Solution 45 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=a11 + 16*a23 + b12 *kap
Expressions
The solution is given through the following expressions:
r60=0
r61=0
r62=0
r63=0
r64=0
r65=0
r66=0
r67=0
r68=0
r69=0
r610=0
r611=0
r612=0
r613=0
r614=0
r615=0
r616=0
r617=0
r618=0
r619=0
r620=0
r621=0
r622=0
r623=0
r624=0
r625=0
r626=0
r627=0
r628=0
r629=0
r630=0
r631=0
r632=0
r633=0
r634=0
r635=0
r636=0
r637=0
r638=0
r639=0
r640=0
r641=0
r642=0
r643=0
r644=0
r645=0
r646=0
r647=0
r648=0
r649=0
r650=0
r651=0
r652=0
r653=0
r654=0
r655=0
r656=0
r657=0
r658=0
r659=0
r660=0
r661=0
r662=0
r663=0
r664=0
r665=0
r666=0
r667=0
r668=0
r669=0
r670=0
r671=0
r672=0
r673=0
1 3 1 2 1 2
-----*a11 *r6260 - ----*a11*a23 *r6260 + -----*a11*b12 *kap*r6260
384 48 384
r674=-------------------------------------------------------------------
3
a23
r675=0
1 3 1 2 1 2
-----*a11 *r6260 + ----*a11*a23 *r6260 + -----*a11*b12 *kap*r6260
384 24 384
r676=-------------------------------------------------------------------
3
a23
r677=0
r678=0
1 3 1 2 1 2
-----*a11 *r6260 - ----*a11*a23 *r6260 + -----*a11*b12 *kap*r6260
384 48 384
r679=-------------------------------------------------------------------
3
a23
r680=0
r681=0
1 3 1 2
- ----*a11 *r6260 - ----*a11*b12 *kap*r6260
32 32
r682=----------------------------------------------
2
a23 *b12
1 5 7 3 2 1 3 2
r683=( - -----*a11 *r6260 + -----*a11 *a23 *r6260 - -----*a11 *b12 *kap*r6260
384 288 192
1 4 7 2 2
+ ----*a11*a23 *r6260 + -----*a11*a23 *b12 *kap*r6260
18 288
1 4 2 3 2
- -----*a11*b12 *kap *r6260)/(a23 *b12 )
384
r684=0
r685=0
r686=0
r687=0
r688=0
r689=0
r690=0
r691=0
r692=0
r693=0
r694=0
r695=0
r696=0
r697=0
r698=0
r699=0
r6100=0
r6101=0
r6102=0
r6103=0
r6104=0
r6105=0
r6106=0
r6107=0
r6108=0
r6109=0
r6110=0
r6111=0
r6112=0
r6113=0
r6114=0
r6115=0
r6116=0
r6117=0
r6118=0
r6119=0
r6120=0
r6121=0
r6122=0
r6123=0
1
---*b12*r6260
8
r6124=---------------
a23
r6125=0
r6126=0
r6127=0
r6128=0
1
---*b12*r6260
8
r6129=---------------
a23
1
r6130= - ---*r6260
4
1
- ---*a11*r6260
8
r6131=------------------
a23
r6132=0
r6133=0
r6134=0
5
r6135= - ---*r6260
4
r6136=0
r6137=0
1 4 1 2 2 1 2 2
r6138=( - -----*a11 *r6260 - ---*a11 *a23 *r6260 - -----*a11 *b12 *kap*r6260
192 9 192
14 4 11 2 2 3
+ ----*a23 *r6260 - ----*a23 *b12 *kap*r6260)/(a23 *b12)
9 72
1 4 17 2 2 1 2 2
r6139=(----*a11 *r6260 + ----*a11 *a23 *r6260 + ----*a11 *b12 *kap*r6260
24 36 24
8 4 11 2 2 2 2
+ ---*a23 *r6260 + ----*a23 *b12 *kap*r6260)/(a23 *b12 )
9 36
r6140=0
r6141=0
r6142=0
r6143=0
r6144=0
r6145=0
r6146=0
r6147=0
r6148=0
r6149=0
r6150=0
r6151=0
r6152=0
r6153=0
r6154=0
r6155=0
r6156=0
r6157=0
r6158=0
r6159=0
1
---*b12*r6260
4
r6160=---------------
a23
r6161=0
r6162=0
r6163=0
r6164=0
1
- ---*a11*r6260
8
r6165=------------------
a23
1
r6166= - ---*r6260
2
1
- ----*a11*r6260
16
r6167=-------------------
a23
r6168=0
r6169=0
7
- ----*a11*r6260
16
r6170=-------------------
a23
r6171=0
r6172=0
3*a11*r6260
r6173=-------------
b12
1 5 59 3 2 1 3 2
r6174=(-----*a11 *r6260 + -----*a11 *a23 *r6260 + -----*a11 *b12 *kap*r6260
192 288 128
37 4 65 2 2
- ----*a11*a23 *r6260 + -----*a11*a23 *b12 *kap*r6260
18 288
1 4 2 3 2
+ -----*a11*b12 *kap *r6260)/(a23 *b12 )
384
r6175=0
r6176=0
r6177=0
r6178=0
r6179=0
r6180=0
1
---*b12*r6260
8
r6181=---------------
a23
r6182=0
r6183=0
1
---*b12*r6260
8
r6184=---------------
a23
r6185=0
1
- ---*a11*r6260
4
r6186=------------------
a23
1
r6187= - ---*r6260
4
r6188=0
r6189=0
5
r6190= - ---*r6260
4
r6191=0
r6192=0
1 2 2
---*a11 *r6260 + 4*a23 *r6260
2
r6193=-------------------------------
a23*b12
7 2 2
- ---*a11 *r6260 - 4*a23 *r6260
4
r6194=----------------------------------
2
b12
r6195=0
r6196=0
1
- ---*a11*r6260
8
r6197=------------------
a23
r6198=0
r6199=0
3
- ---*a11*r6260
8
r6200=------------------
a23
r6201=0
r6202=0
5
---*a11*r6260
2
r6203=---------------
b12
3 3 2
- ----*a11 *r6260 - 4*a11*a23 *r6260
16
r6204=---------------------------------------
2
a23*b12
r6205=0
r6206=0
3 2
---*a11 *r6260
8
r6207=----------------
a23*b12
5 2
- ---*a11 *r6260
4
r6208=-------------------
2
b12
1 3
- ---*a11 *r6260
8
r6209=-------------------
2
a23*b12
r6210=0
r6211=0
r6212=0
r6213=0
r6214=0
r6215=0
r6216=0
r6217=0
r6218=0
r6219=0
r6220=0
r6221=0
r6222=0
r6223=0
r6224=0
r6225=0
r6226=0
r6227=0
r6228=0
r6229=0
r6230=0
r6231=0
r6232=0
r6233=0
r6234=0
r6235=0
r6236=0
r6237=0
r6238=0
r6239=0
r6240=0
r6241=0
r6242=0
r6243=0
r6244=0
r6245=0
r6246=0
1
- ---*b12*r6260
8
r6247=------------------
a23
r6248=0
r6249=0
r6250=0
r6251=0
r6252=0
r6253=0
1
- ---*b12*r6260
8
r6254=------------------
a23
r6255=0
r6256=0
r6257=0
r6258=0
r6259=0
r6261=0
1 3 1 2
----*a11 *r6260 + ----*a11*b12 *kap*r6260
32 32
r6262=-------------------------------------------
2
a23 *b12
1 4 17 2 2 1 2 2
r6263=(-----*a11 *r6260 + ----*a11 *a23 *r6260 + -----*a11 *b12 *kap*r6260
192 72 192
4 4 11 2 2 3
+ ---*a23 *r6260 + ----*a23 *b12 *kap*r6260)/(a23 *b12)
9 72
r6264=0
r6265=0
r6266=0
r6267=0
r6268=0
r6269=0
r6270=0
r6271=0
r6272=0
r6273=0
r6274=0
r6275=0
r6276=0
r6277=0
r6278=0
r6279=0
r6280=0
r6281=0
r6282=0
1
- ---*b12*r6260
4
r6283=------------------
a23
r6284=0
r6285=0
r6286=0
r6287=0
r6288=0
r6289=0
r6290=0
r6291=0
r6292=0
r6293=0
r6294=r6260
1
---*a11*r6260
4
r6295=---------------
a23
r6296=0
1 2 2
- ---*a11 *r6260 - 2*a23 *r6260
8
r6297=----------------------------------
a23*b12
3
- ---*a11*r6260
2
r6298=------------------
b12
r6299=0
r6300=0
r6301=0
r6302=0
r6303=0
1
- ---*b12*r6260
8
r6304=------------------
a23
r6305=0
r6306=0
r6307=0
r6308=0
1
- ---*b12*r6260
8
r6309=------------------
a23
r6310=0
r6311=0
r6312=0
r6313=0
1
---*a11*r6260
4
r6314=---------------
a23
r6315=2*r6260
r6316=0
3
- ---*a11*r6260
2
r6317=------------------
b12
1 2 2
- ---*a11 *r6260 - 4*a23 *r6260
4
r6318=----------------------------------
a23*b12
r6319=0
r6320=0
r6321=0
r6322=0
r6323=0
r6324=0
1
---*a11*r6260
2
r6325=---------------
a23
r6326=0
1 2
- ---*a11 *r6260
4
r6327=-------------------
a23*b12
5
- ---*a11*r6260
2
r6328=------------------
b12
r6329=0
r6330=0
r6331=0
3 2
- ---*a11 *r6260
8
r6332=-------------------
a23*b12
r6333=0
r6334=0
r6335=0
r6336=0
r6337=0
r6338=0
r6339=0
r6340=0
r6341=0
r6342=0
r6343=0
r6344=0
r6345=0
r6346=0
r6347=0
r6348=0
r6349=0
r6350=0
r6351=0
r6352=0
r6353=0
r6354=0
r6355=0
r6356=0
r6357=0
r6358=0
r6359=0
r6360=0
1
- ---*a11*r6260
8
r6361=------------------
a23
r6362= - r6260
r6363=0
r6364=0
r6365=0
1
- ---*a11*r6260
8
r6366=------------------
a23
r6367=0
r6368=0
a11*r6260
r6369=-----------
b12
31 3 2 1 3 2 4 4
r6370=(-----*a11 *a23 *r6260 - -----*a11 *b12 *kap*r6260 + ---*a11*a23 *r6260
144 384 9
25 2 2 1 4 2 3 2
+ -----*a11*a23 *b12 *kap*r6260 - -----*a11*b12 *kap *r6260)/(a23 *b12 )
144 384
r6371=0
r6372=0
r6373=0
r6374=0
r6375=0
r6376=0
r6377=0
r6378=0
r6379=0
r6380=0
r6381=0
1
- ---*a11*r6260
2
r6382=------------------
a23
r6383= - r6260
r6384=0
r6385=0
r6386=0
r6387=0
r6388=0
1 2 1 2
---*a11 *r6260 - ---*b12 *kap*r6260
8 8
r6389=-------------------------------------
a23*b12
19 2 16 2 1 2
- ----*a11 *r6260 - ----*a23 *r6260 - ----*b12 *kap*r6260
12 3 12
r6390=------------------------------------------------------------
2
b12
r6391=0
r6392=0
3
- ---*a11*r6260
8
r6393=------------------
a23
r6394=0
r6395=0
1
- ---*a11*r6260
8
r6396=------------------
a23
r6397=0
r6398=0
2*a11*r6260
r6399=-------------
b12
3 3 2
- ----*a11 *r6260 - 6*a11*a23 *r6260
16
r6400=---------------------------------------
2
a23*b12
r6401=0
r6402=0
3 2
---*a11 *r6260
8
r6403=----------------
a23*b12
9 2
- ---*a11 *r6260
4
r6404=-------------------
2
b12
1 3
- ---*a11 *r6260
4
r6405=-------------------
2
a23*b12
r6406=0
r6407=0
r6408=0
r6409=0
r6410=0
r6411=0
r6412=0
r6413=0
r6414=0
r6415=0
r6416=0
r6417=0
r6418=0
r6419=0
r6420=0
r6421=0
- a11*r6260
r6422=--------------
b12
1 2 1 2
---*a11 *r6260 + ---*b12 *kap*r6260
8 8
r6423=-------------------------------------
a23*b12
r6424=0
r6425=0
r6426=0
r6427=0
r6428=0
r6429=0
r6430=0
r6431=0
1 2
- ---*a11 *r6260
4
r6432=-------------------
a23*b12
- 2*a11*r6260
r6433=----------------
b12
r6434=0
r6435=0
r6436=0
3 2
- ---*a11 *r6260
8
r6437=-------------------
a23*b12
r6438=0
r6439=0
r6440=0
r6441=0
r6442=0
r6443=0
r6444=0
r6445=0
r6446=0
r6447=0
r6448=0
r6449=0
1 3 1 2
---*a11 *r6260 + ---*a11*b12 *kap*r6260
8 8
r6450=-----------------------------------------
2
a23*b12
r6451=0
r6452=0
r6453=0
2
- a11 *r6260
r6454=---------------
2
b12
1 3
- ---*a11 *r6260
8
r6455=-------------------
2
a23*b12
r6456=0
r6457=0
r6458=0
r6459=0
r6460=0
r6461=0
1
a33=---*a11
2
a22=a11
a13=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:
r6260, a11, a23, b12
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 2 2
{r6260,a11 + 4*a23 + b12 *kap,b12,a23,a11}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a11 + 16*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 2
2*u1 *a11 + 2*u1*v2*b12 + 2*u2 *a11 + 4*u2*u3*a23 - 2*u2*v1*b12 + u3 *a11
HAM=---------------------------------------------------------------------------
2
has apart from the Hamiltonian and Casimirs the following 0 first integrals:
And again in machine readable form:
HAM=(2*u1**2*a11 + 2*u1*v2*b12 + 2*u2**2*a11 + 4*u2*u3*a23 - 2*u2*v1*b12 + u3**2
*a11)/2$