Solution 1 to problem e3c2new
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3c2new
Equations
The following unsolved equations remain:
2 2
0=b31 + b32
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
- b31*n2
n1=-----------
b32
- b32*m2
m1=-----------
b31
10 2 4 2
----*b31 *n2 + ---*b32 *n2
3 3
m3=----------------------------
a33*b32
104 2 2 56 2 2
- 16*a33*b32*k1*m2*n2*n3 - -----*b31 *k1*n2*n3 - ----*b32 *k1*n2*n3
3 3
r6=------------------------------------------------------------------------
4
a33 *b32
2 104 2 2 56 2 2
- 16*a33*b32*k1*m2*n2 - -----*b31 *k1*n2 *n3 - ----*b32 *k1*n2 *n3
3 3
r5=----------------------------------------------------------------------
4
a33 *b32
2 104 3 2 56 2 2
16*a33*b31*b32*k1*m2*n2 + -----*b31 *k1*n2 *n3 + ----*b31*b32 *k1*n2 *n3
3 3
r4=---------------------------------------------------------------------------
4 2
a33 *b32
r3=0
r2=0
r1=0
4 8 2 2 112 4
40*b31 *k1*m2*n2 + ---*b31 *b32 *k1*m2*n2 - -----*b32 *k1*m2*n2
3 3
q20=-----------------------------------------------------------------
3 3
a33 *b32
4 8 2 2 112 4
- 40*b31 *k1*m2*n2 - ---*b31 *b32 *k1*m2*n2 + -----*b32 *k1*m2*n2
3 3
q19=--------------------------------------------------------------------
3 2
a33 *b31*b32
64 2 2 64 2 2
8*a33*b32*k1*m2*n3 - ----*b31 *k1*n2 - ----*b32 *k1*n2
3 3
q17=----------------------------------------------------------
3
a33 *b32
3 160 4 2 160 2 2 2
- 8*a33*b32 *k1*m2*n3 + -----*b31 *k1*n2 + -----*b31 *b32 *k1*n2
3 3
q16=---------------------------------------------------------------------
3 2
a33 *b31*b32
2 2 2 2 2 4 2 580 6 2
q15=(4*a33 *b31 *b32 *k1*m2 + 4*a33 *b32 *k1*m2 - -----*b31 *k1*n2
9
704 4 2 2 124 2 4 2 4 2 2
- -----*b31 *b32 *k1*n2 - -----*b31 *b32 *k1*n2 )/(a33 *b31 *b32 )
9 9
5 2 3 2 2 4 2
40*b31 *k1*n2 + 64*b31 *b32 *k1*n2 + 24*b31*b32 *k1*n2
q14=-----------------------------------------------------------
4 3
a33 *b32
2 2
16*b31 *k1*n2
q13=----------------
3
a33 *b32
16 2 8 2
16*a33*b32*k1*m2*n2 - ----*b31 *k1*n2*n3 + ---*b32 *k1*n2*n3
3 3
q12=--------------------------------------------------------------
3
a33 *b32
16*a33*b31*k1*m2*n2 - 8*b31*b32*k1*n2*n3
q11=------------------------------------------
3
a33 *b32
2 2 2 2 2 4 2 256 6 2
q10=(4*a33 *b31 *b32 *k1*m2 + 4*a33 *b32 *k1*m2 - -----*b31 *k1*n2
9
200 4 2 2 56 2 4 2 4 2 2
- -----*b31 *b32 *k1*n2 + ----*b31 *b32 *k1*n2 )/(a33 *b31 *b32 )
9 9
3 2 2 2
16*b31 *k1*n2 + 32*b31*b32 *k1*n2
q9=-------------------------------------
3 2
a33 *b32
2
16*a33*b32*k1*m2*n2 + 8*b31 *k1*n2*n3
q8=---------------------------------------
3
a33 *b31
56 2 80 2
16*a33*b32*k1*m2*n2 + ----*b31 *k1*n2*n3 + ----*b32 *k1*n2*n3
3 3
q7=---------------------------------------------------------------
3
a33 *b32
8*k1*n2*n3
q5=------------
2
a33
- 8*b31*k1*n2*n3
q4=-------------------
2
a33 *b32
2 2
4*k1*n2 - 4*k1*n3
q3=---------------------
2
a33
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2 2 2 2
4*b31 *k1*n2 - 4*b32 *k1*n3
q1=-------------------------------
2 2
a33 *b32
16 4 16 2 2
----*b31 *k1*n2 + ----*b31 *b32 *k1*n2
3 3
p56=----------------------------------------
4
a33 *b32
2 2
- 4*b31 *k1*m2 - 4*b32 *k1*m2
p55=--------------------------------
3
a33
2 3
4*b31 *b32*k1*m2 + 4*b32 *k1*m2
p54=---------------------------------
3
a33 *b31
2 2
- 4*b31 *k1*n3 - 4*b32 *k1*n3
p53=--------------------------------
3
a33
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p52=-----------------------------
3
a33
3 2
- 8*b31 *k1*n2 - 8*b31*b32 *k1*n2
p51=------------------------------------
3
a33 *b32
40 4 80 2 2 40 4
- ----*b31 *k1*n2 - ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
3 3 3
p50=-------------------------------------------------------------
4
a33 *b32
p49=0
4 8 2 2 112 4
40*b31 *k1*n2 + ---*b31 *b32 *k1*n2 - -----*b32 *k1*n2
3 3
p48=--------------------------------------------------------
3 2
a33 *b32
p47=0
p46=0
16 4 32 2 2 16 4
- ----*b31 *k1*n2 - ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
3 3 3
p45=-------------------------------------------------------------
4
a33 *b32
5 8 3 2 112 4
40*b31 *k1*n2 + ---*b31 *b32 *k1*n2 - -----*b31*b32 *k1*n2
3 3
p44=------------------------------------------------------------
3 3
a33 *b32
p43=0
p42=0
16 2 16 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p41=-----------------------------------
2
a33 *b32
8*b32*k1*n3
p40=-------------
2
a33
8*b31*k1*n3
p39=-------------
2
a33
16 2 16 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p38=-----------------------------------
2
a33 *b32
p37=0
40 2 40 2
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p36=-----------------------------------
2
a33 *b32
p35=0
p34=0
p33=0
p32=0
3 2
- 8*b31 *k1*n2 - 8*b31*b32 *k1*n2
p31=------------------------------------
3
a33 *b32
p30=0
p29=0
3 2
8*b31 *k1*n2 + 8*b31*b32 *k1*n2
p28=---------------------------------
3
a33 *b32
2 2
- 8*b31 *k1*n2 - 8*b32 *k1*n2
p27=--------------------------------
3
a33
p26=0
112 2 160 2
-----*b31 *k1*n2 + -----*b32 *k1*n2
3 3
p25=-------------------------------------
2
a33 *b32
16*b31*k1*n2
p24=--------------
2
a33
- 4*k1*m2
p23=------------
a33
8*b32*k1*m2
p22=-------------
a33*b31
4*k1*m2
p21=---------
a33
p20=0
p19=0
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p18=-----------------------------
3
a33
p17=0
p16=0
- 16*b31*k1*n2
p15=-----------------
2
a33
32 2 16 2
- ----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
p14=--------------------------------------
2
a33 *b32
- 4*b32*k1*m2
p13=----------------
a33*b31
- 8*k1*m2
p12=------------
a33
4*b32*k1*m2
p11=-------------
a33*b31
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
- 4*k1*n2
p4=------------
a33
4*b31*k1*n2
p3=-------------
a33*b32
- 4*k1*n2
p2=------------
a33
4*b31*k1*n2
p1=-------------
a33*b32
k125=0
k124=0
k122=0
k121=0
k120=0
k119=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k118=------------------------------
3
a33
k117=0
k116=0
k115=0
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k114=------------------------------
3
a33
k113=0
k112=0
k110=0
k109=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k108=--------------------------
2
a33
k107=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k106=--------------------------
2
a33
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
2 2
8*b31 *k1 + 8*b32 *k1
k95=-----------------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
2 2
8*b31 *k1 + 8*b32 *k1
k84=-----------------------
2
a33
k83=0
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
k64=0
k63=0
k62=0
k61=0
k60=0
k59=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k58=--------------------------
2
a33
k57=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k56=--------------------------
2
a33
k55=0
k54=0
k53=0
k52=0
k51=0
k50=0
k49=0
k48=0
k47=0
k46=0
k45=0
k44=0
k43=0
- 4*b32*k1
k42=-------------
a33
- 8*b31*k1
k41=-------------
a33
4*b32*k1
k40=----------
a33
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k30=0
k29=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k28=--------------------------
2
a33
k27=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k26=--------------------------
2
a33
k25=0
k24=0
k23=0
4*b31*k1
k22=----------
a33
- 8*b32*k1
k21=-------------
a33
- 4*b31*k1
k20=-------------
a33
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
k7=0
k6=0
k5=k1
k4=0
k3=2*k1
k2=0
Parameters
Apart from the condition that they must not vanish to give
a non-trivial solution and a non-singular solution with
non-vanishing denominators, the following parameters are free:
b32, n3, m2, k1, b31, n2, a33
Inequalities
In the following not identically vanishing expressions are shown.
Any auxiliary variables g00?? are used to express that at least
one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3
means that either p4 or p3 or both are non-vanishing.
3 3 3 2 2
{a33 *w123 + 2*a33 *w125 + a33 *w127 - 8*a33 *b31*w087 + 4*a33 *b31*w106
2 2 2 2
- 4*a33 *b31*w108 - 4*a33 *b32*w086 + 4*a33 *b32*w088 - 8*a33 *b32*w107
2 2 2 2
- 2*a33*b31 *w020 - 2*a33*b31 *w022 + 8*a33*b31 *w033 + 8*a33*b31 *w044
2 2 2 2
- 2*a33*b31 *w070 - 2*a33*b31 *w072 - 2*a33*b31 *w100 - 2*a33*b31 *w102
2 2 2 2
- 2*a33*b32 *w020 - 2*a33*b32 *w022 + 8*a33*b32 *w033 + 8*a33*b32 *w044
2 2 2 2
- 2*a33*b32 *w070 - 2*a33*b32 *w072 - 2*a33*b32 *w100 - 2*a33*b32 *w102
3 2 2 3
- 4*b31 *w014 - 4*b31 *b32*w010 - 4*b31*b32 *w014 - 4*b32 *w010,
b32,
n3,
b31,
n2,
a33,
k1,
m2}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(3*a33 *b31*b32*u1 + 3*a33 *b31*b32*u2 + 6*a33 *b31*b32*u3
2 2 2
+ 6*a33*b31 *b32*u3*v1 - 6*a33*b31 *n2*u1 + 6*a33*b31*b32 *u3*v2
+ 6*a33*b31*b32*m2*v2 + 6*a33*b31*b32*n2*u2 + 6*a33*b31*b32*n3*u3
2 3 2
- 6*a33*b32 *m2*v1 + 20*b31 *n2*v3 + 8*b31*b32 *n2*v3)/(6*a33*b31*b32)
has apart from the Hamiltonian and Casimirs the following only first integral:
4 2 3 4 4 2 3 2 2 4 2 3 4
INT=9*a33 *b31 *b32 *u1 + 18*a33 *b31 *b32 *u1 *u2 + 9*a33 *b31 *b32 *u2
3 3 3 2 3 3 3
- 36*a33 *b31 *b32 *u1 *u3*v1 - 72*a33 *b31 *b32 *u1*u2*u3*v2
3 3 3 2 3 3 2 3
+ 36*a33 *b31 *b32 *u2 *u3*v1 + 36*a33 *b31 *b32 *n2*u1
3 3 2 2 3 2 4 2
+ 36*a33 *b31 *b32 *n2*u1*u2 + 36*a33 *b31 *b32 *u1 *u3*v2
3 2 4 3 2 4 2
- 72*a33 *b31 *b32 *u1*u2*u3*v1 - 36*a33 *b31 *b32 *u2 *u3*v2
3 2 3 2 3 2 3
+ 36*a33 *b31 *b32 *m2*u1 *v2 - 72*a33 *b31 *b32 *m2*u1*u2*v1
3 2 3 2 3 2 3 2
- 36*a33 *b31 *b32 *m2*u2 *v2 - 36*a33 *b31 *b32 *n2*u1 *u2
3 2 3 3 3 2 3 2
- 36*a33 *b31 *b32 *n2*u2 - 36*a33 *b31 *b32 *n3*u1 *u3
3 2 3 2 3 4 2
- 36*a33 *b31 *b32 *n3*u2 *u3 + 36*a33 *b31*b32 *m2*u1 *v1
3 4 3 4 2
+ 72*a33 *b31*b32 *m2*u1*u2*v2 - 36*a33 *b31*b32 *m2*u2 *v1
2 4 3 2 2 2 4 3 2 2
- 18*a33 *b31 *b32 *u1 *v1 - 18*a33 *b31 *b32 *u1 *v2
2 4 3 2 2 2 4 3
- 18*a33 *b31 *b32 *u1 *v3 + 72*a33 *b31 *b32 *u1*u3*v1*v3
2 4 3 2 2 2 4 3 2 2
- 18*a33 *b31 *b32 *u2 *v1 - 18*a33 *b31 *b32 *u2 *v2
2 4 3 2 2 2 4 3
- 18*a33 *b31 *b32 *u2 *v3 + 72*a33 *b31 *b32 *u2*u3*v2*v3
2 4 2 2 2 4 2
+ 120*a33 *b31 *b32 *n2*u1 *v3 - 96*a33 *b31 *b32 *n2*u1*u3*v1
2 4 2 2 2 4 2
+ 48*a33 *b31 *b32 *n2*u2 *v3 + 336*a33 *b31 *b32 *n2*u2*u3*v2
2 4 2 2 2 4 2 2
+ 48*a33 *b31 *b32 *n2*u3 *v3 + 36*a33 *b31 *b32*n2 *u1
2 3 3 2 3 3
+ 144*a33 *b31 *b32 *n2*u1*u3*v2 - 144*a33 *b31 *b32 *n2*u2*u3*v1
2 3 3 2 3 2
+ 72*a33 *b31 *b32 *n3*u1*u3*v3 + 144*a33 *b31 *b32 *m2*n2*u1*v2
2 3 2 2 2 3 2
- 72*a33 *b31 *b32 *n2 *u1*u2 - 72*a33 *b31 *b32 *n2*n3*u1*u3
2 2 5 2 2 2 2 5 2 2
- 18*a33 *b31 *b32 *u1 *v1 - 18*a33 *b31 *b32 *u1 *v2
2 2 5 2 2 2 2 5
- 18*a33 *b31 *b32 *u1 *v3 + 72*a33 *b31 *b32 *u1*u3*v1*v3
2 2 5 2 2 2 2 5 2 2
- 18*a33 *b31 *b32 *u2 *v1 - 18*a33 *b31 *b32 *u2 *v2
2 2 5 2 2 2 2 5
- 18*a33 *b31 *b32 *u2 *v3 + 72*a33 *b31 *b32 *u2*u3*v2*v3
2 2 4 2 2 2 4
+ 120*a33 *b31 *b32 *n2*u1 *v3 + 48*a33 *b31 *b32 *n2*u1*u3*v1
2 2 4 2 2 2 4
+ 48*a33 *b31 *b32 *n2*u2 *v3 + 480*a33 *b31 *b32 *n2*u2*u3*v2
2 2 4 2 2 2 4
+ 48*a33 *b31 *b32 *n2*u3 *v3 + 72*a33 *b31 *b32 *n3*u2*u3*v3
2 2 3 2 2 2 2 3 2 2
+ 36*a33 *b31 *b32 *m2 *v1 + 36*a33 *b31 *b32 *m2 *v2
2 2 3 2 2 3
+ 144*a33 *b31 *b32 *m2*n2*u1*v1 + 144*a33 *b31 *b32 *m2*n2*u2*v2
2 2 3 2 2 3 2 2
+ 72*a33 *b31 *b32 *m2*n3*u2*v3 + 36*a33 *b31 *b32 *n2 *u2
2 2 3 2 2 3 2 2
+ 72*a33 *b31 *b32 *n2*n3*u2*u3 - 36*a33 *b31 *b32 *n3 *u1
2 2 3 2 2 2 4
- 36*a33 *b31 *b32 *n3 *u2 + 144*a33 *b31*b32 *m2*n2*u2*v1
2 4 2 5 2 2
- 72*a33 *b31*b32 *m2*n3*u1*v3 + 36*a33 *b32 *m2 *v1
2 5 2 2 7
+ 36*a33 *b32 *m2 *v2 + 360*a33*b31 *n2*u3*v1*v3
6 6
+ 360*a33*b31 *b32*n2*u3*v2*v3 + 360*a33*b31 *m2*n2*v2*v3
5 3 2 5 2 2
- 36*a33*b31 *b32 *u3*v1*v3 - 72*a33*b31 *b32 *n2*u1*v2
5 2 2 5 2
- 72*a33*b31 *b32 *n2*u1*v3 + 72*a33*b31 *b32 *n2*u2*v1*v2
5 2 5
+ 24*a33*b31 *b32 *n2*u3*v1*v3 - 360*a33*b31 *b32*m2*n2*v1*v3
5 2 5 2
+ 480*a33*b31 *b32*n2 *u1*v3 + 144*a33*b31 *b32*n2 *u3*v1
4 4 2 4 3 2
- 36*a33*b31 *b32 *u3*v2*v3 - 36*a33*b31 *b32 *m2*v2*v3
4 3 4 3 2
- 72*a33*b31 *b32 *n2*u1*v1*v2 + 72*a33*b31 *b32 *n2*u2*v1
4 3 2 4 3
+ 72*a33*b31 *b32 *n2*u2*v3 + 24*a33*b31 *b32 *n2*u3*v2*v3
4 3 2 4 2
- 36*a33*b31 *b32 *n3*u3*v3 + 24*a33*b31 *b32 *m2*n2*v2*v3
4 2 2 4 2 2
- 192*a33*b31 *b32 *n2 *u2*v3 + 144*a33*b31 *b32 *n2 *u3*v2
4 2 4 2
+ 168*a33*b31 *b32 *n2*n3*u1*v1 - 48*a33*b31 *b32 *n2*n3*u2*v2
3 5 2 3 4 2
- 36*a33*b31 *b32 *u3*v1*v3 + 36*a33*b31 *b32 *m2*v1*v3
3 4 2 3 4 2
- 72*a33*b31 *b32 *n2*u1*v2 - 72*a33*b31 *b32 *n2*u1*v3
3 4 3 4
+ 72*a33*b31 *b32 *n2*u2*v1*v2 - 336*a33*b31 *b32 *n2*u3*v1*v3
3 3 3 3 2
- 24*a33*b31 *b32 *m2*n2*v1*v3 + 480*a33*b31 *b32 *n2 *u1*v3
3 3 2 3 3
+ 288*a33*b31 *b32 *n2 *u3*v1 - 72*a33*b31 *b32 *n2*n3*u1*v2
3 3 3 2 2
+ 72*a33*b31 *b32 *n2*n3*u2*v1 + 144*a33*b31 *b32 *m2*n2 *v1
2 6 2 2 5 2
- 36*a33*b31 *b32 *u3*v2*v3 - 36*a33*b31 *b32 *m2*v2*v3
2 5 2 5 2
- 72*a33*b31 *b32 *n2*u1*v1*v2 + 72*a33*b31 *b32 *n2*u2*v1
2 5 2 2 5
+ 72*a33*b31 *b32 *n2*u2*v3 - 336*a33*b31 *b32 *n2*u3*v2*v3
2 5 2 2 4
- 36*a33*b31 *b32 *n3*u3*v3 - 336*a33*b31 *b32 *m2*n2*v2*v3
2 4 2 2 4
- 192*a33*b31 *b32 *n2 *u2*v3 + 240*a33*b31 *b32 *n2*n3*u1*v1
2 4 2 3 2
+ 24*a33*b31 *b32 *n2*n3*u2*v2 - 144*a33*b31 *b32 *m2*n2 *v2
2 3 6 2
- 144*a33*b31 *b32 *m2*n2*n3*v3 + 36*a33*b31*b32 *m2*v1*v3
5 7 2
+ 336*a33*b31*b32 *m2*n2*v1*v3 + 360*b31 *n2 *v1*v2
6 2 2 6 2 2 6 2 3
- 48*b31 *b32 *n2*v1 *v3 - 120*b31 *b32 *n2*v2 *v3 + 48*b31 *b32 *n2*v3
6 2 2 6 2 2 5 2 2
- 256*b31 *b32*n2 *v1 - 580*b31 *b32*n2 *v2 + 576*b31 *b32 *n2 *v1*v2
5 2 4 4 2 4 4 2
+ 312*b31 *b32*n2 *n3*v1 - 96*b31 *b32 *n2*v1 *v3 - 240*b31 *b32 *n2*v2 *v3
4 4 3 4 3 2 2 4 3 2 2
+ 48*b31 *b32 *n2*v3 - 200*b31 *b32 *n2 *v1 - 704*b31 *b32 *n2 *v2
4 2 2 4 2 2
- 312*b31 *b32 *n2 *n3*v2 - 312*b31 *b32 *n2*n3 *v3
3 4 2 3 3 2
+ 216*b31 *b32 *n2 *v1*v2 + 168*b31 *b32 *n2 *n3*v1
2 6 2 2 6 2 2 5 2 2
- 48*b31 *b32 *n2*v1 *v3 - 120*b31 *b32 *n2*v2 *v3 + 56*b31 *b32 *n2 *v1
2 5 2 2 2 4 2 2 4 2
- 124*b31 *b32 *n2 *v2 - 168*b31 *b32 *n2 *n3*v2 - 168*b31 *b32 *n2*n3 *v3
2 2 2 4 3 5 2
=((3*(3*((u1 + u2 ) *a33 *b32 + 40*b31 *n2 *v1*v2)
3 2
+ 8*(9*b32*v2 + 7*n3)*b31*b32 *n2 *v1
3 2
+ 8*(24*b32*v2 + 13*n3)*b31 *b32*n2 *v1) - 4
2 2 2 2 2 4
*(6*(5*v2 - 2*v3 + 2*v1 )*b32*v3 + (64*v1 + 145*v2 )*n2)*b31 *b32*n2 +
2 2 2 2
4*(((14*v1 - 31*v2 )*n2 - 6*(2*v1 + 5*v2 )*b32*v3)*b32
4
- 42*(n2*v2 + n3*v3)*n3)*b32 *n2 - 8*(39*(n2*v2 + n3*v3)*n3
2 2 2 2 2 2 2
+ (25*v1 + 88*v2 )*b32*n2 + 6*(5*v2 - v3 + 2*v1 )*b32 *v3)*b31
2 2 2
*b32 *n2)*b31 - 36*(((n2*u2 + n3*u3)*(u1 + u2 )
2 2
- (u1 *v2 - 2*u1*u2*v1 - u2 *v2)*m2
2 2
- (u1 *v2 - 2*u1*u2*v1 - u2 *v2)*b32*u3)*b31*b32
2 2 2 2 2
+ ((u1 *v1 + 2*u1*u2*v2 - u2 *v1)*b32*u3 - (u1 + u2 )*n2*u1)*b31
2 2 2 3 2
- (u1 *v1 + 2*u1*u2*v2 - u2 *v1)*b32 *m2)*a33 *b32 )*b31 + 6*((6*(
2 2 2 2 2 2
(n2*u2 + 2*n3*u3)*n2*u2 + (v1 + v2 )*m2 - (u1 + u2 )*n3
+ 2*(2*(u1*v1 + u2*v2)*n2 + n3*u2*v3)*m2) + (4*(
2 2
((5*u1*v3 + 2*u3*v1)*u1 + 2*(u2 *v3 + 10*u2*u3*v2 + u3 *v3))*n2
+ 3*n3*u2*u3*v3)
2 2 2 2 2
- 3*((v2 + v3 + v1 )*(u1 + u2 ) - 4*(u1*v1 + u2*v2)*u3*v3)*b32)
2 2
*b32)*b31 *b32 - (6*(2*((n2*u2 + n3*u3 - 2*m2*v2)*n2*u1
3
- (2*(u1*v2 - u2*v1)*n2 + n3*u1*v3)*b32*u3)*b31
2 2 2
- (2*(2*n2*u2*v1 - n3*u1*v3)*b31 + (v1 + v2 )*b32*m2)*b32 *m2)
*b32 - (2*(2
2 2
*((5*u1*v3 - 4*u3*v1)*u1 + 2*(u2 *v3 + 7*u2*u3*v2 + u3 *v3))*b32
2
+ 3*n2*u1 )*n2
2 2 2 2 2 2
- 3*((v2 + v3 + v1 )*(u1 + u2 ) - 4*(u1*v1 + u2*v2)*u3*v3)*b32
4 2
)*b31 ))*a33 *b32 + 12*((
5 6
((3*b32*v3 + 28*n2)*b32 *m2 + 30*b31 *n2*u3)*v1
5
+ 30*(b32*u3 + m2)*b31 *n2*v2)*v3 - (
2 2
2*(15*m2*v1*v3 - 20*n2*u1*v3 - 6*n2*u3*v1)*n2 + 3*b32 *u3*v1*v3
2 2 4
- 2*((3*u2*v2 + u3*v3)*v1 - 3*(v2 + v3 )*u1)*b32*n2)*b31 *b32 - (
2*(2*(7*m2*v2 + 4*n2*u2)*v3 - (10*u1*v1 + u2*v2)*n3)*b32*n2
3 2
+ 3*(4*(n2*v2 + n3*v3)*m2*n2 + b32 *u3*v2*v3 ) + (
2 2
2*((3*u1*v1 + 14*u3*v3)*v2 - 3*(v1 + v3 )*u2)*n2
2 2 3
+ 3*(m2*v2 + n3*u3)*v3 )*b32 )*b31*b32 + (
2*(m2*v2*v3 - 8*n2*u2*v3 + 6*n2*u3*v2 + (7*u1*v1 - 2*u2*v2)*n3)*n2
2 2 2 2
- 3*b32 *u3*v2*v3 - (2*((3*u1*v1 - u3*v3)*v2 - 3*(v1 + v3 )*u2)*n2
2 3 2
+ 3*(m2*v2 + n3*u3)*v3 )*b32)*b31 *b32 - (
2*(m2*v1*v3 - 20*n2*u1*v3 - 12*n2*u3*v1 + 3*(u1*v2 - u2*v1)*n3)*b32*n2
3 2 2
+ 3*(b32 *u3*v3 - 4*m2*n2 )*v1 -
2 2 2 2
(2*((3*u2*v2 - 14*u3*v3)*v1 - 3*(v2 + v3 )*u1)*n2 + 3*m2*v1*v3 )*b32
2 2
)*b31 *b32 )*a33*b31
And again in machine readable form:
HAM=(3*a33**2*b31*b32*u1**2 + 3*a33**2*b31*b32*u2**2 + 6*a33**2*b31*b32*u3**2 +
6*a33*b31**2*b32*u3*v1 - 6*a33*b31**2*n2*u1 + 6*a33*b31*b32**2*u3*v2 + 6*a33*b31
*b32*m2*v2 + 6*a33*b31*b32*n2*u2 + 6*a33*b31*b32*n3*u3 - 6*a33*b32**2*m2*v1 + 20
*b31**3*n2*v3 + 8*b31*b32**2*n2*v3)/(6*a33*b31*b32)$
INT=9*a33**4*b31**2*b32**3*u1**4 + 18*a33**4*b31**2*b32**3*u1**2*u2**2 + 9*a33**
4*b31**2*b32**3*u2**4 - 36*a33**3*b31**3*b32**3*u1**2*u3*v1 - 72*a33**3*b31**3*
b32**3*u1*u2*u3*v2 + 36*a33**3*b31**3*b32**3*u2**2*u3*v1 + 36*a33**3*b31**3*b32
**2*n2*u1**3 + 36*a33**3*b31**3*b32**2*n2*u1*u2**2 + 36*a33**3*b31**2*b32**4*u1
**2*u3*v2 - 72*a33**3*b31**2*b32**4*u1*u2*u3*v1 - 36*a33**3*b31**2*b32**4*u2**2*
u3*v2 + 36*a33**3*b31**2*b32**3*m2*u1**2*v2 - 72*a33**3*b31**2*b32**3*m2*u1*u2*
v1 - 36*a33**3*b31**2*b32**3*m2*u2**2*v2 - 36*a33**3*b31**2*b32**3*n2*u1**2*u2 -
36*a33**3*b31**2*b32**3*n2*u2**3 - 36*a33**3*b31**2*b32**3*n3*u1**2*u3 - 36*a33
**3*b31**2*b32**3*n3*u2**2*u3 + 36*a33**3*b31*b32**4*m2*u1**2*v1 + 72*a33**3*b31
*b32**4*m2*u1*u2*v2 - 36*a33**3*b31*b32**4*m2*u2**2*v1 - 18*a33**2*b31**4*b32**3
*u1**2*v1**2 - 18*a33**2*b31**4*b32**3*u1**2*v2**2 - 18*a33**2*b31**4*b32**3*u1
**2*v3**2 + 72*a33**2*b31**4*b32**3*u1*u3*v1*v3 - 18*a33**2*b31**4*b32**3*u2**2*
v1**2 - 18*a33**2*b31**4*b32**3*u2**2*v2**2 - 18*a33**2*b31**4*b32**3*u2**2*v3**
2 + 72*a33**2*b31**4*b32**3*u2*u3*v2*v3 + 120*a33**2*b31**4*b32**2*n2*u1**2*v3 -
96*a33**2*b31**4*b32**2*n2*u1*u3*v1 + 48*a33**2*b31**4*b32**2*n2*u2**2*v3 + 336
*a33**2*b31**4*b32**2*n2*u2*u3*v2 + 48*a33**2*b31**4*b32**2*n2*u3**2*v3 + 36*a33
**2*b31**4*b32*n2**2*u1**2 + 144*a33**2*b31**3*b32**3*n2*u1*u3*v2 - 144*a33**2*
b31**3*b32**3*n2*u2*u3*v1 + 72*a33**2*b31**3*b32**3*n3*u1*u3*v3 + 144*a33**2*b31
**3*b32**2*m2*n2*u1*v2 - 72*a33**2*b31**3*b32**2*n2**2*u1*u2 - 72*a33**2*b31**3*
b32**2*n2*n3*u1*u3 - 18*a33**2*b31**2*b32**5*u1**2*v1**2 - 18*a33**2*b31**2*b32
**5*u1**2*v2**2 - 18*a33**2*b31**2*b32**5*u1**2*v3**2 + 72*a33**2*b31**2*b32**5*
u1*u3*v1*v3 - 18*a33**2*b31**2*b32**5*u2**2*v1**2 - 18*a33**2*b31**2*b32**5*u2**
2*v2**2 - 18*a33**2*b31**2*b32**5*u2**2*v3**2 + 72*a33**2*b31**2*b32**5*u2*u3*v2
*v3 + 120*a33**2*b31**2*b32**4*n2*u1**2*v3 + 48*a33**2*b31**2*b32**4*n2*u1*u3*v1
+ 48*a33**2*b31**2*b32**4*n2*u2**2*v3 + 480*a33**2*b31**2*b32**4*n2*u2*u3*v2 +
48*a33**2*b31**2*b32**4*n2*u3**2*v3 + 72*a33**2*b31**2*b32**4*n3*u2*u3*v3 + 36*
a33**2*b31**2*b32**3*m2**2*v1**2 + 36*a33**2*b31**2*b32**3*m2**2*v2**2 + 144*a33
**2*b31**2*b32**3*m2*n2*u1*v1 + 144*a33**2*b31**2*b32**3*m2*n2*u2*v2 + 72*a33**2
*b31**2*b32**3*m2*n3*u2*v3 + 36*a33**2*b31**2*b32**3*n2**2*u2**2 + 72*a33**2*b31
**2*b32**3*n2*n3*u2*u3 - 36*a33**2*b31**2*b32**3*n3**2*u1**2 - 36*a33**2*b31**2*
b32**3*n3**2*u2**2 + 144*a33**2*b31*b32**4*m2*n2*u2*v1 - 72*a33**2*b31*b32**4*m2
*n3*u1*v3 + 36*a33**2*b32**5*m2**2*v1**2 + 36*a33**2*b32**5*m2**2*v2**2 + 360*
a33*b31**7*n2*u3*v1*v3 + 360*a33*b31**6*b32*n2*u3*v2*v3 + 360*a33*b31**6*m2*n2*
v2*v3 - 36*a33*b31**5*b32**3*u3*v1*v3**2 - 72*a33*b31**5*b32**2*n2*u1*v2**2 - 72
*a33*b31**5*b32**2*n2*u1*v3**2 + 72*a33*b31**5*b32**2*n2*u2*v1*v2 + 24*a33*b31**
5*b32**2*n2*u3*v1*v3 - 360*a33*b31**5*b32*m2*n2*v1*v3 + 480*a33*b31**5*b32*n2**2
*u1*v3 + 144*a33*b31**5*b32*n2**2*u3*v1 - 36*a33*b31**4*b32**4*u3*v2*v3**2 - 36*
a33*b31**4*b32**3*m2*v2*v3**2 - 72*a33*b31**4*b32**3*n2*u1*v1*v2 + 72*a33*b31**4
*b32**3*n2*u2*v1**2 + 72*a33*b31**4*b32**3*n2*u2*v3**2 + 24*a33*b31**4*b32**3*n2
*u3*v2*v3 - 36*a33*b31**4*b32**3*n3*u3*v3**2 + 24*a33*b31**4*b32**2*m2*n2*v2*v3
- 192*a33*b31**4*b32**2*n2**2*u2*v3 + 144*a33*b31**4*b32**2*n2**2*u3*v2 + 168*
a33*b31**4*b32**2*n2*n3*u1*v1 - 48*a33*b31**4*b32**2*n2*n3*u2*v2 - 36*a33*b31**3
*b32**5*u3*v1*v3**2 + 36*a33*b31**3*b32**4*m2*v1*v3**2 - 72*a33*b31**3*b32**4*n2
*u1*v2**2 - 72*a33*b31**3*b32**4*n2*u1*v3**2 + 72*a33*b31**3*b32**4*n2*u2*v1*v2
- 336*a33*b31**3*b32**4*n2*u3*v1*v3 - 24*a33*b31**3*b32**3*m2*n2*v1*v3 + 480*a33
*b31**3*b32**3*n2**2*u1*v3 + 288*a33*b31**3*b32**3*n2**2*u3*v1 - 72*a33*b31**3*
b32**3*n2*n3*u1*v2 + 72*a33*b31**3*b32**3*n2*n3*u2*v1 + 144*a33*b31**3*b32**2*m2
*n2**2*v1 - 36*a33*b31**2*b32**6*u3*v2*v3**2 - 36*a33*b31**2*b32**5*m2*v2*v3**2
- 72*a33*b31**2*b32**5*n2*u1*v1*v2 + 72*a33*b31**2*b32**5*n2*u2*v1**2 + 72*a33*
b31**2*b32**5*n2*u2*v3**2 - 336*a33*b31**2*b32**5*n2*u3*v2*v3 - 36*a33*b31**2*
b32**5*n3*u3*v3**2 - 336*a33*b31**2*b32**4*m2*n2*v2*v3 - 192*a33*b31**2*b32**4*
n2**2*u2*v3 + 240*a33*b31**2*b32**4*n2*n3*u1*v1 + 24*a33*b31**2*b32**4*n2*n3*u2*
v2 - 144*a33*b31**2*b32**3*m2*n2**2*v2 - 144*a33*b31**2*b32**3*m2*n2*n3*v3 + 36*
a33*b31*b32**6*m2*v1*v3**2 + 336*a33*b31*b32**5*m2*n2*v1*v3 + 360*b31**7*n2**2*
v1*v2 - 48*b31**6*b32**2*n2*v1**2*v3 - 120*b31**6*b32**2*n2*v2**2*v3 + 48*b31**6
*b32**2*n2*v3**3 - 256*b31**6*b32*n2**2*v1**2 - 580*b31**6*b32*n2**2*v2**2 + 576
*b31**5*b32**2*n2**2*v1*v2 + 312*b31**5*b32*n2**2*n3*v1 - 96*b31**4*b32**4*n2*v1
**2*v3 - 240*b31**4*b32**4*n2*v2**2*v3 + 48*b31**4*b32**4*n2*v3**3 - 200*b31**4*
b32**3*n2**2*v1**2 - 704*b31**4*b32**3*n2**2*v2**2 - 312*b31**4*b32**2*n2**2*n3*
v2 - 312*b31**4*b32**2*n2*n3**2*v3 + 216*b31**3*b32**4*n2**2*v1*v2 + 168*b31**3*
b32**3*n2**2*n3*v1 - 48*b31**2*b32**6*n2*v1**2*v3 - 120*b31**2*b32**6*n2*v2**2*
v3 + 56*b31**2*b32**5*n2**2*v1**2 - 124*b31**2*b32**5*n2**2*v2**2 - 168*b31**2*
b32**4*n2**2*n3*v2 - 168*b31**2*b32**4*n2*n3**2*v3$