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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 322 14 "Work and Force" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 432 "We inves
tigate the relationship between force and energy (work). For energy-co
nserving systems one can formulate the equations of motion by making u
se of scalar quantities that represent the kinetic and potential energ
y for the particle in the force field. For non-conservative forces the
 work integral has to be calculated for the individual particle trajec
tories using a line integral. Both cases are considered in this worksh
eet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "
We begin with the conservative case. Take the example of a two-centre \+
electrostatic potential as experienced in a diatomic molecule by the v
alence electron (e.g., the H2+ molecule, cf. R.L. Greene, " }{TEXT 
257 30 "Classical Mechanics with Maple" }{TEXT -1 47 " (Springer 1994)
, pp. 88-93, and M.L. de Jong, " }{TEXT 256 37 "Introduction to Comput
ational Physics" }{TEXT -1 238 " (Addison-Wesley 1991), chapter 10). T
he problem of a spaceship, or asteroid, or small planet moving in the \+
field of two nearly stationary stars would be calculated in a complete
ly analogous way. Two protons are kept at a fixed distance 2" }{TEXT 
258 1 "d" }{TEXT -1 11 " along the " }{TEXT 259 1 "z" }{TEXT -1 28 " a
xis, one located at (0,0,-" }{TEXT 260 1 "d" }{TEXT -1 21 "), the othe
r at (0,0," }{TEXT 261 1 "d" }{TEXT -1 31 "). The electron is located \+
at (" }{TEXT 262 1 "x" }{TEXT -1 1 "," }{TEXT 263 1 "y" }{TEXT -1 1 ",
" }{TEXT 264 1 "z" }{TEXT -1 161 ") and being of opposite charge exper
iences an attractive force. It is convenient to express the electron d
istance from either proton in terms of its coordinate (" }{TEXT 267 1 
"x" }{TEXT -1 1 "," }{TEXT 268 1 "y" }{TEXT -1 1 "," }{TEXT 269 1 "z" 
}{TEXT -1 21 ") using coordinates, " }{TEXT 266 1 "r" }{TEXT -1 6 "1 a
nd " }{TEXT 265 1 "r" }{TEXT -1 15 "2 respectively." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r1:=sqrt(x^2+y^2+(z-d)^2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G*$-%%sqrtG6#,,*$)%\"xG\"\"#
\"\"\"\"\"\"*$)%\"yGF-F.F/*$)%\"zGF-F.F/*&F5F/%\"dGF/!\"#*$)F7F-F.F/F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r2:=sqrt(x^2+y^2+(z+d)^
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G*$-%%sqrtG6#,,*$)%\"xG\"
\"#\"\"\"\"\"\"*$)%\"yGF-F.F/*$)%\"zGF-F.F/*&F5F/%\"dGF/F-*$)F7F-F.F/F
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The electrotatic potential e
nergy in MKSA (SI) units " }{TEXT 19 3 "Usi" }{TEXT -1 155 " is then g
iven in terms of the charges for the protons and the electron, Q and q
 respectively, the dielectric constant epsilon[0] and the two distance
s as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f[0]:=q*Q/(4*Pi*ep
silon[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"!,$*&*&%\"q
G\"\"\"%\"QGF,\"\"\"*&%#PiG\"\"\"&%(epsilonGF&\"\"\"!\"\"#F,\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Usi:=-f[0]*(1/r1+1/r2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$UsiG,$*&*(%\"qG\"\"\"%\"QGF),&*&\"
\"\"F-*$-%%sqrtG6#,,*$)%\"xG\"\"#F-F)*$)%\"yGF6F-F)*$)%\"zGF6F-F)*&F<F
)%\"dGF)!\"#*$)F>F6F-F)F-!\"\"F)*&F-F-*$-F06#,,F3F)F7F)F:F)F=F6F@F)F-F
BF)F)F-*&%#PiG\"\"\"&%(epsilonG6#\"\"!\"\"\"FB#!\"\"\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 257 "For plotting as well as calculations it \+
is more convenient to measure energy in units in which the dielectric \+
constant, the charge units and a characteristic length scale are absor
bed into a single constant. The characteristic length scale can be cho
sen as " }{TEXT 270 1 "d" }{TEXT -1 10 ", (or as 2" }{TEXT 271 1 "d" }
{TEXT -1 97 "), i.e., the equilibrium separation of the nuclei in the \+
molecule. In these units we simply have:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "U:=unapply(subs(d=1,-1/r1-1/r2),x,y,z);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"UGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG
F*,&*&\"\"\"F0*$-%%sqrtG6#,,*$)9$\"\"#F0\"\"\"*$)9%F9F0F:*$)9&F9F0F:F@
!\"#F:F:F0!\"\"!\"\"*&F0F0*$-F36#,,F6F:F;F:F>F:F@F9F:F:F0FBFCF*F*F*" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 366 "There are various ways to visua
lize this function. It is possible to make cuts such that a 2D graph c
onveys some information. Two of the coordinates have to be fixed for t
his purpose, and keep in mind that all distances are measured in terms
 of d which has been set to unity, i.e., all coordinates are measured \+
as multiples of the half-separation between the nuclei." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We take cuts along t
he " }{TEXT 275 1 "x" }{TEXT -1 7 " axis (" }{TEXT 274 1 "x" }{TEXT 
-1 15 "=0) with fixed " }{TEXT 273 2 "y " }{TEXT -1 24 "(treated as a \+
parameter " }{TEXT 279 1 "y" }{TEXT -1 32 "0) and display as a functio
n of " }{TEXT 272 1 "z" }{TEXT -1 111 ". We restrict the range of the \+
ordinate to avoid a huge scaling from the singularity in the potential
 at (0,0,-" }{TEXT 277 1 "d" }{TEXT -1 11 ") and (0,0," }{TEXT 276 1 "
d" }{TEXT -1 15 ") respectively." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "P1:=plot(U(0,0,z),z=-3..3,-5..0,color=blue):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P2:=plot(U(0,0.5,z),z=-3..3,color=r
ed):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "P3:=plot(U(0,1,z),z=-3..3,c
olor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(\{P1,P2,P3\});
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "It is straightforward to read off \+
from this diagram the " }{TEXT 278 1 "z" }{TEXT -1 67 " component of t
he force for the electron when it is located at (0, " }{TEXT 280 1 "y
" }{TEXT -1 2 "0," }{TEXT 281 2 " z" }{TEXT -1 26 ") for the three val
ues of " }{TEXT 282 1 "y" }{TEXT -1 164 "0 (0, 0.5 and 1) represented \+
by the three curves above. Due to the superposition principle we can t
reat these potentials as 1D potentials for the motion with fixed " }
{TEXT 283 1 "x" }{TEXT -1 6 "0 and " }{TEXT 284 1 "y" }{TEXT -1 79 "0,
 and thus the force is simply given as the negative of the derivative \+
w.r.t. " }{TEXT 285 1 "z" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "P1f:=plot(-diff(U(0,0,z),z),z=-3..3,-5..5,color=blue,
discont=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "P2f:=plot(-diff(U
(0,0.5,z),z),z=-3..3,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"P3f:=plot(-diff(U(0,1,z),z),z=-3..3,color=green):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "display(\{P1f,P2f,P3f\});" }}{PARA 13 "" 1 
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45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "We added
 an option in the graph for " }{TEXT 286 1 "x" }{TEXT -1 292 "0=0 to a
void a vertical line showing up at the location of the singularity (in
correct joining of the asymptotic behaviour done by Maple without the \+
option). Note that the sign of the force changes, and that the locatio
n of the equilibrium points (intercept with zero) depends on the value
 of " }{TEXT 287 1 "y" }{TEXT -1 7 "0. For " }{TEXT 288 1 "y" }{TEXT 
-1 70 "0=0 the behaviour is qualitatively different as a singularity a
ppears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
246 "How can the full 3D behaviour be displayed? The correct approach \+
would be to switch to cylindrical (polar) coordinates and make use of \+
axial symmetry w.r.t. the z axis. We prefer to keep Cartesian coordina
tes however. Thus we keep the constraint " }{TEXT 291 1 "x" }{TEXT -1 
38 "0=0, and display the potential in the " }{TEXT 289 1 "y" }{TEXT 
-1 1 "-" }{TEXT 290 1 "z" }{TEXT -1 50 " plane using a surface plot an
d a contour diagram." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 20 "Since we called the " }{TEXT 19 5 "plots" }{TEXT -1 44 
" package already, contourplot is available.." }}}{EXCHG {PARA 0 "> " 
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Fag[mFag[mFeeuF[fu7&Ffg[mFfg[mF_fuFbfu7&Ffg[mFfg[mFhfuFbfu7&Fdg[mFdg[m
F`h[mFfh[m7&Ffh[mFag[mFiguFfgu7&Fdg[mFdg[mFag[mFfh[m7&Fgg[mF`h[mFbgu7$
F]gu$!1YrJj,AeAF-7&Fgg[mFfg[mFhfuF]j[m7&Fag[mFdg[mF[euFaeu-Fafal6&Fcfa
lFdfalFh_vFh_vFefal-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"zG%\"
yG-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "The force acting o
n the particle is given as the negative of the gradient. We need to lo
ad the " }{TEXT 19 6 "linalg" }{TEXT -1 20 " package and obtain:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "F:=grad(-U(x,y,z),[x,y,z]);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%\"FG-%'vectorG6#7%,&*&%\"xG\"\"\"*$),,*$)F+\"\"#F,
\"\"\"*$)%\"yGF2F,F3*$)%\"zGF2F,F3F9!\"#F3F3#\"\"$F2F,!\"\"!\"\"*&F+F,
*$),,F0F3F4F3F7F3F9F2F3F3#\"\"$F2F,F=F>,&*&F6F,*$)F/#\"\"$F2F,F=F>*&F6
F,*$)FB#\"\"$F2F,F=F>,&*&,&F9F2F:F3F,*$)F/#\"\"$F2F,F=#F>F2*&,&F9F2F2F
3F,*$)FB#\"\"$F2F,F=FW" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Note t
hat one could not simply evaluate F:=-grad(U...) as this would amount \+
to the multiplication of a vector by a scalar (namely by -1). Maple re
quires an " }{TEXT 19 5 "evalm" }{TEXT -1 59 " wrapped around such a c
ommand to be able to evaluate. The " }{TEXT 19 5 "evalm" }{TEXT -1 57 
" procedure is used for matrix and vector multiplications." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "To confirm our pr
evious result for the z component of the force we graph:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(subs(x=0,y=0.5,F[3]),z=-3..3,-
5..5,color=red);" }}{PARA 13 "" 1 "" {GLPLOT2D 402 301 301 {PLOTDATA 
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-$\"1eU+8\\(4+&F-7$$\"1,++vuY)o$F-$\"1PBSZW'om(F-7$$\"1)******4FL(\\F-
$\"1Y$=Cl.)H5F17$$\"1)***\\P$>=g&F-$\"16tF*G^Y8\"F17$$\"1)****\\d6.B'F
-$\"1Us<Y8)R?\"F17$$\"1)\\iS@OBQ'F-$\"1#f4*4!\\I@\"F17$$\"1**\\7`3OMlF
-$\"1V:[7FS=7F17$$\"1*\\(=#\\&Q'o'F-$\"1A4_Kgm>7F17$$\"1***\\785%QoF-$
\"1jTPxGX;7F17$$\"1**\\P4%fC9(F-$\"1nFsVe.&>\"F17$$\"1++](o3lW(F-$\"1M
ygJ'*3^6F17$$\"1***\\iX)p@\")F-$\"1Jv*G?`#*f*F-7$$\"1*****\\A))oz)F-$
\"1b\"RxMH7H'F-7$$\"1)**\\PCK-5*F-$\"1Fn#*p&e/Q%F-7$$\"1(***\\iid.%*F-
$\"1Ju;feXfAF-7$$\"1(**\\7G?pq*F-$!1FBqA#p!G7Fdy7$$\"1+++Ik-,5F1$!1RM.
\"GnEO#F-7$$\"1++vy[TL5F1$!1t8^3+.q[F-7$$\"1++]FL!e1\"F1$!1;<6%Qr=G(F-
7$$\"1++Dw<>)4\"F1$!1+#y4e^F^*F-7$$\"1+++D-eI6F1$!1))pM\"**y%\\6F17$$
\"1+](=sx#*=\"F1$!1\\T^DU'>V\"F17$$\"1++v=_(zC\"F1$!16Dif9X5;F17$$\"1+
D\"G:h+G\"F1$!18zc9I>n;F17$$\"1+](o3Z@J\"F1$!1YMpj-)))p\"F17$$\"1]i!R0
!>G8F1$!1w?jS8W1<F17$$\"1+v$4-LUM\"F1$!1;b'oH&44<F17$$\"1](oz)fFg8F1$!
1`$=H4+tq\"F17$$\"1+++b*=jP\"F1$!1v@scF],<F17$$\"1+](=o*pO9F1$!1%Q![\\
#p4l\"F17$$\"1++v3/3(\\\"F1$!1Z:g1`^p:F17$$\"1++vB4JB;F1$!1lOPlvQf8F17
$$\"1+++DVsY<F1$!1a\\s&Q4^:\"F17$$\"1++v=n#f(=F1$!1F2()f\\n$p*F-7$$\"1
+++!)RO+?F1$!1&fW([g*y@)F-7$$\"1++]_!>w7#F1$!1$=P_[/D*pF-7$$\"1++v)Q?Q
D#F1$!1$owuj1*4gF-7$$\"1+++5jypBF1$!1z%R$=\")=q_F-7$$\"1++]Ujp-DF1$!1C
L/8\"RRd%F-7$$\"1+++gEd@EF1$!1%R7KB(zfSF-7$$\"1++v3'>$[FF1$!1+'QE0H3g$
F-7$$\"1++D6EjpGF1$!17Yv[:lIKF-7$$\"\"$F*$!1y@:v2K$*GF--%+AXESLABELSG6
$Q\"z6\"%!G-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-%%VIEWG6$;F(F_gl;$!\"&
F*$\"\"&F*" 1 2 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "We can demonstrate that the force \+
field is curl-free:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "curl
(F,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F
'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We can demonstrate by examp
le that the line integral is independent of the chosen path. We choose
 two paths in the " }{TEXT 293 1 "y" }{TEXT -1 1 "-" }{TEXT 294 1 "z" 
}{TEXT -1 257 " plane for which the potential was shown in the contour
 diagram. Let us calculate the work to reach the location [0,1,1] from
 the origin [0,0,0] in two different ways. Once through a straight-lin
e connection, and once through an arc of a circle with radius " }
{TEXT 19 9 "sqrt(2)/2" }{TEXT -1 30 " around the point [0,0.5,0.5]." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "To set \+
up the line integral to calculate the work done by the force on the pa
rticle as it traverses the trajectory " }{TEXT 295 1 "r" }{TEXT -1 1 "
(" }{TEXT 296 1 "t" }{TEXT -1 203 ") we use a parameter representation
 of the path. If the path is differentiable with respect to its parame
ter (e.g., time in physics), then the line integral can be reduced to \+
a definite Riemann integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 10 "W =  int (" }{TEXT 315 1 "F" }{TEXT -1 4 " . d
" }{TEXT 316 1 "r" }{TEXT -1 9 ") =  int(" }{TEXT 317 1 "F" }{TEXT -1 
1 "(" }{TEXT 318 1 "r" }{TEXT -1 1 "(" }{TEXT 319 1 "t" }{TEXT -1 6 ")
) . d" }{TEXT 320 1 "r" }{TEXT -1 2 "/d" }{TEXT 321 1 "t" }{TEXT -1 
15 ", t = t0 .. t1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "For the straight-line connection we have:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r_in:=[0,0,0]; r_fin:=[0,1,1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%r_inG7%\"\"!F&F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&r_finG7%\"\"!\"\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "rSL:=evalm(r_in+t*(r_fin-r_in));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$rSLG-%'vectorG6#7%\"\"!%\"tGF*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 23 "rSL[1], rSL[2], rSL[3];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"\"!%\"tGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 
"The above does not work properly without the call to evalm, and the o
peration below does not work without the map construct, once " }{TEXT 
19 3 "rSL" }{TEXT -1 13 " is a vector." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "vSL:=map(diff,rSL,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$vSLG-%'vectorG6#7%\"\"!\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "whattype(rSL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'s
ymbolG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "In this parameter repre
sentation of the straight line the endpoints are realized for " }
{TEXT 298 1 "t" }{TEXT -1 7 "=0 and " }{TEXT 297 1 "t" }{TEXT -1 101 "
=1 respectively. It is important to evaluate the force along the traje
ctory, i.e., to substitute for " }{TEXT 299 1 "x" }{TEXT -1 1 "," }
{TEXT 300 1 "y" }{TEXT -1 1 "," }{TEXT 301 1 "z" }{TEXT -1 39 " in the
 force assignment the values of " }{TEXT 302 1 "x" }{TEXT -1 1 "(" }
{TEXT 303 1 "t" }{TEXT -1 3 "), " }{TEXT 304 1 "y" }{TEXT -1 1 "(" }
{TEXT 305 1 "t" }{TEXT -1 3 "), " }{TEXT 306 1 "z" }{TEXT -1 1 "(" }
{TEXT 307 1 "t" }{TEXT -1 124 "), so that the definite integration mak
es sense, and the force evaluations happen at the correct positions as
 a function of " }{TEXT 308 1 "t" }{TEXT -1 180 ", namely along the ch
osen trajectory. This is the most important aspect of the line integra
l to be understood by the beginning physicist (and also the most commo
n source of error)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 124 "The op(F) rather than F in the substitution is importa
nt. It is needed for the substitution to take place in all components:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(x=rSL[1],y=rSL[2],
z=rSL[3],op(F));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"
!,&*&%\"tG\"\"\"*$),(\"\"\"F/*$)F*\"\"#F+F2F*!\"##\"\"$F2F+!\"\"!\"\"*
&F*F+*$),(F/F/F0F2F*F2#\"\"$F2F+F6F7,&*&,&F*F2F3F/F+*$)F.#\"\"$F2F+F6#
F7F2*&,&F*F2F2F/F+*$)F;#\"\"$F2F+F6FE" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 11 "We use the " }{TEXT 19 7 "dotprod" }{TEXT -1 20 " procedu
re from the " }{TEXT 19 6 "linalg" }{TEXT -1 46 " package to calculate
 the dot product between " }{TEXT 309 1 "F" }{TEXT -1 1 "(" }{TEXT 
310 1 "r" }{TEXT -1 1 "(" }{TEXT 311 1 "t" }{TEXT -1 8 ")) and d" }
{TEXT 312 1 "r" }{TEXT -1 2 "/d" }{TEXT 313 1 "t" }{TEXT -1 26 ", and \+
then integrate over " }{TEXT 314 1 "t" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "W1:=int(dotprod(subs(x=rSL[1],y=rSL
[2],z=rSL[3],op(F)),vSL),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#W1G,&*$-%%sqrtG6#\"\"&\"\"\"#\"\"\"F*!\"\"F-" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 135 "We can check whether this result agrees with the di
fference between the potential energies evaluated at the two endpoints
 respectively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "U(op(r_fi
n))-U(op(r_in));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$-%%sqr
tG6#\"\"&\"\"\"#!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
evalf(%,4); evalf(W1,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%Gb!\"%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%Gb!\"%" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 105 "To see how the first of the two numbers came about lo
ok at the contributions to the potential difference:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(U(op(r_fin)),4); evalf(U(op(r_in)),
4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%Z9!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!\"#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "The
 amount of work to move the electron from location [0,0,0] to [0,1,1] \+
is negative, which means that on average the force was acting more aga
inst the motion. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 127 "Try simpler trajectories for which the force vector is a
ligned with the position vector, such as moving from [0,0,0] to [0,0,1
-" }{TEXT 324 1 "c" }{TEXT -1 98 "], i.e., from the unstable equilibri
um point towards one of the protons, e.g., the one located at " }
{TEXT 323 1 "z" }{TEXT -1 70 "=1. Also see what happens as your path m
oves across the singularity..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "The potential energy at the initial posit
ion " }{TEXT 19 12 "r_in=[0,0,0]" }{TEXT -1 28 " is more attractive th
an at " }{TEXT 19 13 "r_fin=[0,1,1]" }{TEXT -1 366 ". The difference i
n potential energy between the final and initial points of the traject
ory is therefore positive. It costs energy to move from [0,0,0] to [0,
1,1]. This is consistent with the previous finding that the work is ne
gative: we have to invest energy to move the electron from [0,0,0] to \+
[0,1,1]. The electron can trade kinetic energy to achieve this goal." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now we \+
repeat the calculation for a circular trajectory:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "rC:=evalm(sqrt(2)/2*[0,cos(t),sin(t)]+[0,1/2
,1/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rCG-%'vectorG6#7%\"\"!,&
*&-%%sqrtG6#\"\"#\"\"\"-%$cosG6#%\"tG\"\"\"#F5F/F6F5,&*&F,F0-%$sinGF3F
5F6F6F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "rC[1];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "plot([rC[2],rC[3],t=Pi/4..5/4*Pi],scaling=CONSTRAINED);" }}
{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CURVESG6$7S7$
$\"0C([**********!#:$\"1w70++++5F*7$$\"16!*)G>ghk*!#;$\"1'y<OHSI.\"F*7
$$\"1!G(QOm]?$*F0$\"1*\\qK'3wf5F*7$$\"1X$3KG;g$*)F0$\"1=bl%>Lu3\"F*7$$
\"1L844(R6`)F0$\"1TjkCfi76F*7$$\"1'Q$))y'RA6)F0$\"1\")3F&yK\\8\"F*7$$
\"19grz\"*H6xF0$\"16))z@41`6F*7$$\"1Nm7bW5&G(F0$\"1Ky*4)e;p6F*7$$\"1oB
%)>ePMoF0$\"1))3&Rr)*G=\"F*7$$\"19ZjlW)pP'F0$\"1$352&)pN>\"F*7$$\"1h\"
pk7$=+fF0$\"1a$H'zMN,7F*7$$\"1&fhL#>mwaF0$\"1pcoe$)\\07F*7$$\"1jCL1*))
y*\\F0$\"1.Gn\\n527F*7$$\"1XjQyF;<XF0$\"1P@ZkjX07F*7$$\"1j-9<F)f0%F0$
\"1fCwNpx+7F*7$$\"1kO'Ga\\1k$F0$\"1l-+?w\"R>\"F*7$$\"1.&*3,$oJ:$F0$\"1
=rZ5Hc#=\"F*7$$\"1Kk:y:NZFF0$\"1D*\\JVl-<\"F*7$$\"1%z(R>6e&G#F0$\"1SHY
98$H:\"F*7$$\"1XPD0^y()=F0$\"1*)[5+S$\\8\"F*7$$\"10#z?6Y]Y\"F0$\"1T!*Q
>fS76F*7$$\"1i*on_\\v2\"F0$\"1#*H=())R$)3\"F*7$$\"1)f2v6Cm!p!#<$\"1WOL
p6ig5F*7$$\"1W\"[H#HMDNFdr$\"1XSPsk#H.\"F*7$$\"1n8r\"\\#Hfx!#>$\"1w5)p
su2+\"F*7$$!1(obN\"))\\oKFdr$\"11#)>kLB]'*F07$$!1#H76&*RQ(fFdr$\"1t\\9
fwz?$*F07$$!1#)GlBi!pm)Fdr$\"1V%3]v'RZ*)F07$$!15bC)p!y=6F0$\"11l:#y#3W
&)F07$$!1%4/nc?'Q8F0$\"1e2Wg$*)R8)F07$$!1!ollI5__\"F0$\"1QIQdpECxF07$$
!1IQKNrO,<F0$\"1`gz)zukD(F07$$!1J2k7'46$=F0$\"1$>;O)fXEoF07$$!1-j#>B+#
R>F0$\"1s0'QxF#fjF07$$!1Agw59f4?F0$\"1zi3_9RIfF07$$!1u]'Q\"eCc?F0$\"10
O*)**pfdaF07$$!1?Jm\\&f52#F0$\"1uK.#\\43,&F07$$!1=*p1OMj0#F0$\"1uB+j/x
VXF07$$!1[0yPl97?F0$\"1Ld,,f1*3%F07$$!1$))3u.HX$>F0$\"1^z&=5Qrh$F07$$!
1//l3\"y(H=F0$\"1z%ojKr&oJF07$$!1?S)>$Ro#p\"F0$\"1x**R)p**yr#F07$$!1;?
)[%))QF:F0$\"1o69%*p&4G#F07$$!1Ayg2>O]8F0$\"1W\\s*zq)*)=F07$$!15R!*zns
=6F0$\"1]%\\18CeX\"F07$$!1U@LgQ=k))Fdr$\"19cGmB3#3\"F07$$!1$GVz\"\\SOh
Fdr$\"1C\"z%*\\TM+(Fdr7$$!1GG:*>>RH$Fdr$\"1iN5E:!o_$Fdr7$$\"1!GiL;#zj]
!#D$!1#zt%=Fzj]Fhz-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Fb[l-%(SCALINGG6#%,
CONSTRAINEDG-%+AXESLABELSG6$%!GFj[l-%%VIEWG6$%(DEFAULTGF^\\l" 1 2 0 1 
0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "map(simplify,subs(t=Pi/4,op(rC)));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!\"\"\"F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "map(simplify,subs(t=5*Pi/4,op(rC)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 181 "With the chosen parametrisation the trajectory st
arts at 5/4 Pi, and ends at Pi/4. However, it passes through the singu
larity at [0,0,1], and thus we integrate the other way around:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "map(simplify,subs(t=-3*Pi/4,op(rC)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "t0:=-3*Pi/4: t1:=Pi/4:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "vC:=map(diff,rC,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#vCG-%'vectorG6#7%\"\"!,$*&-%%sqrtG6#\"\"#\"\"\"-%$sinG6#%\"tG\"\"
\"#!\"\"F/,$*&F,F0-%$cosGF3F5#F5F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "W2:=int(expand(dotprod(subs(x=rC[1],y=rC[2],z=rC[3],o
p(F)),vC)),t=t0..t1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#W2G-%$intG
6$,2*&*&-%$sinG6#-%*conjugateG6#%\"tG\"\"\"-%$cosGF0\"\"\"F5*$),,*$)F3
\"\"#F5F;*&-%%sqrtG6#F;F5F3F2F;F;F2*$)-F,F0F;F5F;*&F=F5FBF2!\"##\"\"$F
;F5!\"\"\"\"%*&*&F+F5F3F5F5*$),,\"#5F2F9F;F<F;F@F;FC\"\"'#\"\"$F;F5FGF
H*&*&F=F5F+F5F5*$)F8#\"\"$F;F5FGF;*&*&F=F5F+F5F5*$)FM#\"\"$F;F5FGF;*&*
&-F4F-F2FBF5F5*$)F8#\"\"$F;F5FG!\"%*&*&F=F5FjnF5F5*$)F8#\"\"$F;F5FGF;*
&*&FjnF5FBF5F5*$)FM#\"\"$F;F5FGF_o*&*&F=F5FjnF5F5*$)FM#\"\"$F;F5FG!\"'
/F1;,$%#PiG#!\"$FH,$Ffp#F2FH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "
Maple has a hard time doing the integral in closed form. We see that t
he integrand is non-trivial in the next command. There is no difficult
y in integrating numerically if the singularity is avoided." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "arg:=expand(dotprod(subs(x=r
C[1],y=rC[2],z=rC[3],op(F)),vC)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "evalf(Int(arg,t=t0..t1),4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$!%Gb!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "We a
re satisfied that the answer agrees with the floating-point evaluation
 of the previous work calculation for the straight-line trajectory and
 the negative of the potential difference." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 748 "Now that we established the fa
ct that the potential energy allows to calculate the work done on part
icles by a curl-free force, we illustrate how one uses the contour dia
gram to read off information about the force. We fixed a plane when gr
aphing the equipotential contours, and thus we choose to illustrate on
ly the force components in that plane. The force is given by the gradi
ent of the potential (up to a minus sign). The direction of the gradie
nt vector originating at any point is orthogonal to an equipotential c
ontour passing through that point. The plots package in Maple contains
 a command that allows to draw a set of arrows on a discrete mesh for \+
any given potential function in two dimensions. To avoid the singulari
ty we look at the " }{TEXT 326 1 "y" }{TEXT -1 1 "-" }{TEXT 325 1 "z" 
}{TEXT -1 18 " plane defined by " }{TEXT 327 1 "x" }{TEXT -1 6 "0=0.5:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "PL1:=gradplot(-U(0.5,y,
z),z=-3..3,y=-2..2,axes=boxed,color=red,grid=[16,24],arrows=thick):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "PL2:=contourplot(U(0.5,y,
z),z=-3..3,y=-2..2,axes=boxed,scaling=CONSTRAINED,grid=[40,40],contour
s=[-2,-1.5,-1,-0.5],filled=true,coloring=[white,blue]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(PL1,PL2);" }}{PARA 13 "" 1 
"" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%)POLYGONSG6]cl7*7$$!1w$>F6z=
*H!#:$!1/SBf$oK*>F*7$$!1zfVI9w/IF*$!1J74D,?%*>F*7$$!1J=gT9++IF*$!16Dut
Av'*>F*7$$!1(p(p\")3]5IF*$!1c.WmbX0?F*7$$!1]N'G*3u0IF*$!1O;4:x+3?F*7$$
!1$onFXT_*HF*$!1\"z$RAWI**>F*7$$!1NN$RY\"[!*HF*$!1s]/rl&=+#F*F'7*7$$!1
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Z=*4?F*7$$!1H3,_,Y%f#F*$!1>\"e()GN))*>F*7$$!10oea&)R)e#F*$!15(**zXF:+#
F*FL7*7$$!1&))eATF9>#F*$!19$\\qR]#*)>F*7$$!1%=&*G#3*)3AF*$!1oZdd.\"G*>
F*7$$!1\"p`dq*G,AF*$!1ga)>Y/b*>F*7$$!1i=JYJP7AF*$!1S`u]DS4?F*7$$!1p.<H
?x/AF*$!1Lg:bm47?F*7$$!1)>7')e)o$>#F*$!1`hRm&)>)*>F*7$$!1/2Zru3'=#F*$!
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "The arrows give a \+
good impression of the variation of the two force components in the " 
}{TEXT 328 1 "y" }{TEXT -1 1 "-" }{TEXT 329 1 "z" }{TEXT -1 45 " plane
. No information is conveyed about the " }{TEXT 330 1 "z" }{TEXT -1 
24 " component of the force." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 357 "The diagram illustrates the fact that it is no
t entirely obvious that the work integral is path-independent: while w
alking different trajectories from one endpoint to another one encount
ers force vectors pointing in different directions. Think about a path
 where the electron is first accelerated towards the proton located in
 the upper half-plane (at [0,0," }{TEXT 333 1 "d" }{TEXT -1 271 "]), a
nd then swings around towards the final location [0,1,1] working again
st the force field. A very different experience as compared to the str
aight-line or circular paths calculated before. Thus, the curl-free na
ture of conservative forces is a very powerful property." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "The potential in \+
the present case is given by a superposition of two central potentials
 with respect to the positions of the nuclei [0,0," }{TEXT 332 1 "d" }
{TEXT -1 12 "] and [0,0,-" }{TEXT 331 1 "d" }{TEXT -1 134 "] respectiv
ely. Central potentials play an important role due to their spherical \+
symmetry (the potential depends only on the distance " }{TEXT 334 1 "r
" }{TEXT -1 146 ", not on the spherical polar angles theta and phi). T
his symmetry implies angular momentum conservation, both in classical \+
and in quantum physics." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "0 0 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 }