to be crossed. One massive cable made of woven grass is first installed over the canyon to link these anchors. Two additional cables acting as guard rails are then added above and on both sides of the first cable. They are then linked to the first cable with smaller gauge secondary suspension cables all of the same length. Given that the first massive cable is the footpath, the secondary suspension cables must be at an angle to the vertical sufficient to allow easy and secure crossing of the bridge. This angle must not be too large, otherwise the guard rails would be too low to be useful. Contrary to the four-cable suspension bridges which have a footpath made with plaited branches that can be used by livestock, these three-cable bridges are designed for the exclusive use of humans. Miniature versions of these bridges can sometimes be seen in playgrounds. Under normal conditions, only small relative displacements between the three main cables take place, and the secondary suspension cables are under tension because they support a part of the footpath’s weight. The analysis of this case is an extension of the analysis in Sec. III except that there are three strings to consider, and we have to consider two-dimensional motion of the cable elements because each of them will move in a plane perpendicular to the main cables. Another extension is that we now have to take gravity into account. Because gravity is a constant force, it can be incorporated by a redefinition of the zero of their vertical components. For this reason the differential equations are not affected by such a change. The analysis given in Appendix B shows that the overall motion decomposes into six independent quantities, each of them obeying its own differential equation separately from the others. Four of these quantities obey the ordinary wave equation, and the two others obey the Klein–Gordon equation. If the members in pairs of secondary cables form right angles when at rest, the quantities obeying the ordinary wave equation are the displacements of each guard rail perpendicular to the plane containing it and the footpath when at rest. P. Gravel and C. Gauthier
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Also obeying this equation in this case are the displacements of the curve defined by the centers of mass of all the pairs, each of which are made of one guard rail element and the corresponding footpath element in the same plane perpendicular to the main cables. The two quantities obeying the Klein–Gordon equation are the distances between the footpath and each of the guard rails. V. SYMMETRY BREAKING AND MASS
The study of the symmetries of the system formed by the two parallel strings and the elastic strip between them allows us to explain the origin of the inertial mass associated with this system. We first consider each of the two strings separately. Each string can be seen as made of infinitesimal length elements joined by elastic links. If the elasticity coefficient of these links is zero, then their motions are independent of one another. This independence means that the motion of each of these elements is subject to no restriction in the three-dimensional space whose origin coincides with the string element at rest. We say that the element has three degrees of freedom, and that its motions’ tangent space is three-dimensional. In this case the inertial mass of each of the elements is a physically irrelevant parameter because it can be eliminated from its equation of motion. If the elasticity coefficient of the links between infinitesimal elements of each string is nonzero, these elements lose their degree of freedom in the direction defined by the string because the tension is uniform within the string. Because we limit ourselves to linear elastic forces, the infinitesimal elements will move only in directions perpendicular to the straight line of the string at rest. The tangent space, that is, the space of all possible velocities of the particle, will now be two-dimensional. The motions are described by Eq. 1, where the inertial mass of the string elements partially determines the value of the parameter c. The mass parameter cannot be removed from the equations any longer, and its presence can be seen to depend on the existence of the elastic links between the string elements. The value of the mass parameter is thus a measure of the element’s resistance to follow the trajectory imposed by the elastic links. The original three-dimensional isotropy in the freedom of each string’s element is reduced or broken by the elastic links to a two-dimensional isotropic freedom. This symmetry breaking implies that a mass must be associated with these elements. We have seen that the vibrations of two coupled parallel strings decouple into two types of waves moving along them. The waves of one of these types correspond to motion in which the two strings move in unison and are described by Eq. 6b. This equation is the same as Eq. 1 and describes waves propagating along the system of two strings and the elastic strip. These waves are similar to the waves in each of the strings alone because the motion of the corresponding pairs of elements on the two strings has the same freedom as the elements of each string, in this case with respect to the straight line equidistant to the two strings at rest. Because the two strings have the same linear density and a combined tension that is twice that of each string, the speed at which waves move along this system is the same as the speed in each string. The other type of waves in the two coupled strings is given by Eq. 6a and corresponds to vibrations of the two strings moving in opposition. The symmetry breaking generated by the elastic strip between the strings deter449
Am. J. Phys., Vol. 79, No. 5, May 2011
mines a new inertial mass whose value is a function of the strip’s Young modulus. By gradually decreasing the width of the elastic strip, the waves of the latter type disappear, as well as the new mass associated with the Young modulus of the strip. The new mass is thus directly related to the existence of the elastic strip and is a measure of the resistance to crosswise stretching of the elastic strip associated with the opposed motion of the two strings. These observations suggest that inertial mass is associated with a reduction of spatial symmetry due to an interaction, where the mass is associated with the quantity that is losing symmetry. VI. LINK BETWEEN CLASSICAL AND QUANTUM APPLICATIONS
We note that in the description of the coupling between vibrating strings the parameter in the Klein–Gordon equation is related to the Young modulus Y of the medium carrying out the coupling. In quantum mechanics this parameter is related to the mass of the spin zero particle. To see how these notions relate, we return to the system of the two elastically linked parallel strings. If the Young modulus of the elastic strip is zero, there is no connection between the strings. In this case, each energy eigenstate of the system is unique and invariant under rotations of the strings around the straight line which is coplanar and equidistant from the two strings at rest. This property applies to the system’s lowest energy state. If the strings are connected and form a coherent system, then each string loses its freedom to oscillate separately, but the two strings can still form an oscillating system. Assuming that the central line around which the two strings can rotate is collinear with the x-axis in three-dimensional space, then the plane containing the two strings and the elastic strip can form any angle with the x- y plane. Any one of these angles corresponds to the system’s lowest energy state, which is thus degenerate. The system must select one of these angles to fix its lowest energy state. As soon as this choice is made, the system is no longer invariant under rotations of the two strings. The introduction of the medium with a nonzero Young modulus between the strings thus breaks the rotational symmetry of the system’s lowest energy state. In quantum field theory this kind of symmetry breaking is called spontaneous, because it does not require an explicit 7 mass term in the Lagrangian to violate symmetry. No potential is necessary to describe the motion of infinitesimal elements of string when they are not linked to form a string under tension. If the elastic link between these elements is introduced, the description of the string as a whole requires a potential with a unique minimum, such as V y , z = a y 2 + z2 + b y 2 + z22 ,
7
with a , b 0. If we take into account the elastic link between the two parallel strings, then the description of the motion of the whole physical system needs a potential with an infinity of states of lowest energy, such as Eq. 7 with a 0 and b 0. This potential expresses the fact that the system of two strings has a stable equilibrium configuration where both strings tend to stay away from the straight line that is equidistant from each string’s rest position. The potential is repulsive when the strings are close to one another and attractive when they are far apart. To describe the position r x , t of one of these two strings with respect to its position at rest, we assume that in the P. Gravel and C. Gauthier
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vicinity of the minimum of the potential energy at r = r 0, that is, when r x , t y x , t 2 + z x , t 2 r 0, the elastic force of the link between the coupled strings is linear in the radial distance between r and r 0. Due to the tensions in the strings and elastic strip, the system will be in its state of lowest energy when the strings are parallel at the distance r = r 0 from the x-axis. In this case y x , t = r 0 cos and z x , t = r 0 sin for a constant angle between the elastic strip and the x- y plane. The symmetry breaking corresponding to a specific choice by the system of its actual lowest energy configuration is linked to the choice of a value for . Assume that this symmetry breaks for =0. We use u and v to represent the position of the string element at x in a Cartesian coordinate system centered at the position of this global-symmetry breaking minimum. It follows that y = r 0 + u and z = v. The potential energy part of the Lagrangian of the system then takes the form
Y
V =
=
2
Yr 20
2
=
r 0 + u, v − r 0,0 2
1+
Yr 20
2 Y
8r 20
1+
2ur 0 + u2 + v2 r 20
1 2ur 0 + u2 + v2 2
r 20
8a
2
−1
8b
2
−1
4r 20u2 + 4 r 0uu2 + v2 + u2 + v22 .
8c
8d
Near its minimum r 0 , 0, this potential is Y 2 Y Y u + u u 2 + v 2 + 2 u 2 + v2 2 . 2 2r 0 8r 0
9
Any motion in the radial u direction near the bottom of the potential corresponds to widthwise stretching or compression of the elastic strip and confers a Klein–Gordon mass to the u variable through the Yu 2 / 2 term. Motion restricted to the circular bottom of the potential corresponds to transverse movements of the elastic strip. These motions do not confer any mass to the variable v which obeys the ordinary wave equation. No energy is transferred to the elastic strip by such movements, because the elastic strip does not resist transverse deformations. The other terms in the potential function correspond to the interaction terms between the fields u and v . This situation is similar to the potential-induced global spontaneous symmetry breaking of the Goldstone process described in Appendix C. One way to interpret the repulsive part of V with a 0 and b 0 is by assigning a negative sign to the inertial mass associated with the elastic strip through the Klein–Gordon equation. If we define the inertial mass of a system as the one associated with its global potential, this mass is positive because the potential V with a 0 and b 0 is attractive at large distances. VII. CONCLUDING REMARKS
The quantization of linear waves in an infinite vibrating string or along a set of uncoupled vibrating strings leads to a massless quantum of vibration. We have seen that when the medium between these strings has a nonzero Young modu450
Am. J. Phys., Vol. 79, No. 5, May 2011
lus, it is possible to generate a massive scalar quantum of vibration. The new mass stems from the coupling between linear waves. Hence the notion of mass becomes related to the more fundamental notion of coupling between linear waves. In this context, no mass exists unless there is a coupling, and the stronger the coupling, the larger the mass.
APPENDIX A: TWO COUPLED VIBRATING STRINGS
To obtain the equations describing the motion of the two coupled strings, the infinite strings are replaced by stretched strings of finite length L fixed at both ends. The two strings are assumed to have constant linear density and to be under tension T . We also assume that the strings are connected along their length by a strip of massless material having an elasticity characterized by Young modulus Y . The amplitude of waves is assumed to be sufficiently small for the tension in the strings to remain uniform and constant in time. The equations describing the vibrations result from the replacement of each of the two string segments with N material points placed at intervals of length = L / N + 1, and each point has mass = . These material points are connected together and to the fixed ends of the strings by linear and massless springs which are assumed to be perfectly elastic. The tension in these small massless springs is equal to T . The motion of the material points is described using a Cartesian coordinate system whose x-axis is collinear with one of the strings when at rest. The origin coincides with the left fixed point of this string. The other string is placed parallel to the x-axis at y = y 1, where y 1 is a constant. The positions of the two sets of material points at rest are given by
A1
xij = ie 1 + jy je2 ,
where for the first string y j = y 0 =0 and where i = 0 , 1 , . . . , N + 1, j =0, 1, and e1 and e2 are the unit vectors along the xand y -axis, respectively. The points x0 j and x N +1, j, j =0,1, correspond to the fixed end points of the two strings. The material point at xij is denoted by i , j . Let uij t be the displacement of the point i , j parallel to the y -axis relative to xij at time t . The functions uij t are assumed to be twice differentiable. We assume that the coupling can be expressed by a link between members of pairs of material points having the same value of their first index. Hence, we assume that the points i , 0 and i , 1, i = 1 , 2 , . . . , N , are connected by a massless spring with a given stiffness constant. This constant represents the resistance to transverse displacement due to the medium between the string segments of length and is denoted by k . We can determine the equation of motion for the material point 1,0 from the forces acting on it. These forces result from the displacement from their equilibrium positions of 1,0 and its neighboring points, that is, the points 0,0, 2,0, and 1,1 see Fig. 1. For small displacements the transverse force on the point 1,0 due to the fixed point 0,0 is given by − Tu 10 / . The force due to the point 2,0 is equal to T u20 − u10 / . The force due to the point 1,1 equals k −u10 + u11. The equation of motion of the point 1,0 follows from Newton’s second law. We find
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Fig. 1. Schematic of the system of two strings with an elastic band between them.
u¨ 10 =
T
− 2 u10 + u20 + k − u10 + u11 ,
A2
where u¨ 10 = d 2u10t / dt 2. The equations of motion of the other material points are obtained similarly. We thus arrive at the following system of differential-difference equations, for j =0,1, u¨ 1 j =
u¨ 2 j =
T
j
T
− 2 u1 j + u2 j + − 1
k
u1 j − 2 u2 j + u3 j + − 1 j
Fig. 2. Motion in a cross section of a three-rope Inca suspension bridge. The white dot inside the large black dot indicates that the x-axis is perpendicular to the plane of the sheet.
=
− u10 + u11
A3a
+
k
k
4
4
t u0 + u1 2 − xu0 + u1 2
k
Y
4
4
2
t u0 − u1 2 − xu0 − u1 2 − u0 − u12 . A4b
− u20 + u21 A3b APPENDIX B: THREE COUPLED VIBRATING STRINGS
]
u¨ N −1, j =
T
u N −2, j − 2 u N −1, j
+ u N , j + − 1 j
T
u¨ N , j =
k
− u N −1,0 + u N −1,1
u N −1, j − 2 u N , j + − 1 j
k
A3c
− u N ,0 + u N ,1 . A3d
Equations A3a and A3d differ from the others because they include the fixed end points of the two strings. Now we let the number of material points approach infinity and the common distance between successive points of the same string approach zero on both strings, such that / = and k / are constant. The value of the constant k / can be obtained by observing that it corresponds to Y when =1 so that k / = Y . In this limit each relation of Eq. A3, other than the first and the last, transforms into one of the two partial differential Eqs. 5a and 5b. The dependent variables u0 and u1 are the functions associated with ui0 and ui1, respectively, for i = 1 , 2 , . . . , N . Observe that Eq. 5 can also be deduced from the Lagrangian L=
2 −
451
2
2
k
u 0 − u1 2
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Bk + uk − u0
Bk =
Bk + uk − u0 − B .
Bk + uk − u0
B1
We let uk − u0 = k = yk , zk and approximate Eq. B1 to first order as
2
2
Y
− z-plane, and b = b y , b z to designate the corresponding unit vectors. The differences between the equilibrium and the displaced position of guard rail k =1,2 and the footpath are
2
t u0 + t u1 − xu0 + xu1
2
As before, we set up a Cartesian coordinate system see Fig. 2. Assuming the cables to be straight and parallel when at rest, we will use the horizontal x-axis to represent the equilibrium position of the footpath. The y -axis is also horizontal. The z-axis is vertical and the x − z-plane passes through the line that is equidistant to the two guard rails at rest. We denote by j =0,1,2 the footpath and the two guard rail cables, respectively. The two-component vectors ui; j = u yi ; j , u zi ; j, j =0 ,1 ,2, denote small displacements from their equilibrium positions of the cable elements at x = i. These elements are assumed not to be immediate neighbors of the anchoring points, that is, i 0 and i N +1. For simplicity, the absence of the first index and the semicolon in the indices of a quantity will be understood to mean that we are using the value of the quantity that corresponds to the plane at x = i for any i = 1 , 2 , . . . , N . We shall use two constant vectors B1 = − 1 , 2 and B2 = 1 , 2 of length B in each x = i plane parallel to the y
A4a
Bk =
Bk + k 1 −
B
Bk + k
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B2a 451
+
2 2 − 1 k 2 1 yk + 2 2 zk + yk + zk
21 + 22
Bk + k
− 1 k 1 yk + 2 zk 21 + 22
u¨ i;0 =
T
0
k
0
k
u0 − u1 · b−b−
ui−1;2 − 2 ui;2 + ui+1;2 +
k
u0 − u2 · b+b+ .
B3c
u0 − u1 · b−b−
+ u0 − u2 · b+b+
u¨ i;2 =
B2b
B2c
B3b
.
ui−1;0 − 2 ui;0 + ui+1;0 −
ui−1;1 − 2 ui;1 + ui+1;1 +
−1 /2
What is left after a first-order approximation in the s is the projection of k parallel to Bk , namely k · bk , where b1 = b− and b2 = b+ are the components of the stretching vector of the distance between guard rail k and the footpath. The equations for the forces on each of the three cables are then found to be
u¨ i;1 =
= Bk + k 1 − 1
B3a
Here and T are the tensions in the guard rails and the footpath, respectively, while and 0 are the linear densities, and 0, multiplied by in the guard rails and the footpath respectively. The terms in and T are the first-order approximations of the total restoring forces due to the tensions in the three main cables, whose form for the first guard rail is
ui−1;1 − 2 ui;1 + ui+1;1
2 + u yi −1;1 − 2 u yi ;1 + u yi +1;12 + u zi −1;1 − 2 u zi ;1 + u zi +1;12 1/2
The limits of the differential-difference Eq. B3 as goes to zero while keeping constant / l, 0 / l, and k / lead to a system of three coupled partial differential equations: 2 t u 0
= C 2 x2u0 − A2 u0 − u1 · b−b− + u0 − u2 · b+b+
B5a 2 t u 1
=
c2 x2u1
=
2 c2 x u2
2
− u1 − u0 · b−b−
B5b
B4
.
L u1 − u0+ −
L u2 − u0− −
2
− u2 − u0 · b+b+ ,
B5c
where c = / , C = T / 0, = Y / and A = Y / 0. If the three cables are made of the same material, it is reasonable to assume that the constants c = / and C = T / 0 are very close to one another. In these square roots the tensions serve to keep the bridge more or less straight against gravity, which means that they must be proportional to the linear density to obtain the same shape. At the same time, the linear density increases proportionally to the area of the cross section of the cable. The numbers c and C should thus be approximately the same. Here we shall assume that they are identical. Note that there is no reason why the constants and A should behave the same way. We now write the displacement vectors ui in terms of their components in the b− , b+ basis of the planes orthogonal to the three main cables to obtain the following system of six scalar differential equations
Lu1 = 0
B6a
Lu2− = 0
B6b
+
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Am. J. Phys., Vol. 79, No. 5, May 2011
2 + A2 A2
2 + A2
u2 − u0 + = 0
B6c
u1 − u0 − = 0
B6d
L u1 − u0− = − 2 + A2 u1 − u0−
B6e
L u2 − u0+ = − 2 + A2 u2 − u0+ ,
B6f
2 t u 2
A2
where L = 2t − c2 x2 is the wave equation operator with speed c. These equations are decoupled in terms of the six different scalar arguments of L. As we can see, the quantities u1 − u0− and u2 − u0+, which represent the stretching of the secondary cables parallel to their rest orientation, do not obey wave equations, but instead obey Klein–Gordon equations.
APPENDIX C: THE GOLDSTONE MODEL
In the Goldstone model we start with a complex wave function x = 1 x + i 2 x / 2 and the Lagrangian density L x
= h x x − V ,
C1
where V is a potential, x is the complex conjugate of x, and there is a summation over repeated indices using the Minkowski metric h = h =diag1,−1,−1,−1. In the simplest case the self-interaction potential of the field is the quartic function
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V = 2 x 2 + x 4 ,
C2
L x
where 0 and 2 are arbitrary real parameters. The Lagrangian density given by Eq. C1 is invariant under global rotations about the origin of the plane containing the values of x
x → x = xexpi
C3a
x → x = xexp− i ,
C3b
in which is a real constant. If 2 0, the minimum of V is unique and located at 0 =0. This minimum position remains unchanged under the phase transformations of x and x in Eq. C3. The existence of this minimum does not break the original rotational symmetry of the Lagrangian. The mass associated with the field x is given by Eq. C2 as the coefficient 2 of the quadratic term. If 2 0, the potential V assumes the shape of a sombrero whose minimum corresponds to the set of complex values 0 of x satisfying
0 =
r 0
2 ,
C4
where r 0 = −2 / 2. This lowest state of energy is infinitely degenerate. Any system described by L x will settle around one of these energetically equivalent minima, none of which is invariant under a rotation around the origin in the plane containing the values of . The symmetry of the Lagrangian is said to be spontaneously broken when the system is in one of these minima. If we select the real positive minimum position 0 = r 0 / 2, and expand x around 0, we can write
x =
1
2 r 0 + x + i x ,
C5
where x and x are real fields expressing the deviation of x from its state of lowest energy 0. In terms of these new fields, Eq. C1 becomes
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Am. J. Phys., Vol. 79, No. 5, May 2011
=
1 2 + −
1 2 1 4
1
x x − 2r 20 2 x 2
x x − r 0 x 2 x + 2 x
2
x + 2 x 2 ,
C6
where we have omitted a constant term which is of no consequence. The first three terms of Eq. C6 correspond to the free Lagrangian density, and the last two terms are associated with the interactions of the fields x and x . We deduce that x corresponds to a massless field and that x is a
field of the Klein–Gordon type with mass 2r 20. The field x describes radial displacements close to the bottom of the valley of the potential, and the field x represents displacements restricted to the circular bottom of the valley. The free part of the Lagrangian shows that, when quantized, the field x yields a massless boson, and x yields a boson of
mass 2r 20. This process is the Goldstone phenomenon: A continuous global symmetry implies the existence of a massless field. The similarity between the pairs 1 , 2 and u , v of Sec. VI is clear, as is that of the process through which u and 1 acquire mass while v and 2 remain massless. a
Electronic mail:
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[email protected] 1 H. Kragh, “Erwin Schrödinger and the wave equation: The crucial phase,” Centaurus 26, 154–197 1982. 2 Wolfrang Pauli, in Wissenschaftlicher Briefwechsel, edited by A. Hermann, K. v. Meyenn, and V. F. Weisskopt Springer, New York, 1979 , Vol. I, p. 356. 3 H. Kragh, “Equation with the many fathers. The Klein-Gordon equation in 1926,” Am. J. Phys. 52, 1024–1033 1984. 4 P. M. Morse and H. Feshbach, Methods of Theoretical Physics McGrawHill, New York, 1953 , Vol. I, p. 138. 5 F. Mandl and G. Shaw, Quantum Field Theory John Wiley & Sons, New York, 1984 , p. 43. 6 D. W. Gade, “Bridge types in the central Andes,” Ann. Assoc. Am. Geogr. 62, 94–109 1972. 7 K. Moriyasu, An Elementary Primer for Gauge Theory World Scientific, Singapore, 1983, p. 96.
b
P. Gravel and C. Gauthier
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