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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 30 "Classical Mechanics with \+
Maple" }}{PARA 256 "" 0 "" {TEXT -1 63 "Section 5.1: The Analytic Meth
od Using the Lagrangian Equations" }}{PARA 19 "" 0 "" {TEXT -1 41 "Dr.
 Harald Kammerer\nmaple@jademountain.de" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 14 "Initialisation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "libname:=\"C
:/mylib/m6dynlib\",\"C:/mylib/m6dynfig\",libname:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 80 "with(linalg):with(plots):with(plottools):wit
h(dynamics);with(figures_chapter_5);" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 24 "5.1: The Analytic Method" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 130 "Up to now we have considered mass particles, systems of mass p
articles and rigid bodies. Now we consider systems of rigid bodies. " 
}{TEXT -1 159 "Now we consider different methods to get the equation o
f motion for rigid bodies and systems of them. Here we compare two usu
al methods to get these equations." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 36 "5.1.1 Review of the Synthetic Method" }}{PARA 0 "" 0 "" {TEXT 
-1 292 "The synthetic method is well known in mechanics. To get the eq
uation which describes a mechanical model, one cuts all connection bet
ween the single parts of the system and all supports and considers the
 equilibrium of the free bodies. We have done this for the case of sin
gle rigid bodies in " }{HYPERLNK 17 "section 4.2.4" 1 "sec-04.mws" "eo
m" }{TEXT -1 59 ". We don't need the details at this point. Repeat exa
mple: " }{HYPERLNK 17 "cuboid and cylinder on a slope" 1 "sec-04.mws" 
"example 1" }{TEXT 256 73 " for details. Later in this section we cons
ider some additional examples." }}}{SECT 0 {PARA 258 "" 0 "" {TEXT -1 
41 "5.1.2 Introduction to the Analytic Method" }}{PARA 0 "" 0 "" 
{TEXT -1 180 "For clarity we derive the following relations only for s
ystems of mass particles. Then we needn't consider rotations. The resu
lts for the case of rigid bodies are analogous. As in " }{HYPERLNK 17 
"section 4.1.1" 1 "sec-04.mws" "4.1.1" }{TEXT -1 104 " the considered \+
mass particles are held together by boundaries. Their postition vector
s are the vectors " }{XPPEDIT 18 0 "r[i]" "6#&%\"rG6#%\"iG" }{TEXT -1 
2 ", " }{TEXT 257 1 "i" }{TEXT -1 10 "=1, 2, .. " }{TEXT 258 1 "n" }
{TEXT -1 35 ". The situation is shown in Fig. 1." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "display(Fig_5_1()[1],scaling=constrained,axes=
none,title=\"Figure 1\");" }}}{PARA 0 "" 0 "" {TEXT -1 43 "The resulta
nt force acting on the particle " }{TEXT 259 1 "i" }{TEXT -1 22 " shou
ld be denoted as " }{XPPEDIT 18 0 "F[i]" "6#&%\"FG6#%\"iG" }{TEXT -1 
72 " (notice that this is a vector). For the mass particle the we know
 from " }{HYPERLNK 17 "Newton's second law" 1 "sec-03.mws" "newton 2" 
}}{PARA 257 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "F[i]-m*a[i] = 0:" "6
#/,&&%\"FG6#%\"iG\"\"\"*&%\"mGF)&%\"aG6#F(F)!\"\"\"\"!" }}{PARA 0 "" 
0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[i];" "6#&%\"aG6#%\"iG" }
{TEXT -1 38 " is the acceleration of the particle: " }{XPPEDIT 18 0 "a
[i] := diff(r[i],`$`(t,2));" "6#>&%\"aG6#%\"iG-%%diffG6$&%\"rG6#F'-%\"
$G6$%\"tG\"\"#" }{TEXT -1 40 ". Now we define an virtual displacement \+
" }{XPPEDIT 18 0 "delta r[i]" "6#*&%&deltaG\"\"\"&%\"rG6#%\"iGF%" }
{TEXT -1 228 " for every mass particle. The conditions for this virtua
l displacements are that they are small and with the boundaries of the
 system compatible, but apart from that without restrictions. The mult
iplication of the equation with " }{XPPEDIT 18 0 "delta r[i]" "6#*&%&d
eltaG\"\"\"&%\"rG6#%\"iGF%" }{TEXT -1 54 " and addition of all equatio
ns yiels the principle of " }{TEXT 260 10 "d'Alembert" }}{PARA 257 "" 
0 "" {XPPEDIT 18 0 "sum((F[i]-m[i]*a[i])*delta*r[i],i = 1 .. n) = 0;" 
"6#/-%$sumG6$*(,&&%\"FG6#%\"iG\"\"\"*&&%\"mG6#F,F-&%\"aG6#F,F-!\"\"F-%
&deltaGF-&%\"rG6#F,F-/F,;F-%\"nG\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 34 "
In a prior section we had defined " }{HYPERLNK 17 "active" 1 "sec-03.m
ws" "active forces" }{TEXT -1 5 " and " }{HYPERLNK 17 "passive forces
" 1 "sec-03.mws" "passive forces" }{TEXT -1 243 ". We know that passiv
e forces which are caused by a boundary are always orthogonal to the b
oundary. And because the virtual displacements have to be compatible t
o the boundaries of the system they don't do work. So we needn't regar
d them here." }}{PARA 0 "" 0 "" {TEXT -1 121 "To describe the position
 of a mass particle in the three dimensional space we need three coord
inates. In general we need " }{TEXT 261 2 "3n" }{TEXT -1 44 " conditio
ns to describe the position of all " }{TEXT 262 1 "n" }{TEXT -1 99 " p
articles. The boundaries between the individual particles cause the nu
mber of degrees of freedom " }{TEXT 263 1 "f" }{TEXT -1 37 " of the to
tal system to be less than " }{TEXT 264 2 "3n" }{TEXT -1 23 ". Consequ
ently we need " }{XPPEDIT 18 0 "3n-f" "6#,&*&\"\"$\"\"\"%\"nGF&F&%\"fG
!\"\"" }{TEXT -1 49 " boundary conditions to describe the system. The \+
" }{TEXT 265 22 "Lagrangian coordinates" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "q[i]" "6#&%\"qG6#%\"iG" }{TEXT 266 8 ", i=1..f" }{TEXT -1 41 " are \+
the coordinates which belong to the " }{TEXT 267 1 "f" }{TEXT -1 59 " \+
degrees of freedom. Between them and the position vectors " }{XPPEDIT 
18 0 "r[i]" "6#&%\"rG6#%\"iG" }{TEXT -1 37 " there are transformation \+
conditions " }{XPPEDIT 18 0 "r[i]=r[i]" "6#/&%\"rG6#%\"iG&F%6#F'" }
{TEXT -1 1 "(" }{XPPEDIT 18 0 "q[1]" "6#&%\"qG6#\"\"\"" }{TEXT -1 4 ",
..," }{XPPEDIT 18 0 "q[f]" "6#&%\"qG6#%\"fG" }{TEXT -1 147 ") which we
 assume to be known. In general, these conditions can be dependent on \+
velocity and time. But in these course we don't consider this case." }
}{SECT 0 {PARA 5 "" 0 "EX 1" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "In Fig. 2 there are two connected cuboids. We define the \+
" }{TEXT 268 3 "x,y" }{TEXT -1 119 "-coordinate system, which we use t
o define the position vectors for both bodies. Additionally we define \+
the coordinate " }{TEXT 269 1 "s" }{TEXT -1 33 " starting from the ori
gin of the " }{TEXT 270 3 "x,y" }{TEXT -1 39 "-coordinate system, as s
hown in Fig. 2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "display(F
ig_5_2(),scaling=constrained,axes=none,title=\"Figure 2\");" }}}{PARA 
0 "" 0 "" {TEXT -1 72 "The total system has two bodies and 1 degree of
 freedom. The coordinate " }{TEXT 271 1 "s" }{TEXT -1 42 " is the Lagr
angian coordinate. So we need " }{XPPEDIT 18 0 "3*n-f=5" "6#/,&*&\"\"$
\"\"\"%\"nGF'F'%\"fG!\"\"\"\"&" }{TEXT -1 73 " boundary conditions. Be
cause we have a planar system we can assume that " }{XPPEDIT 18 0 "z[1
]=0" "6#/&%\"zG6#\"\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2]
=0" "6#/&%\"zG6#\"\"#\"\"!" }{TEXT -1 6 ". The " }{TEXT 273 1 "x" }
{TEXT -1 44 "-coordinate from the position of the cuboid " }{TEXT 284 
1 "1" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "x[1] = -L-D+s;" "6#/&%\"xG6#
\"\"\",(%\"LG!\"\"%\"DGF*%\"sGF'" }{TEXT -1 6 ". Its " }{TEXT 272 1 "y
" }{TEXT -1 15 "-coordinate is " }{XPPEDIT 18 0 "y[1]=0" "6#/&%\"yG6#
\"\"\"\"\"!" }{TEXT -1 29 ". The position of the cuboid " }{TEXT 285 
1 "2" }{TEXT -1 9 " has the " }{TEXT 274 1 "x" }{TEXT -1 12 "-coordina
te " }{XPPEDIT 18 0 "x[2]=s*cos(alpha)" "6#/&%\"xG6#\"\"#*&%\"sG\"\"\"
-%$cosG6#%&alphaGF*" }{TEXT -1 9 " and the " }{TEXT 275 1 "y" }{TEXT 
-1 12 "-coordinate " }{XPPEDIT 18 0 "y[2]=-s*sin(alpha)" "6#/&%\"yG6#
\"\"#,$*&%\"sG\"\"\"-%$sinG6#%&alphaGF+!\"\"" }{TEXT -1 60 ". In the i
nitial position which is shown in Fig. 2 there is " }{XPPEDIT 18 0 "s[
0] = D/2;" "6#/&%\"sG6#\"\"!*&%\"DG\"\"\"\"\"#!\"\"" }{TEXT -1 43 ". S
o we get the equations of transformation" }}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "r[1] := vector(3,[-L-D+s, 0, 0]);" "6#>&%\"rG6#\"\"\"-%
'vectorG6$\"\"$7%,(%\"LG!\"\"%\"DGF/%\"sGF'\"\"!F2" }}}{PARA 0 "" 0 "
" {TEXT -1 15 "for the cuboid " }{TEXT 286 1 "1" }{TEXT -1 4 " and" }}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "r[2]:=vector(3,[s*cos(alpha),-
s*sin(alpha),0]);" "6#>&%\"rG6#\"\"#-%'vectorG6$\"\"$7%*&%\"sG\"\"\"-%
$cosG6#%&alphaGF/,$*&F.F/-%$sinG6#F3F/!\"\"\"\"!" }}}{PARA 0 "" 0 "" 
{TEXT -1 15 "for the cuboid " }{TEXT 287 1 "2" }{TEXT -1 1 "." }}}
{PARA 0 "" 0 "" {TEXT -1 70 "Now we use the formula of Taylor to write
 for the virtual displacement" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "delt
a r[i]:=sum(diff(r[i],q[k])*delta q[k],k=1..f):" "6#>*&%&deltaG\"\"\"&
%\"rG6#%\"iGF&-%$sumG6$*(-%%diffG6$&F(6#F*&%\"qG6#%\"kGF&F%F&&F56#F7F&
/F7;F&%\"fG" }}{PARA 0 "" 0 "" {TEXT -1 39 "With this we get for the e
quation above" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum(F[i]-m*a[i],i=1.
.n)*sum(diff(r[i],q[k])*delta*q[k],k = 1 .. f)=0" "6#/*&-%$sumG6$,&&%
\"FG6#%\"iG\"\"\"*&%\"mGF-&%\"aG6#F,F-!\"\"/F,;F-%\"nGF--F&6$*(-%%diff
G6$&%\"rG6#F,&%\"qG6#%\"kGF-%&deltaGF-&FA6#FCF-/FC;F-%\"fGF-\"\"!" }}
{PARA 0 "" 0 "" {TEXT -1 23 "Some rearranging yields" }}{PARA 257 "" 
0 "" {XPPEDIT 18 0 "sum(sum(F[i]*diff(r[i],q[i])-m[i]*a[i]*diff(r[i],q
[k]),i=1..n)*delta q[k],k=1..f)=0" "6#/-%$sumG6$*(-F%6$,&*&&%\"FG6#%\"
iG\"\"\"-%%diffG6$&%\"rG6#F/&%\"qG6#F/F0F0*(&%\"mG6#F/F0&%\"aG6#F/F0-F
26$&F56#F/&F86#%\"kGF0!\"\"/F/;F0%\"nGF0%&deltaGF0&F86#FGF0/FG;F0%\"fG
\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 69 "The virtual displacements can be
 choosen freely. So we can set every " }{XPPEDIT 18 0 "delta r[k]" "6#
*&%&deltaG\"\"\"&%\"rG6#%\"kGF%" }{TEXT -1 41 " to zero except one of \+
them, for example " }{XPPEDIT 18 0 "delta r[1]" "6#*&%&deltaG\"\"\"&%
\"rG6#F%F%" }{TEXT -1 16 ". Remember that " }{XPPEDIT 18 0 "F[i]" "6#&
%\"FG6#%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "r[i]" "6#&%\"rG6#%\"iG
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT 
-1 73 " are vectors. Write the resulting equation with use of coordina
tes we get" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum(F[x[i]]*diff(x[i],q
[1])+F[y[i]]*diff(y[i],q[1])+F[z[i]]*diff(y[i],q[1])-m[i]*(a[x[i]]*dif
f(x[i],q[1])+a[y[i]]*diff(y[i],q[1])+a[z[i]]*diff(z[i],q[1])),i = 1 ..
 n)*delta*q[1] = 0;" "6#/*(-%$sumG6$,**&&%\"FG6#&%\"xG6#%\"iG\"\"\"-%%
diffG6$&F.6#F0&%\"qG6#F1F1F1*&&F+6#&%\"yG6#F0F1-F36$&F>6#F0&F86#F1F1F1
*&&F+6#&%\"zG6#F0F1-F36$&F>6#F0&F86#F1F1F1*&&%\"mG6#F0F1,(*&&%\"aG6#&F
.6#F0F1-F36$&F.6#F0&F86#F1F1F1*&&FY6#&F>6#F0F1-F36$&F>6#F0&F86#F1F1F1*
&&FY6#&FJ6#F0F1-F36$&FJ6#F0&F86#F1F1F1F1!\"\"/F0;F1%\"nGF1%&deltaGF1&F
86#F1F1\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 26 "where we use the notation
 " }{XPPEDIT 18 0 "diff(x[i],t$2)=a[x[i]]" "6#/-%%diffG6$&%\"xG6#%\"iG
-%\"$G6$%\"tG\"\"#&%\"aG6#&F(6#F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "di
ff(y[i],t$2)=a[y[i]]" "6#/-%%diffG6$&%\"yG6#%\"iG-%\"$G6$%\"tG\"\"#&%
\"aG6#&F(6#F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(z[i],t$2)=a[z[
i]]" "6#/-%%diffG6$&%\"zG6#%\"iG-%\"$G6$%\"tG\"\"#&%\"aG6#&F(6#F*" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "We need some additional \+
relations. But we don't prove these relations here." }}{PARA 0 "" 0 "
" {TEXT -1 2 "1:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "a[x[i]]*diff(x[i]
,q[1])=diff(v[x[i]]*diff(x[i],q[1]),t)-v[x[i]]*diff(diff(x[1],q[1]),t)
" "6#/*&&%\"aG6#&%\"xG6#%\"iG\"\"\"-%%diffG6$&F)6#F+&%\"qG6#F,F,,&-F.6
$*&&%\"vG6#&F)6#F+F,-F.6$&F)6#F+&F36#F,F,%\"tGF,*&&F:6#&F)6#F+F,-F.6$-
F.6$&F)6#F,&F36#F,FDF,!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "with " }{XPPEDIT 18 0 "v[x[i]]=diff(x[i],t)" "6#/&%\"vG6#&%\"xG6
#%\"iG-%%diffG6$&F(6#F*%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 2 "2:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(x[i],q[1])=dif
f(v[x[i]],v[q[1]])" "6#/-%%diffG6$&%\"xG6#%\"iG&%\"qG6#\"\"\"-F%6$&%\"
vG6#&F(6#F*&F26#&F,6#F." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "
with " }{XPPEDIT 18 0 "v[q[1]] = diff(q[1],t);" "6#/&%\"vG6#&%\"qG6#\"
\"\"-%%diffG6$&F(6#F*%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 2 "3:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(diff(x[i],q[1]),t)=d
iff(v[x[i]],q[1])" "6#/-%%diffG6$-F%6$&%\"xG6#%\"iG&%\"qG6#\"\"\"%\"tG
-F%6$&%\"vG6#&F*6#F,&F.6#F0" }}{PARA 0 "" 0 "" {TEXT -1 33 "With use o
f 2 and 3 we get for 1:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "a[x[i]]*di
ff(x[i],q[1])=diff((v[x[i]]*diff(v[x[i]],v[q[1]])),t)-v[x[i]]*diff(v[x
[i]],q[1])" "6#/*&&%\"aG6#&%\"xG6#%\"iG\"\"\"-%%diffG6$&F)6#F+&%\"qG6#
F,F,,&-F.6$*&&%\"vG6#&F)6#F+F,-F.6$&F:6#&F)6#F+&F:6#&F36#F,F,%\"tGF,*&
&F:6#&F)6#F+F,-F.6$&F:6#&F)6#F+&F36#F,F,!\"\"" }}{PARA 257 "" 0 "" 
{XPPEDIT 18 0 "a[x[i]]*diff(x[i],q[1])=diff(diff(v[x[i]]**2/2,v[q[1]])
,t)-diff(v[x[i]]**2/2,q[1])" "6#/*&&%\"aG6#&%\"xG6#%\"iG\"\"\"-%%diffG
6$&F)6#F+&%\"qG6#F,F,,&-F.6$-F.6$*&&%\"vG6#&F)6#F+\"\"#F@!\"\"&F<6#&F3
6#F,%\"tGF,-F.6$*&&F<6#&F)6#F+F@F@FA&F36#F,FA" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "The congruent results we get for " }{TEXT 276 1 "y" }
{TEXT -1 5 " and " }{TEXT 277 1 "z" }{TEXT -1 28 ". Using these result
s yields" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "sum(m[i]*(diff(diff((v[x[
i]]**2+v[y[i]]**2+v[z[i]]**2)/2,v[q[1]]),t)-diff((v[x[i]]**2+v[y[i]]**
2+v[z[i]]**2)/2,q[1])),i=1..n)*delta q[1]=(sum(F[x[i]]*diff(x[i],q[1])
+F[y[i]]*diff(y[i],q[1])+F[z[i]]*diff(z[i],q[1]),i=1..n)*delta q[1]" "
6#/*(-%$sumG6$*&&%\"mG6#%\"iG\"\"\",&-%%diffG6$-F06$*&,(*$&%\"vG6#&%\"
xG6#F,\"\"#F-*$&F86#&%\"yG6#F,F=F-*$&F86#&%\"zG6#F,F=F-F-F=!\"\"&F86#&
%\"qG6#F-%\"tGF--F06$*&,(*$&F86#&F;6#F,F=F-*$&F86#&FB6#F,F=F-*$&F86#&F
H6#F,F=F-F-F=FJ&FN6#F-FJF-/F,;F-%\"nGF-%&deltaGF-&FN6#F-F-*(-F&6$,(*&&
%\"FG6#&F;6#F,F--F06$&F;6#F,&FN6#F-F-F-*&&F\\p6#&FB6#F,F--F06$&FB6#F,&
FN6#F-F-F-*&&F\\p6#&FH6#F,F--F06$&FH6#F,&FN6#F-F-F-/F,;F-FboF-FcoF-&FN
6#F-F-" }}{PARA 0 "" 0 "" {TEXT -1 8 "The term" }}{PARA 257 "" 0 "" 
{XPPEDIT 18 0 "T = sum(m[i]*(v[x[i]]^2+v[y[i]]^2+v[z[i]]^2),i = 1 .. n
)/2;" "6#/%\"TG*&-%$sumG6$*&&%\"mG6#%\"iG\"\"\",(*$&%\"vG6#&%\"xG6#F-
\"\"#F.*$&F26#&%\"yG6#F-F7F.*$&F26#&%\"zG6#F-F7F.F./F-;F.%\"nGF.F7!\"
\"" }}{PARA 0 "" 0 "" {TEXT -1 100 "is the total kinetic energy of the
 system of mass particles. But consider that the position vectors " }
{XPPEDIT 18 0 "r[i]" "6#&%\"rG6#%\"iG" }{TEXT -1 59 " must be expresse
d as functions of the Lagrnge coordinates " }{XPPEDIT 18 0 "q[1] .. q[
f]" "6#;&%\"qG6#\"\"\"&F%6#%\"fG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The same precedure can be
 used for the virtual displacements " }{XPPEDIT 18 0 "delta q[2] .. de
lta q[f]" "6#;*&%&deltaG\"\"\"&%\"qG6#\"\"#F&*&F%F&&F(6#%\"fGF&" }
{TEXT -1 20 ". Altogether we get " }{TEXT 278 1 "f" }{TEXT -1 10 " rel
ations" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(diff(T,v[q[k]]),t)-dif
f(T,q[k]) = Q[k];" "6#/,&-%%diffG6$-F&6$%\"TG&%\"vG6#&%\"qG6#%\"kG%\"t
G\"\"\"-F&6$F*&F/6#F1!\"\"&%\"QG6#F1" }{TEXT -1 4 " ;  " }{XPPEDIT 18 
0 "k=1..f" "6#/%\"kG;\"\"\"%\"fG" }}{PARA 0 "" 0 "" {TEXT -1 17 "We ca
ll them the " }{TEXT 279 20 "Lagrangian equations" }{TEXT -1 33 ". The
 right side of the equations" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "Q[k]=
sum(F[x[i]]*diff(x[i],q[k])+F[y[i]]*diff(y[i],q[k])+F[z[i]]*diff(z[i],
q[k]),i = 1 .. n)" "6#/&%\"QG6#%\"kG-%$sumG6$,(*&&%\"FG6#&%\"xG6#%\"iG
\"\"\"-%%diffG6$&F16#F3&%\"qG6#F'F4F4*&&F.6#&%\"yG6#F3F4-F66$&FA6#F3&F
;6#F'F4F4*&&F.6#&%\"zG6#F3F4-F66$&FM6#F3&F;6#F'F4F4/F3;F4%\"nG" }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "Q[k]=sum(F[i]*diff(r[i],q[k]),i=1..n)
" "6#/&%\"QG6#%\"kG-%$sumG6$*&&%\"FG6#%\"iG\"\"\"-%%diffG6$&%\"rG6#F/&
%\"qG6#F'F0/F/;F0%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 13 "(notice that " 
}{XPPEDIT 18 0 "F[i]" "6#&%\"FG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "r[i]" "6#&%\"rG6#%\"iG" }{TEXT -1 47 " in the last relations are
 vectors) are called " }{TEXT 280 18 "generalized forces" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We \+
have mentioned that " }{XPPEDIT 18 0 "F[i]" "6#&%\"FG6#%\"iG" }{TEXT 
-1 51 " is the resultant of all acting forces on particle " }{TEXT 
281 1 "i" }{TEXT -1 57 " without consideration of the passive forces. \+
So we have " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "F[i] = F[i,conservativ
e]+F[i,nonconservative];" "6#/&%\"FG6#%\"iG,&&F%6$F'%-conservativeG\"
\"\"&F%6$F'%0nonconservativeGF," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 144 "with the definition from section 3.2 for conservative an
d non-conservative forces. For the conservative forces we have with th
e total potential " }{TEXT 282 1 "V" }{TEXT -1 0 "" }}{PARA 257 "" 0 "
" {XPPEDIT 18 0 "F[x[i,cons]] = -diff(V,x[i]),F[y[i,cons]] = -diff(V,y
[i]),F[z[i,cons]] = -diff(V,z[i]);" "6%/&%\"FG6#&%\"xG6$%\"iG%%consG,$
-%%diffG6$%\"VG&F(6#F*!\"\"/&F%6#&%\"yG6$F*F+,$-F.6$F0&F86#F*F3/&F%6#&
%\"zG6$F*F+,$-F.6$F0&FC6#F*F3" }}{PARA 0 "" 0 "" {TEXT -1 47 "Use thes
e relation in the equation above we get" }}{PARA 257 "" 0 "" {XPPEDIT 
18 0 "Q[k,conservative]=-sum((diff(V,x[i])*diff(x[i],q[k])+diff(V,y[i]
)*diff(y[i],q[k])+diff(V,z[i])*diff(z[i],q[k])),i=1..n)" "6#/&%\"QG6$%
\"kG%-conservativeG,$-%$sumG6$,(*&-%%diffG6$%\"VG&%\"xG6#%\"iG\"\"\"-F
06$&F46#F6&%\"qG6#F'F7F7*&-F06$F2&%\"yG6#F6F7-F06$&FC6#F6&F=6#F'F7F7*&
-F06$F2&%\"zG6#F6F7-F06$&FO6#F6&F=6#F'F7F7/F6;F7%\"nG!\"\"" }}{PARA 
257 "" 0 "" {XPPEDIT 18 0 "Q[k,conservative]=-diff(V,q[k])" "6#/&%\"QG
6$%\"kG%-conservativeG,$-%%diffG6$%\"VG&%\"qG6#F'!\"\"" }}{PARA 0 "" 
0 "" {TEXT -1 14 "So we get the " }{TEXT 283 20 "Lagrangian equations
" }{TEXT -1 23 " in the following form:" }}{PARA 257 "" 0 "" {XPPEDIT 
18 0 "diff(diff(T,v[q[k]]),t)-diff(T,q[k])+diff(V,q[k]) = Q[k,nc];" "6
#/,(-%%diffG6$-F&6$%\"TG&%\"vG6#&%\"qG6#%\"kG%\"tG\"\"\"-F&6$F*&F/6#F1
!\"\"-F&6$%\"VG&F/6#F1F3&%\"QG6$F1%#ncG" }{TEXT -1 6 ",     " }
{XPPEDIT 18 0 "k=1..f" "6#/%\"kG;\"\"\"%\"fG" }}{PARA 0 "" 0 "" {TEXT 
-1 6 "where " }{XPPEDIT 18 0 "v[q[k]]=diff(q[k],t)" "6#/&%\"vG6#&%\"qG
6#%\"kG-%%diffG6$&F(6#F*%\"tG" }{TEXT -1 5 ".The " }{XPPEDIT 18 0 "Q[k
,nc]" "6#&%\"QG6$%\"kG%#ncG" }{TEXT -1 48 " are the generalized forces
, which follows from " }{XPPEDIT 18 0 "Q[k] = sum(F[i]*diff(r[i],q[k])
,i = 1 .. n)" "6#/&%\"QG6#%\"kG-%$sumG6$*&&%\"FG6#%\"iG\"\"\"-%%diffG6
$&%\"rG6#F/&%\"qG6#F'F0/F/;F0%\"nG" }{TEXT -1 39 " when only the non-c
onservative forces " }{XPPEDIT 18 0 "F[i,nonconservative]" "6#&%\"FG6$
%\"iG%0nonconservativeG" }{TEXT -1 16 " are considered." }}{PARA 0 "" 
0 "" {TEXT -1 292 "At last we point out that the Lagrangian equations \+
are also valid for rigid bodies and systems of them when in the kineti
c energy the part of the rotations is taken into account. It is not ne
cessary to derive this fact in detail. We will see this in the followi
ng continuation of the example " }{HYPERLNK 17 "above." 1 "" "EX 1" }}
{SECT 0 {PARA 5 "" 0 "" {TEXT -1 24 "Example 1 (continuation)" }}
{PARA 0 "" 0 "" {TEXT -1 100 "For clarity  we don't use the prepared f
unctions. The total kinetic energy of the system is given by" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "T:=m[1]*(v[x[1]]**2+v[y[1]]*
*2+v[z[1]]**2)/2+m[2]*(v[x[2]]**2+v[y[2]]**2+v[z[2]]**2)/2;" }}}{PARA 
0 "" 0 "" {TEXT -1 47 "From the transformation equations above we know
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(r[1]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(r[2]);" }}}{PARA 0 "" 0 "" 
{TEXT -1 50 "For the velocities we get from these the relations" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rel1:=v[x[1]]=v,v[y[1]]=0,v[
z[1]]=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "rel2:=v[x[2]]=v
*cos(alpha),v[y[2]]=-v*sin(alpha),v[z[2]]=0;" }}}{PARA 0 "" 0 "" 
{TEXT -1 5 "with " }{XPPEDIT 18 0 "v=diff(s,t)" "6#/%\"vG-%%diffG6$%\"
sG%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "and for the ki
netic energy we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Ts:=s
ubs(\{rel1,rel2\},T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ts
:=simplify(Ts,trig);" }}}{PARA 0 "" 0 "" {TEXT -1 164 "In this example
 we assume that there is no friction. Then there are only conservative
 forces. We have only the potential of gravity. The total potential is
 given by" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "V:=m[1]*g*y[1]+
m[2]*g*y[2];" }}}{PARA 0 "" 0 "" {TEXT -1 52 "And by use of the transf
ormation equation it follows" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "Vs:=subs(\{y[1]=r[1][2],y[2]=r[2][2]\},V);" }}}{PARA 0 "" 0 "" 
{TEXT -1 152 "Here we see that it is usually possible to formulate the
 relations for the kinetic energy and the potential direct by use of t
he Lagrangian coordinates." }}{PARA 0 "" 0 "" {TEXT -1 75 "Because we \+
have only one degree of freedom, the Lagrangian equation is here" }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(diff(Ts,v[s]),t)-diff(Ts,s)+diff
(Vs,s) = 0;" "6#/,(-%%diffG6$-F&6$%#TsG&%\"vG6#%\"sG%\"tG\"\"\"-F&6$F*
F.!\"\"-F&6$%#VsGF.F0\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 29 "The necessa
ry derivatives are" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L1:=di
ff(Ts,v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L2:=diff(Ts,s)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L3:=diff(Vs,s);" }}}
{PARA 0 "" 0 "" {TEXT -1 34 "At last we need the derivative of " }
{TEXT 288 2 "L1" }{TEXT -1 28 " with respect to time. With " }
{XPPEDIT 18 0 "a:=diff(v,t)" "6#>%\"aG-%%diffG6$%\"vG%\"tG" }{TEXT -1 
7 " we get" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "L1a:=a*(m[1]+m
[2]);" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Altogether this yields" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "EOMa:=L1a-L2+L3=0;" }}}
{PARA 0 "" 0 "" {TEXT -1 22 "or with some rewriting" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "EOM:=subs(\{a=diff(s(t),t$2),s=s(t)\},EOMa)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 125 "In the next section we \+
consider two more examples to illustrate the difference between the sy
nthetic and the analytic method." }}}}}{MARK "0 0 0" 14 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }