Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
2 2
0=a12 + a13
Expressions
The solution is given through the following expressions:
3 3 1 2 2
r10=( - ----*a12 *b11*n2*n3*r350 - ----*a12 *a13*b11*n3 *r350
64 32
13 2 3 4
+ ----*a12*a13 *b11*n2*n3*r350 + ---*a13 *m1*n3*r350
64 2
9 3 2 5
- ----*a13 *b11*n3 *r350)/(a13 *b11)
32
1 3 3 3
r11=( - ----*a12 *b11*n2*n3*r350 + ---*a12*a13 *m1*n2*r350
32 2
9 2 1 3 2 4
- ----*a12*a13 *b11*n2*n3*r350 - ---*a13 *b11*n2 *r350)/(a12*a13 *b11)
32 4
1 5 1 4
r12=( - ----*a12 *b11*n1*n3*r350 + ----*a12 *a13*b11*n1*n2*r350
64 64
17 3 2 1 2 3
- ----*a12 *a13 *b11*n1*n3*r350 + ----*a12 *a13 *b11*n1*n2*r350
64 64
3 5 1 4
+ ---*a12*a13 *m1*n1*r350 - ---*a12*a13 *b11*n1*n3*r350
2 2
1 5 6
- ---*a13 *b11*n1*n2*r350)/(a12*a13 *b11)
4
r13=0
r14=0
r15=0
1 3 1 2
r20=(-----*a12 *b11*n2*r350 + -----*a12 *a13*b11*n3*r350
192 192
1 2 1 3 5
+ -----*a12*a13 *b11*n2*r350 + -----*a13 *b11*n3*r350)/a13
192 192
r21=0
35 2 17 2
-----*a12 *b11*n1*r350 + ----*a13 *b11*n1*r350
192 64
r23=------------------------------------------------
4
a13
1 3 19 2
- ----*a12 *b11*n1*r350 - -----*a12*a13 *b11*n1*r350
64 192
r24=-------------------------------------------------------
5
a13
1 5 1 4
r25=( - -----*a12 *b11*n3*r350 + -----*a12 *a13*b11*n2*r350
192 192
1 3 3 1 3 2
- ----*a12 *a13 *m1*r350 - ----*a12 *a13 *b11*n3*r350
32 16
1 2 3 15 5
+ -----*a12 *a13 *b11*n2*r350 + ----*a12*a13 *m1*r350
192 32
35 4 1 5 6
- -----*a12*a13 *b11*n3*r350 - ---*a13 *b11*n2*r350)/(a12*a13 )
192 8
r26=0
1 3 17 2
- ----*a12 *n1*r350 - ----*a12*a13 *n1*r350
32 32
r27=----------------------------------------------
4
a13
1 2 9 2
- ----*a12 *n3*r350 - ----*a13 *n3*r350
16 16
r28=------------------------------------------
3
a13
3 3
- ---*a12*n2*r350 - ---*a13*n3*r350
2 2
r29=--------------------------------------
a13*b11
1 3 17 2
----*a12 *n1*r350 + ----*a12*a13 *n1*r350
32 32
r210=-------------------------------------------
4
a13
1 2 9 2
- ----*a12 *n2*r350 - ----*a13 *n2*r350
16 16
r212=------------------------------------------
3
a13
r213=0
3 3
- ---*a12*n2*r350 - ---*a13*n3*r350
2 2
r214=--------------------------------------
a13*b11
1
3*a12*m1*r350 - ---*b11*n2*r350
2
r215=---------------------------------
a12*b11
3 3 3 13 2
----*a12 *b11*n3*r350 + 3*a12*a13 *m1*r350 - ----*a12*a13 *b11*n3*r350
32 32
r216=------------------------------------------------------------------------
4
a13 *b11
1 4 17 2 2 1 4
----*a12 *n1*r350 + ----*a12 *a13 *n1*r350 + ---*a13 *n1*r350
32 32 2
r217=---------------------------------------------------------------
5
a13
r219=0
3 3
- ---*a12*n2*r350 - ---*a13*n3*r350
2 2
r220=--------------------------------------
a13*b11
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
13 2 2 7 2 2
-----*a12 *b11 *r350 + ----*a13 *b11 *r350
288 96
r37=--------------------------------------------
4
a13
11 3 2 1 2 2
-----*a12 *b11 *r350 + ----*a12*a13 *b11 *r350
288 96
r38=------------------------------------------------
5
a13
r39=0
r310=0
r311=0
r312=0
1 2 3 2
----*a12 *b11*r350 + ----*a13 *b11*r350
48 16
r313=-----------------------------------------
3
a13
1 3 3 2
- ----*a12 *b11*r350 - ----*a12*a13 *b11*r350
48 16
r314=------------------------------------------------
4
a13
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
r321=0
r322=0
1 3 3 2
----*a12 *b11*r350 + ----*a12*a13 *b11*r350
48 16
r323=---------------------------------------------
4
a13
1 2 3 2
----*a12 *b11*r350 + ----*a13 *b11*r350
48 16
r324=-----------------------------------------
3
a13
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
1 2 3 2
- ----*a12 *b11*r350 - ----*a13 *b11*r350
48 16
r335=--------------------------------------------
3
a13
r336=0
1 2 3 2
- ----*a12 *b11*r350 - ----*a13 *b11*r350
48 16
r337=--------------------------------------------
3
a13
r338=0
r339=0
1 4 1 2 2 3 4
----*a12 *b11*r350 + ----*a12 *a13 *b11*r350 - ----*a13 *b11*r350
96 12 32
r340=-------------------------------------------------------------------
5
a13
r341=0
r342=0
1 2 9 2
- ---*a12 *r350 - ---*a13 *r350
8 8
r343=----------------------------------
2
a13
r344=0
r345=0
r346=0
1 3 9 2
- ---*a12 *r350 - ---*a12*a13 *r350
8 8
r347=--------------------------------------
3
a13
r348=0
r349=0
a12*r350
r351=----------
a13
r352=0
r353=0
r354=0
r355=0
1
- ---*b11*n1
6
m3=---------------
a13
1
---*a12*b11*n1
6
m2=----------------
2
a13
c33=0
c23=0
c22=0
1 2
- ----*b11
36
c13=--------------
a13
1 2
----*a12*b11
36
c12=---------------
2
a13
b33=0
b31=0
b21=0
b13=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:
r350, m1, n3, n1, n2, b11, a12, a13
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.
{b11,a13,r350,a12,{n3,n2,n1,m1}}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2 3 2 2 2
HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13 + u1*v1*a13 *b11 + u1*a13 *n1 + u2*a13 *n2
2 1 2 1 2 2
+ u3*a13 *n3 + ----*v1*v2*a12*b11 - ----*v1*v3*a13*b11 + v1*a13 *m1
18 18
1 1 2
+ ---*v2*a12*b11*n1 - ---*v3*a13*b11*n1)/a13
6 6
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 5 2 6
FI=u1 *v2*a12 *a13 *b11 + u1 *v3*a12*a13 *b11
2 3 2 5 3 6
+ u1 *( - ---*a12 *a13 *n2 - ---*a12*a13 *n3)
2 2
1 4 3 9 2 5
+ u1*u2*v1*( - ---*a12 *a13 *b11 - ---*a12 *a13 *b11)
8 8
1 3 4 9 6
+ u1*u3*v1*( - ---*a12 *a13 *b11 - ---*a12*a13 *b11)
8 8
2 1 5 2 1 3 3 2 3 5 2
+ u1*v1 *(----*a12 *a13*b11 + ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
96 12 32
1 5 17 3 3 1 5
+ u1*v1*(----*a12 *a13*b11*n1 + ----*a12 *a13 *b11*n1 + ---*a12*a13 *b11*n1)
32 32 2
2 1 3 3 2 3 5 2
+ u1*v2 *( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
48 16
3 4 2 2 5 13 2 4
+ u1*v2*(----*a12 *a13 *b11*n3 + 3*a12 *a13 *m1 - ----*a12 *a13 *b11*n3)
32 32
2 1 3 3 2 3 5 2
+ u1*v3 *( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
48 16
6 1 6
+ u1*v3*(3*a12*a13 *m1 - ---*a13 *b11*n2)
2
2 3 2 5 3 6
+ u2 *( - ---*a12 *a13 *n2 - ---*a12*a13 *n3)
2 2
1 3 3 2 3 5 2
+ u2*v1*v2*(----*a12 *a13 *b11 + ----*a12*a13 *b11 )
48 16
1 4 2 2 3 2 4 2
+ u2*v1*v3*(----*a12 *a13 *b11 + ----*a12 *a13 *b11 )
48 16
1 3 3 9 5
+ u2*v1*( - ----*a12 *a13 *b11*n2 - ----*a12*a13 *b11*n2)
16 16
1 4 2 17 2 4
+ u2*v3*(----*a12 *a13 *b11*n1 + ----*a12 *a13 *b11*n1)
32 32
2 3 2 5 3 6
+ u3 *( - ---*a12 *a13 *n2 - ---*a12*a13 *n3)
2 2
1 4 2 2 3 2 4 2
+ u3*v1*v2*( - ----*a12 *a13 *b11 - ----*a12 *a13 *b11 )
48 16
1 3 3 2 3 5 2
+ u3*v1*v3*(----*a12 *a13 *b11 + ----*a12*a13 *b11 )
48 16
1 3 3 9 5
+ u3*v1*( - ----*a12 *a13 *b11*n3 - ----*a12*a13 *b11*n3)
16 16
1 4 2 17 2 4
+ u3*v2*( - ----*a12 *a13 *b11*n1 - ----*a12 *a13 *b11*n1)
32 32
2 11 4 3 1 2 3 3
+ v1 *v2*(-----*a12 *a13*b11 + ----*a12 *a13 *b11 )
288 96
2 13 3 2 3 7 4 3 2
+ v1 *v3*(-----*a12 *a13 *b11 + ----*a12*a13 *b11 ) + v1 *(
288 96
1 5 2 1 4 2 1 3 3
- -----*a12 *b11 *n3 + -----*a12 *a13*b11 *n2 - ----*a12 *a13 *b11*m1
192 192 32
1 3 2 2 1 2 3 2 15 5
- ----*a12 *a13 *b11 *n3 + -----*a12 *a13 *b11 *n2 + ----*a12*a13 *b11*m1
16 192 32
35 4 2 1 5 2
- -----*a12*a13 *b11 *n3 - ---*a13 *b11 *n2)
192 8
1 4 2 19 2 3 2
+ v1*v2*( - ----*a12 *a13*b11 *n1 - -----*a12 *a13 *b11 *n1)
64 192
35 3 2 2 17 4 2
+ v1*v3*(-----*a12 *a13 *b11 *n1 + ----*a12*a13 *b11 *n1) + v1*(
192 64
1 5 1 4
- ----*a12 *b11*n1*n3 + ----*a12 *a13*b11*n1*n2
64 64
17 3 2 1 2 3
- ----*a12 *a13 *b11*n1*n3 + ----*a12 *a13 *b11*n1*n2
64 64
3 5 1 4 1 5
+ ---*a12*a13 *m1*n1 - ---*a12*a13 *b11*n1*n3 - ---*a13 *b11*n1*n2) + v2*
2 2 4
1 3 2 3 5 9 4
( - ----*a12 *a13 *b11*n2*n3 + ---*a12*a13 *m1*n2 - ----*a12*a13 *b11*n2*n3
32 2 32
1 5 2 2 1 4 2 1 3 2 2
- ---*a13 *b11*n2 ) + v3 *(-----*a12 *a13*b11 *n2 + -----*a12 *a13 *b11 *n3
4 192 192
1 2 3 2 1 4 2
+ -----*a12 *a13 *b11 *n2 + -----*a12*a13 *b11 *n3) + v3*(
192 192
3 4 1 3 2 2
- ----*a12 *a13*b11*n2*n3 - ----*a12 *a13 *b11*n3
64 32
13 2 3 3 5 9 4 2
+ ----*a12 *a13 *b11*n2*n3 + ---*a12*a13 *m1*n3 - ----*a12*a13 *b11*n3 )
64 2 32
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u1*u2*a12*a13**2 + 2*u1*u3*a13**3 + u1*v1*a13**2*b11 + u1*a13**2*n1 + u2*
a13**2*n2 + u3*a13**2*n3 + 1/18*v1*v2*a12*b11**2 - 1/18*v1*v3*a13*b11**2 + v1*
a13**2*m1 + 1/6*v2*a12*b11*n1 - 1/6*v3*a13*b11*n1)/a13**2$
FI=u1**2*v2*a12**2*a13**5*b11 + u1**2*v3*a12*a13**6*b11 + u1**2*( - 3/2*a12**2*
a13**5*n2 - 3/2*a12*a13**6*n3) + u1*u2*v1*( - 1/8*a12**4*a13**3*b11 - 9/8*a12**2
*a13**5*b11) + u1*u3*v1*( - 1/8*a12**3*a13**4*b11 - 9/8*a12*a13**6*b11) + u1*v1
**2*(1/96*a12**5*a13*b11**2 + 1/12*a12**3*a13**3*b11**2 - 3/32*a12*a13**5*b11**2
) + u1*v1*(1/32*a12**5*a13*b11*n1 + 17/32*a12**3*a13**3*b11*n1 + 1/2*a12*a13**5*
b11*n1) + u1*v2**2*( - 1/48*a12**3*a13**3*b11**2 - 3/16*a12*a13**5*b11**2) + u1*
v2*(3/32*a12**4*a13**2*b11*n3 + 3*a12**2*a13**5*m1 - 13/32*a12**2*a13**4*b11*n3)
+ u1*v3**2*( - 1/48*a12**3*a13**3*b11**2 - 3/16*a12*a13**5*b11**2) + u1*v3*(3*
a12*a13**6*m1 - 1/2*a13**6*b11*n2) + u2**2*( - 3/2*a12**2*a13**5*n2 - 3/2*a12*
a13**6*n3) + u2*v1*v2*(1/48*a12**3*a13**3*b11**2 + 3/16*a12*a13**5*b11**2) + u2*
v1*v3*(1/48*a12**4*a13**2*b11**2 + 3/16*a12**2*a13**4*b11**2) + u2*v1*( - 1/16*
a12**3*a13**3*b11*n2 - 9/16*a12*a13**5*b11*n2) + u2*v3*(1/32*a12**4*a13**2*b11*
n1 + 17/32*a12**2*a13**4*b11*n1) + u3**2*( - 3/2*a12**2*a13**5*n2 - 3/2*a12*a13
**6*n3) + u3*v1*v2*( - 1/48*a12**4*a13**2*b11**2 - 3/16*a12**2*a13**4*b11**2) +
u3*v1*v3*(1/48*a12**3*a13**3*b11**2 + 3/16*a12*a13**5*b11**2) + u3*v1*( - 1/16*
a12**3*a13**3*b11*n3 - 9/16*a12*a13**5*b11*n3) + u3*v2*( - 1/32*a12**4*a13**2*
b11*n1 - 17/32*a12**2*a13**4*b11*n1) + v1**2*v2*(11/288*a12**4*a13*b11**3 + 1/96
*a12**2*a13**3*b11**3) + v1**2*v3*(13/288*a12**3*a13**2*b11**3 + 7/96*a12*a13**4
*b11**3) + v1**2*( - 1/192*a12**5*b11**2*n3 + 1/192*a12**4*a13*b11**2*n2 - 1/32*
a12**3*a13**3*b11*m1 - 1/16*a12**3*a13**2*b11**2*n3 + 1/192*a12**2*a13**3*b11**2
*n2 + 15/32*a12*a13**5*b11*m1 - 35/192*a12*a13**4*b11**2*n3 - 1/8*a13**5*b11**2*
n2) + v1*v2*( - 1/64*a12**4*a13*b11**2*n1 - 19/192*a12**2*a13**3*b11**2*n1) + v1
*v3*(35/192*a12**3*a13**2*b11**2*n1 + 17/64*a12*a13**4*b11**2*n1) + v1*( - 1/64*
a12**5*b11*n1*n3 + 1/64*a12**4*a13*b11*n1*n2 - 17/64*a12**3*a13**2*b11*n1*n3 + 1
/64*a12**2*a13**3*b11*n1*n2 + 3/2*a12*a13**5*m1*n1 - 1/2*a12*a13**4*b11*n1*n3 -
1/4*a13**5*b11*n1*n2) + v2*( - 1/32*a12**3*a13**2*b11*n2*n3 + 3/2*a12*a13**5*m1*
n2 - 9/32*a12*a13**4*b11*n2*n3 - 1/4*a13**5*b11*n2**2) + v3**2*(1/192*a12**4*a13
*b11**2*n2 + 1/192*a12**3*a13**2*b11**2*n3 + 1/192*a12**2*a13**3*b11**2*n2 + 1/
192*a12*a13**4*b11**2*n3) + v3*( - 3/64*a12**4*a13*b11*n2*n3 - 1/32*a12**3*a13**
2*b11*n3**2 + 13/64*a12**2*a13**3*b11*n2*n3 + 3/2*a12*a13**5*m1*n3 - 9/32*a12*
a13**4*b11*n3**2)$