{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Unit 9: Calculus of Variat ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "C hapter 48: Variational Mechanics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "Section 48.4: the spherical pendulum" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "C opyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ ts reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elec tronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United Stat es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(Cal cvar):\nwith(Partials):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " with(linalg):\nwith(plots):\nwith(plottools):\nread(`pvac.txt`):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 9 "The Model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 410 "A pendulum not constrained to move in a \+ vertical plane becomes a spherical pendulum. The pendulum bob is free to traverse the surface of a sphere centered at the pendulum's suppor t. In fact, the spherical pendulum is the simplest pendulum to constr uct, since it requires only a piece of string, a bob, and a support. \+ There is no need for a mechanism to constrain the oscillations to a si ngle vertical plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 329 "If the spherical pendulum is started with motion in a \+ single vertical plane, subsequent oscillations will remain in that pla ne. Hence, three-dimensional motion requires an initial velocity with a component not in the plane formed by the equilibrium position, the \+ support, and the raised bob. As before, we will ignore friction." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "Except f or its moment of inertia, a small sphere such as a marble or ball-bear ing rolling in a spherical bowl would be another example of the spher ical pendulum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Let us put our pendulum support at the origin, and descr ibe the location of the bob in spherical coordinates. If the length o f the pendulum is " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 37 ", then \+ the coordinates of the bob are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x(t)=r*cos(u)*sin(v) " "6#/-%\"xG6#%\"tG*(%\"rG\"\"\"-%$cosG6#%\"uGF*-%$sinG6#%\"vGF*" } {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) =r*sin(u)*sin(v)" "6#/-%\"yG6#%\"tG*(%\"rG\"\"\"-%$sinG6#%\"uGF*-F,6#% \"vGF*" }{TEXT -1 2 " " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z(t)=r*cos(v)" "6#/-%\"zG6#%\"tG*&%\"rG\"\"\"-%$cosG6#%\"vGF*" } {TEXT -1 12 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 4 " \+ in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 47 " measures rotation counterclockwise around the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 34 "-axis, starting from the positive " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 11 "-axis, and " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#P iG" }{TEXT -1 38 " measures the angle from the positive " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 86 "-axis to the radius vector. (See Fig ure 22.8, and equations (22.10) in Section 22.3.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "The Lagrangian" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The coordinates of the bob are given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "xt := r*cos(u(t))*sin(v(t));\nyt := r*sin(u(t))*sin(v(t));\nzt := r*cos(v(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The radius vector from the origin (where the support is located) to the bob is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R := vector([xt,yt,zt]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The velocity \+ vector for the bob is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`R'` := map(diff,R,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "A bob of mass " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 25 " the n has kinetic energy " }{XPPEDIT 18 0 "T=1/2" "6#/%\"TG*&\"\"\"F&\"\"# !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m*v^2" "6#*&%\"mG\"\"\"*$%\"vG \"\"#F%" }{TEXT -1 4 ", or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "T := collect(simplify(m/2*innerprod (`R'`,`R'`)),[m,r]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Ordinary motions of the spherical pendulum find the bob below the " }{XPPEDIT 18 0 "xy" "6#%#xyG" } {TEXT -1 13 "-plane where " }{XPPEDIT 18 0 "phi " 0 "" {MPLTEXT 1 0 12 "V := m*g*zt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "means the lowest potential is when the bob is at equilibrium, \+ that is, when the bob is straight down. Raising the bob increases the potential energy since the potential goes to zero when the bob is lif ted to the " }{XPPEDIT 18 0 "xy" "6#%#xyG" }{TEXT -1 126 "-plane. Bec ause of the simplicity of the expression for the potential, we accept \+ this definition, and write the Lagrangian as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L := T-V;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "The Equations of Motio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Using Maple's " }{TEXT 256 14 "Euler_Lagrange" }{TEXT -1 19 " comm and, we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "q := Euler_Lagrange(L,t,[u(t),v(t)]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We have obtained both equations of motion, namely" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L[u]-d/dt" "6#,&&%\"LG6#%\"uG\"\"\"*&%\"dGF(%#dtG!\"\"F ," }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[`u'`]=0 " "6#/&%\"LG6#%#u'G\"\"! " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L[v]-d/dt " "6#,&&%\"LG6#%\"vG\"\"\" *&%\"dGF(%#dtG!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[`v'`]=0" "6#/ &%\"LG6#%#v'G\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 229 "In addition, we have two first integrals . One will be the constancy of the total energy, and one is the integ rated form of the first Euler-Lagrange equation. If we remove both fi rst integrals, we are left with the two equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q1 := remov e(has,q,[K[1],K[2]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The absence and presence of " } {XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 110 ", the gravitational constan t, distinguishes one equation from the other, so we single out the two equations by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Q2 := op(select(has,q1,g)) = 0;\nQ1 := op(remove (has,q1,g)) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "To see which equation is the " } {XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 28 " equation, and which is the \+ " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 85 " equation, we construct t he Euler-Lagrange equations from first principles using the " }{TEXT 257 5 "pdiff" }{TEXT -1 33 " command. For example, the term " } {XPPEDIT 18 0 "L[u]" "6#&%\"LG6#%\"uG" }{TEXT -1 8 " in the " } {XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 12 " equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pdiff( L,u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "because " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 42 " coes not explicitly contain the variable " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 21 ". Consequently, the " }{XPPEDIT 18 0 "u" "6#% \"uG" }{TEXT -1 18 " equation is just " }{XPPEDIT 18 0 "d/dt" "6#*&%\" dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[`u'`]=0" "6#/&% \"LG6#%#u'G\"\"!" }{TEXT -1 43 ", and the expanded form of this equati on is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "pdiff(L,u(t)) - diff(pdiff(L,diff(u(t),t)),t) = 0;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "Hence, the equation without " }{XPPEDIT 18 0 "g" "6#%\" gG" }{TEXT -1 8 " is the " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 65 " equation, and that is the equation which has the first integral " } {XPPEDIT 18 0 "L[`u'`]=c" "6#/&%\"LG6#%#u'G%\"cG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 17 " equation is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p diff(L,v(t)) - diff(pdiff(L,diff(v(t),t)),t) = 0;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "and \+ it contains the constant " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Maple 's version of the first integral " }{XPPEDIT 18 0 "L[`u'`]=c" "6#/&%\" LG6#%#u'G%\"cG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pdiff(L,diff(u(t),t)) = c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 72 "which is not the simplest form possible. Maple is prog rammed to prefer " }{XPPEDIT 18 0 "1-cos^2*``(x)" "6#,&\"\"\"F$*&%$cos G\"\"#-%!G6#%\"xGF$!\"\"" }{TEXT -1 16 " to the simpler " }{XPPEDIT 18 0 "sin^2*``(x)" "6#*&%$sinG\"\"#-%!G6#%\"xG\"\"\"" }{TEXT -1 157 ", and any effort to coerce the latter form is generally wasted because, at the first opportunity, Maple will revert to the former. However, \+ although writing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`u'`*sin^2*``(v)=c/m/r^2" "6#/*(%#u'G\" \"\"*$%$sinG\"\"#F&-%!G6#%\"vGF&*(%\"cGF&%\"mG!\"\"*$%\"rGF)F1" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "would lead to the elimination of " }{XPPEDIT 18 0 "u" "6# %\"uG" }{TEXT -1 8 " in the " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 138 " equation, and hence to a quadrature in terms of elliptic functio ns, we take the more pragmatic approach of generating a numeric soluti on." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Example 48. 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Let us take a time-scale for which " }{XPPEDIT 18 0 "g=1" "6#/%\"g G\"\"\"" }{TEXT -1 15 ", and set both " }{XPPEDIT 18 0 "r=1" "6#/%\"rG \"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "m=1" "6#/%\"mG\"\"\"" } {TEXT -1 38 ". The Euler-Lagrange equations become" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Q3 := sub s(r=1,m=1,g=1,Q1);\nQ4 := subs(r=1,m=1,g=1,Q2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We can obtain a numeric solution via the syntax" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "QQ := dsolve(\{Q3,Q4 , u(0)=0,v(0)=3*Pi/4, D(u)(0)=1/2,D(v)(0)=0\}, \{u(t),v(t)\},numeric); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We chose initial conditions corresponding to the bob being raised through an angle of " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\" \"\"\"\"%!\"\"" }{TEXT -1 8 " in the " }{XPPEDIT 18 0 "xz" "6#%#xzG" } {TEXT -1 52 "-plane, then being given an initial velocity in the " } {XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 11 "-direction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "Maple has merely writ ten a procedure, which, when invoked, generates the desired numeric so lution. To extract this numeric solution, we write the following func tions which give the values for " }{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "U := s -> subs(QQ(s),u(t));\nV := s -> subs(QQ(s),v(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "F igure 48.11 contains graphs of " }{XPPEDIT 18 0 "u(t)" "6#-%\"uG6#%\"t G" }{TEXT -1 14 " in black and " }{XPPEDIT 18 0 "v(t)" "6#-%\"vG6#%\"t G" }{TEXT -1 39 " in cyan. It indicates that the angle " }{XPPEDIT 18 0 "u=theta" "6#/%\"uG%&thetaG" }{TEXT -1 36 " is an increasing func tion of time, " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 114 ". Thus, t he bob tends to rotate in a counterclockwise direction. The height of the bob is governed by the angle " }{XPPEDIT 18 0 "v=phi" "6#/%\"vG%$ phiG" }{TEXT -1 157 ", and this angle remains bounded away from zero. \+ Hence, the bob does not descend to equilibrium, but rather, oscillate s between maximum and minimum heights." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "plot([U,V],0..15,color =[black,cyan],labels=[t,``], xtickmarks=7, ytickmarks=8, labelfont=[TI MES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Figure 48.12, another way of obser ving the motion, is a parametric plot of " }{XPPEDIT 18 0 "v(t)" "6#-% \"vG6#%\"tG" }{TEXT -1 9 " against " }{XPPEDIT 18 0 "u(t)" "6#-%\"uG6# %\"tG" }{TEXT -1 17 ". It shows that " }{XPPEDIT 18 0 "u(t)" "6#-%\"u G6#%\"tG" }{TEXT -1 61 ", represented by the horizontal axis, is incre asing, whereas " }{XPPEDIT 18 0 "v(t)" "6#-%\"vG6#%\"tG" }{TEXT -1 95 ", represented by the vertical axis, undergoes a bounded oscillation w ith a nonzero lower bound." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "plot([U,V,0..15],color=black, labe ls=[u,`v `], view=[0..15,0..3], xtickmarks=7, ytickmarks=3, labelfont =[TIMES,ITALIC,12], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 260 "The actual motion of the bob takes place in three dimensions. Hence, Figures 48 .13 and 48.14 show the motion as a space curve. In Figure 48.13, the \+ space curve is viewed from directly above, so the curve seen is the ph ysical trajectory's projection onto the " }{XPPEDIT 18 0 "xy" "6#%#xyG " }{TEXT -1 99 "-plane. The initial point of the trajectory is near t he center of the right edge of the graph, at " }{XPPEDIT 18 0 "(x,y)=( 1/sqrt(2),0)" "6#/6$%\"xG%\"yG6$*&\"\"\"F)-%%sqrtG6#\"\"#!\"\"\"\"!" } {TEXT -1 110 ". The figure reveals a looping motion, with the long as is of the loops precessing counterclockwise about the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 6 "-axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "spacecurve([cos('U'(t))*s in('V'(t)),sin('U'(t))*sin('V'(t)),cos('V'(t))],t=0..50, color=black, \+ axes=boxed, numpoints=200, orientation=[-90,0], labels=[x,y,z],labelfo nt=[TIMES,ITALIC,12],tickmarks=[7,7,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The single qu ote marks in the " }{TEXT 258 10 "spacecurve" }{TEXT -1 59 " command a re required because U and V represent procedures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Figure 48.14 shows the tr ajectory of the bob in space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "spacecurve([cos('U'(t))*sin ('V'(t)),sin('U'(t))*sin('V'(t)),cos('V'(t))],t=0..30, color=black, ax es=boxed, numpoints=200, scaling=constrained, labels=[`x `,` y`,`z \+ `], labelfont=[TIMES,ITALIC,12], tickmarks=[7,7,[-1,-.7]], view=[-.71. .0.71,-.71..0.71,-1..-.6], orientation=[-45,45]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }