Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=9*a23 + a33
Expressions
The solution is given through the following expressions:
9 2 3 2
- ----*a23 *b32*r26 + ----*a33 *b32*r26
64 64
r20=------------------------------------------
2
a23*a33
3
----*b32*r26
16
r21=--------------
a33
r23=0
r24=0
9 11 9 10 2 37 9 2
r25=(---*a23 *c23*r26 - ----*a23 *b32 *r26 + ----*a23 *a33 *c23*r26
8 32 8
63 8 2 2 29 7 4
- ----*a23 *a33 *b32 *r26 + ----*a23 *a33 *c23*r26
64 4
61 6 4 2 21 5 6
- ----*a23 *a33 *b32 *r26 + ----*a23 *a33 *c23*r26
48 4
23 4 6 2 13 3 8
- ----*a23 *a33 *b32 *r26 + ----*a23 *a33 *c23*r26
32 8
5 2 8 2 1 10 1 10 2
- ----*a23 *a33 *b32 *r26 + ---*a23*a33 *c23*r26 - -----*a33 *b32 *r26)/
32 8 192
9 2 10 7 4 5 6 3 8
(a23 *a33 *b32 + ----*a23 *a33 *b32 + 4*a23 *a33 *b32 + 2*a23 *a33 *b32
3
1 10
+ ---*a23*a33 *b32)
3
3
---*a23*r26
2
r27=-------------
a33
r28=0
9 3 9 2
---*a23 *r26 + ---*a23*a33 *r26
4 4
r29=---------------------------------
2
a33 *b32
3 2 1 2
- ---*a23 *r26 - ---*a33 *r26
4 4
r210=--------------------------------
a23*a33
r212=0
3 2 3 2
- ---*a23 *r26 - ---*a33 *r26
2 2
r213=--------------------------------
a33*b32
9 4 3 2 2 3 4
---*a23 *r26 + ---*a23 *a33 *r26 + ---*a33 *r26
8 2 8
r214=-------------------------------------------------
2
a23*a33 *b32
r215=0
r216=0
9 2 5 2
---*a23 *r26 + ---*a33 *r26
8 8
r217=-----------------------------
2
a33
r218=0
r219=0
11 3 10 2 21 9 2 29 8 2 2
c33=( - 3*a23 *c23 + ----*a23 *b32 - ----*a23 *a33 *c23 + ----*a23 *a33 *b32
16 2 48
122 7 4 35 6 4 2 23 5 6
- -----*a23 *a33 *c23 + ----*a23 *a33 *b32 - ----*a23 *a33 *c23
9 48 3
173 4 6 2 5 3 8 5 2 8 2
+ -----*a23 *a33 *b32 - ---*a23 *a33 *c23 + ----*a23 *a33 *b32
432 3 54
1 10 1 10 2 10 10 8 3
- ----*a23*a33 *c23 + -----*a33 *b32 )/(a23 *a33 + ----*a23 *a33
18 216 3
6 5 4 7 1 2 9
+ 4*a23 *a33 + 2*a23 *a33 + ---*a23 *a33 )
3
3 8 2 5 7 2 5 6 2 2
c22=(----*a23 *b32 - ---*a23 *a33 *c23 + ----*a23 *a33 *b32
16 6 12
11 5 4 5 4 4 2 7 3 6
- ----*a23 *a33 *c23 + ----*a23 *a33 *b32 - ---*a23 *a33 *c23
6 16 6
19 2 6 2 1 8 1 8 2 8
+ -----*a23 *a33 *b32 - ---*a23*a33 *c23 + -----*a33 *b32 )/(a23 *a33
216 6 216
7 6 3 5 4 5 1 2 7
+ ---*a23 *a33 + ---*a23 *a33 + ---*a23 *a33 )
3 3 3
c13=0
c12=0
3 2 1 2
- ---*a23 *b32 + ---*a33 *b32
2 6
b33=--------------------------------
a23*a33
b31=0
b21=0
1
---*a33*b32
6
b11=-------------
a23
3 2 1 2
---*a23 + ---*a33
2 2
a22=---------------------
a33
2 2
3*a23 + a33
a11=---------------
a33
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:
r26, c23, a33, b32, a23
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.
2
{a22,a11,64*a23*c23 - 3*b32 ,a33,r26,a23,b32}
Relevance for the application:
Modulo the following equation:
2 2
0=9*a23 + a33
the system of equations related to the Hamiltonian HAM:
2 12 10 2 46 8 4 6 6 4 8
HAM=(u1 *(3*a23 + 11*a23 *a33 + ----*a23 *a33 + 10*a23 *a33 + 3*a23 *a33
3
1 2 10 1 9 2 5 7 4
+ ---*a23 *a33 ) + u1*v1*(---*a23 *a33 *b32 + ---*a23 *a33 *b32
3 6 9
2 5 6 1 3 8 1 10 2
+ ---*a23 *a33 *b32 + ---*a23 *a33 *b32 + ----*a23*a33 *b32) + u2 *(
3 3 18
3 12 11 10 2 23 8 4 6 6
---*a23 + ----*a23 *a33 + ----*a23 *a33 + 5*a23 *a33
2 2 3
3 4 8 1 2 10
+ ---*a23 *a33 + ---*a23 *a33 ) + u2*u3
2 6
11 20 9 3 7 5 5 7 2 3 9
*(2*a23 *a33 + ----*a23 *a33 + 8*a23 *a33 + 4*a23 *a33 + ---*a23 *a33 )
3 3
2
+ u3
10 2 10 8 4 6 6 4 8 1 2 10
*(a23 *a33 + ----*a23 *a33 + 4*a23 *a33 + 2*a23 *a33 + ---*a23 *a33 )
3 3
10 10 8 3 6 5
+ u3*v2*(a23 *a33*b32 + ----*a23 *a33 *b32 + 4*a23 *a33 *b32
3
4 7 1 2 9 3 11
+ 2*a23 *a33 *b32 + ---*a23 *a33 *b32) + u3*v3*( - ---*a23 *b32
3 2
29 9 2 49 7 4 7 5 6
- ----*a23 *a33 *b32 - ----*a23 *a33 *b32 - ---*a23 *a33 *b32
6 9 3
1 3 8 1 10 2 3 10 2
- ---*a23 *a33 *b32 + ----*a23*a33 *b32) + v2 *(----*a23 *b32
6 18 16
5 9 2 29 8 2 2 8 7 4
- ---*a23 *a33 *c23 + ----*a23 *a33 *b32 - ---*a23 *a33 *c23
6 48 3
35 6 4 2 5 6 173 4 6 2
+ ----*a23 *a33 *b32 - 3*a23 *a33 *c23 + -----*a23 *a33 *b32
48 432
4 3 8 5 2 8 2 1 10
- ---*a23 *a33 *c23 + ----*a23 *a33 *b32 - ---*a23*a33 *c23
3 54 6
1 10 2 10 20 8 3
+ -----*a33 *b32 ) + v2*v3*(2*a23 *a33*c23 + ----*a23 *a33 *c23
216 3
6 5 4 7 2 2 9 2
+ 8*a23 *a33 *c23 + 4*a23 *a33 *c23 + ---*a23 *a33 *c23) + v3 *(
3
11 3 10 2 21 9 2
- 3*a23 *c23 + ----*a23 *b32 - ----*a23 *a33 *c23
16 2
29 8 2 2 122 7 4 35 6 4 2
+ ----*a23 *a33 *b32 - -----*a23 *a33 *c23 + ----*a23 *a33 *b32
48 9 48
23 5 6 173 4 6 2 5 3 8
- ----*a23 *a33 *c23 + -----*a23 *a33 *b32 - ---*a23 *a33 *c23
3 432 3
5 2 8 2 1 10 1 10 2
+ ----*a23 *a33 *b32 - ----*a23*a33 *c23 + -----*a33 *b32 ))/(
54 18 216
10 10 8 3 6 5 4 7 1 2 9
a23 *a33 + ----*a23 *a33 + 4*a23 *a33 + 2*a23 *a33 + ---*a23 *a33 )
3 3
has apart from the Hamiltonian and Casimirs only the following first integral:
9 11 35 9 2 79 7 4
FI=u1*v1*(---*a23 *b32 + ----*a23 *a33 *b32 + ----*a23 *a33 *b32
8 8 12
19 5 6 13 3 8 5 10 2
+ ----*a23 *a33 *b32 + ----*a23 *a33 *b32 + ----*a23*a33 *b32) + u2
4 8 24
9 12 21 10 2 79 8 4 19 6 6
*(---*a23 + ----*a23 *a33 + ----*a23 *a33 + ----*a23 *a33
8 4 8 2
39 4 8 5 2 10 1 12 3 11
+ ----*a23 *a33 + ---*a23 *a33 + ---*a33 ) + u2*u3*( - ---*a23 *a33
8 4 8 2
13 9 3 7 5 5 7 7 3 9
- ----*a23 *a33 - 11*a23 *a33 - 9*a23 *a33 - ---*a23 *a33
2 2
1 11 3 10 11 8 3
- ---*a23*a33 ) + u2*v3*( - ---*a23 *a33*b32 - ----*a23 *a33 *b32
2 4 4
23 6 5 5 4 7 3 2 9
- ----*a23 *a33 *b32 - ---*a23 *a33 *b32 - ---*a23 *a33 *b32
6 2 4
1 11 2 9 12 39 10 2 33 8 4
- ----*a33 *b32) + u3 *(---*a23 + ----*a23 *a33 + ----*a23 *a33
12 4 4 2
27 6 6 21 4 8 3 2 10
+ ----*a23 *a33 + ----*a23 *a33 + ---*a23 *a33 ) + u3*v2*(
2 4 4
3 10 8 3 6 5 4 7
---*a23 *a33*b32 + 5*a23 *a33 *b32 + 6*a23 *a33 *b32 + 3*a23 *a33 *b32
2
1 2 9 9 2 10 7 4
+ ---*a23 *a33 *b32) + u3*v3*(a23 *a33 *b32 + ----*a23 *a33 *b32
2 3
5 6 3 8 1 10 2
+ 4*a23 *a33 *b32 + 2*a23 *a33 *b32 + ---*a23*a33 *b32) + v1 *(
3
9 11 9 10 2 37 9 2 63 8 2 2
---*a23 *c23 - ----*a23 *b32 + ----*a23 *a33 *c23 - ----*a23 *a33 *b32
8 32 8 64
29 7 4 61 6 4 2 21 5 6
+ ----*a23 *a33 *c23 - ----*a23 *a33 *b32 + ----*a23 *a33 *c23
4 48 4
23 4 6 2 13 3 8 5 2 8 2
- ----*a23 *a33 *b32 + ----*a23 *a33 *c23 - ----*a23 *a33 *b32
32 8 32
1 10 1 10 2 3 9 2
+ ---*a23*a33 *c23 - -----*a33 *b32 ) + v2*v3*(----*a23 *a33*b32
8 192 16
5 7 3 2 3 5 5 2 3 3 7 2
+ ---*a23 *a33 *b32 + ---*a23 *a33 *b32 + ---*a23 *a33 *b32
8 4 8
1 9 2 2 9 10 2 27 8 2 2
+ ----*a23*a33 *b32 ) + v3 *( - ----*a23 *b32 - ----*a23 *a33 *b32
16 64 64
13 6 4 2 3 4 6 2 3 2 8 2
- ----*a23 *a33 *b32 - ----*a23 *a33 *b32 + ----*a23 *a33 *b32
32 32 64
1 10 2
+ ----*a33 *b32 )
64
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(3*a23**12 + 11*a23**10*a33**2 + 46/3*a23**8*a33**4 + 10*a23**6*a33**
6 + 3*a23**4*a33**8 + 1/3*a23**2*a33**10) + u1*v1*(1/6*a23**9*a33**2*b32 + 5/9*
a23**7*a33**4*b32 + 2/3*a23**5*a33**6*b32 + 1/3*a23**3*a33**8*b32 + 1/18*a23*a33
**10*b32) + u2**2*(3/2*a23**12 + 11/2*a23**10*a33**2 + 23/3*a23**8*a33**4 + 5*
a23**6*a33**6 + 3/2*a23**4*a33**8 + 1/6*a23**2*a33**10) + u2*u3*(2*a23**11*a33 +
20/3*a23**9*a33**3 + 8*a23**7*a33**5 + 4*a23**5*a33**7 + 2/3*a23**3*a33**9) +
u3**2*(a23**10*a33**2 + 10/3*a23**8*a33**4 + 4*a23**6*a33**6 + 2*a23**4*a33**8 +
1/3*a23**2*a33**10) + u3*v2*(a23**10*a33*b32 + 10/3*a23**8*a33**3*b32 + 4*a23**
6*a33**5*b32 + 2*a23**4*a33**7*b32 + 1/3*a23**2*a33**9*b32) + u3*v3*( - 3/2*a23
**11*b32 - 29/6*a23**9*a33**2*b32 - 49/9*a23**7*a33**4*b32 - 7/3*a23**5*a33**6*
b32 - 1/6*a23**3*a33**8*b32 + 1/18*a23*a33**10*b32) + v2**2*(3/16*a23**10*b32**2
- 5/6*a23**9*a33**2*c23 + 29/48*a23**8*a33**2*b32**2 - 8/3*a23**7*a33**4*c23 +
35/48*a23**6*a33**4*b32**2 - 3*a23**5*a33**6*c23 + 173/432*a23**4*a33**6*b32**2
- 4/3*a23**3*a33**8*c23 + 5/54*a23**2*a33**8*b32**2 - 1/6*a23*a33**10*c23 + 1/
216*a33**10*b32**2) + v2*v3*(2*a23**10*a33*c23 + 20/3*a23**8*a33**3*c23 + 8*a23
**6*a33**5*c23 + 4*a23**4*a33**7*c23 + 2/3*a23**2*a33**9*c23) + v3**2*( - 3*a23
**11*c23 + 3/16*a23**10*b32**2 - 21/2*a23**9*a33**2*c23 + 29/48*a23**8*a33**2*
b32**2 - 122/9*a23**7*a33**4*c23 + 35/48*a23**6*a33**4*b32**2 - 23/3*a23**5*a33
**6*c23 + 173/432*a23**4*a33**6*b32**2 - 5/3*a23**3*a33**8*c23 + 5/54*a23**2*a33
**8*b32**2 - 1/18*a23*a33**10*c23 + 1/216*a33**10*b32**2))/(a23**10*a33 + 10/3*
a23**8*a33**3 + 4*a23**6*a33**5 + 2*a23**4*a33**7 + 1/3*a23**2*a33**9)$
FI=u1*v1*(9/8*a23**11*b32 + 35/8*a23**9*a33**2*b32 + 79/12*a23**7*a33**4*b32 +
19/4*a23**5*a33**6*b32 + 13/8*a23**3*a33**8*b32 + 5/24*a23*a33**10*b32) + u2**2*
(9/8*a23**12 + 21/4*a23**10*a33**2 + 79/8*a23**8*a33**4 + 19/2*a23**6*a33**6 +
39/8*a23**4*a33**8 + 5/4*a23**2*a33**10 + 1/8*a33**12) + u2*u3*( - 3/2*a23**11*
a33 - 13/2*a23**9*a33**3 - 11*a23**7*a33**5 - 9*a23**5*a33**7 - 7/2*a23**3*a33**
9 - 1/2*a23*a33**11) + u2*v3*( - 3/4*a23**10*a33*b32 - 11/4*a23**8*a33**3*b32 -
23/6*a23**6*a33**5*b32 - 5/2*a23**4*a33**7*b32 - 3/4*a23**2*a33**9*b32 - 1/12*
a33**11*b32) + u3**2*(9/4*a23**12 + 39/4*a23**10*a33**2 + 33/2*a23**8*a33**4 +
27/2*a23**6*a33**6 + 21/4*a23**4*a33**8 + 3/4*a23**2*a33**10) + u3*v2*(3/2*a23**
10*a33*b32 + 5*a23**8*a33**3*b32 + 6*a23**6*a33**5*b32 + 3*a23**4*a33**7*b32 + 1
/2*a23**2*a33**9*b32) + u3*v3*(a23**9*a33**2*b32 + 10/3*a23**7*a33**4*b32 + 4*
a23**5*a33**6*b32 + 2*a23**3*a33**8*b32 + 1/3*a23*a33**10*b32) + v1**2*(9/8*a23
**11*c23 - 9/32*a23**10*b32**2 + 37/8*a23**9*a33**2*c23 - 63/64*a23**8*a33**2*
b32**2 + 29/4*a23**7*a33**4*c23 - 61/48*a23**6*a33**4*b32**2 + 21/4*a23**5*a33**
6*c23 - 23/32*a23**4*a33**6*b32**2 + 13/8*a23**3*a33**8*c23 - 5/32*a23**2*a33**8
*b32**2 + 1/8*a23*a33**10*c23 - 1/192*a33**10*b32**2) + v2*v3*(3/16*a23**9*a33*
b32**2 + 5/8*a23**7*a33**3*b32**2 + 3/4*a23**5*a33**5*b32**2 + 3/8*a23**3*a33**7
*b32**2 + 1/16*a23*a33**9*b32**2) + v3**2*( - 9/64*a23**10*b32**2 - 27/64*a23**8
*a33**2*b32**2 - 13/32*a23**6*a33**4*b32**2 - 3/32*a23**4*a33**6*b32**2 + 3/64*
a23**2*a33**8*b32**2 + 1/64*a33**10*b32**2)$