Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r61=0
r62=r613
r63=0
r615
r64=------
2
r65=0
r66=0
r67=0
r68=0
r69=0
r610=0
r611=0
r612=0
r614=0
r616=0
r617=0
r618=0
r619=0
r620=0
r621=0
r615
r622=------
2
r623=0
r624=0
r625=0
r626=0
b33*r684
r628=----------
b23
r629=0
b33*r695
r630=----------
b23
r631=0
r632=0
r633=0
r634=0
r635=0
r636=0
r637=0
r638=0
b33*r695
r639=----------
b23
r640=0
r641=0
r642=0
r643=0
r644=0
r645=0
r646=0
r647=0
r648=0
2 2 2
a33*b23*r684 + b23 *r6336 + b33 *r6140 - b33 *r6336
r649=-----------------------------------------------------
2
b23
r650=0
a33*r695 + b23*r6338
r651=----------------------
b23
r652=0
r653=0
r654=0
r655=0
r656=0
r657=0
a33*r695 + b23*r6338
r658=----------------------
b23
r659=0
r660=0
r661=0
r662=0
r663=0
1
---*b33*r6130
2
r664=---------------
a23
r665=0
r666=0
r667=0
r668=0
r669=0
r670=0
r671=0
r672=0
r673=0
1 1
---*a23*r6361 + ---*a33*r6130
2 4
r674=-------------------------------
a23
r675=0
r676=0
r677=0
r678=0
r679=0
r680=0
r681=0
r682=0
r683=0
r685=0
r686=r695
r687=0
r688=0
r689=0
r690=0
r691=0
r692=0
r693=0
r694=0
r696=0
r697=0
r698=0
r699=0
r6100=0
r6101=0
r6102=0
r6103=0
r6104=0
2*a23*r684 + 2*b33*r6140 - 2*b33*r6336
r6105=----------------------------------------
b23
r6106=0
2*a23*r695
r6107=------------
b23
r6108=0
r6109=0
r6110=0
r6111=0
r6112=0
r6113=0
2*a23*r695
r6114=------------
b23
r6115=0
r6116=0
r6117=0
r6118=0
2 2 1 3
4*a23 *b33*r6140 - 4*a23 *b33*r6336 + ---*b23 *r6130
2
r6120=------------------------------------------------------
2
a23*b23
r6121=0
r6122=0
r6123=0
r6124=0
r6125=0
r6126=0
r6127=0
r6128=0
r6129=0
r6131=0
r6132=0
r6133=0
r6134=0
r6135=0
r6136=0
r6137=0
r6138=0
r6139=0
r6141=0
r6142=r6338
r6143=0
r6144=0
r6145=0
r6146=0
r6147=0
r6148=0
r6149=r6338
r6150=0
r6151=0
r6152=0
r6153=0
r6154=0
2 2
r6155=(4*a23 *b23*r6140 - 4*a23 *b23*r6336 - 2*a23*a33*b33*r6140
1 2 2
+ 2*a23*a33*b33*r6336 + ---*b23 *b33*r6130)/(a23*b23 )
2
r6156=0
r6157=0
r6158=0
r6159=0
r6160=0
r6161=0
r6162=0
r6163=0
r6164=0
2 2 2
4*a23 *r6140 - 4*a23 *r6336 + b23 *r6361
r6165=------------------------------------------
2
b23
r6166=0
r6167=0
r6168=0
r6169=0
r6171=0
r6172=0
r6173=0
r6174=0
1 2
- 2*a23*a33*r6140 + 2*a23*a33*r6336 + ---*b23 *r6130
2
r6175=-------------------------------------------------------
a23*b23
r6176=0
r6177=0
r6178=0
r6179=0
r6180=0
r6181=0
r6182=0
r6183=0
r6184=0
2
- 4*a23*a33*r6140 + 4*a23*a33*r6336 + b23 *r6130
r6185=---------------------------------------------------
2
b23
r6186=0
r6187=0
r6188=0
r6189=0
r6190=0
r6191=0
r6192=0
r6193=0
r6195
2 2 1 2 1 2
a23*a33 *r6140 - a23*a33 *r6336 + ---*a23*b23 *r6361 - ---*a33*b23 *r6130
2 4
=---------------------------------------------------------------------------
2
a23*b23
r6196=0
r6197=0
r6198=0
r6199=0
r6200=0
r6201=0
r6202=0
r6203=0
r6204=0
r6205=0
r6206=0
r6207=0
r6208=0
r6209=0
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
r6247=0
r6248=0
r6249=0
r6250=0
r6251=0
r6252=0
r6253=0
r6254=0
r6255=0
r6256=0
r6257=0
r6258=0
r6259=0
r6260=0
r6261=0
r6262=0
r6263=0
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
r6283=0
r6284=0
r6285=0
r6286=0
r6287=0
r6288=0
r6289=0
r6290=0
r6291=0
r6292=0
r6293=0
r6294=0
r6295=0
r6296=0
r6297=0
r6298=0
r6299=0
r6300=0
r6301=0
r6302=0
r6303=0
r6304=0
r6305=0
r6306=0
r6307=0
r6308=0
r6309=0
r6310=0
r6311=0
r6312=0
r6313=0
r6314=0
r6315=0
r6316=0
r6317=0
r6318=0
r6319=0
r6320=0
r6321=0
r6322=0
r6323=0
r6324=0
r6325=0
r6326=0
r6327=0
r6328=0
r6329=0
r6330=0
r6331=0
r6332=0
r6333=0
r6334=0
r6335=0
r6337=0
r6339=0
r6340=0
r6341=0
r6342=0
r6343=0
r6344=0
r6345=r6338
r6346=0
r6347=0
r6348=0
r6349=0
1 2
- 2*a23*a33*b33*r6140 + 2*a23*a33*b33*r6336 + ---*b23 *b33*r6130
2
r6351=-------------------------------------------------------------------
2
a23*b23
r6352=0
r6353=0
r6354=0
r6355=0
r6356=0
r6357=0
r6358=0
r6359=0
r6360=0
r6362=0
r6363=0
r6364=0
r6365=0
r6366=0
r6367=0
r6368=0
r6369=0
r6370=0
1 2
- 2*a23*a33*r6140 + 2*a23*a33*r6336 + ---*b23 *r6130
2
r6371=-------------------------------------------------------
a23*b23
r6372=0
r6373=0
r6374=0
r6375=0
r6376=0
r6377=0
r6378=0
r6379=0
r6380=0
2
- 4*a23*a33*r6140 + 4*a23*a33*r6336 + b23 *r6130
r6381=---------------------------------------------------
2
b23
r6382=0
r6383=0
r6384=0
r6385=0
r6387=0
r6388=0
r6389=0
r6390=0
r6391
2 2 2 1 2
2*a23*a33 *r6140 - 2*a23*a33 *r6336 + a23*b23 *r6361 - ---*a33*b23 *r6130
2
=---------------------------------------------------------------------------
2
a23*b23
r6392=0
r6393=0
r6394=0
r6395=0
r6396=0
r6397=0
r6398=0
r6399=0
r6401=0
r6402=0
r6403=0
r6404=0
r6405=0
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
r6422=0
r6423=0
r6424=0
r6425=0
r6426=0
r6427=0
r6428=0
r6429=0
r6430=0
r6431=0
r6432=0
r6433=0
r6434=0
r6435=0
r6436=0
r6437=0
r6438=0
r6439=0
r6440=0
r6441
2 2 1 2 1 2
a23*a33 *r6140 - a23*a33 *r6336 + ---*a23*b23 *r6361 - ---*a33*b23 *r6130
2 4
=---------------------------------------------------------------------------
2
a23*b23
r6442=0
r6443=0
r6444=0
r6445=0
r6447=0
r6448=0
r6449=0
r6450=0
r6451=0
r6452=0
r6453=0
r6455=0
r6456=0
r6457=0
r6458=0
r6459=0
r6460=0
c23=0
b32=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:
r60, r613, r615, r684, r695, r6130, r6140, r6336, r6338,
r6361, a33, b33, c33, a23, b23
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.
6 5 4 2 4 2
{{5*b23 *r6361 - b23 *c33*r6130 + 10*b23 *c33 *r6140 - 10*b23 *c33 *r6336,
5 4 4
3*b23 *r6130 - 30*b23 *c33*r6140 + 30*b23 *c33*r6336,
2
b23 *r6130,
r6361,
2
24*c33 *r6338,
r6338,
r6336,
6 5 4 2 4 2
15*b23 *r6361 - 3*b23 *c33*r6130 + 30*b23 *c33 *r6140 - 30*b23 *c33 *r6336,
6 5 5
9*b23 *r6130 - 90*b23 *c33*r6140 + 90*b23 *c33*r6336,
3 7 3 7 3 7
-----*b23 *r6140 - -----*b23 *r6336 + ----*b23 *r6361,
128 128 32
7 7
15*b23 *r6140 - 15*b23 *r6336,
8 2
2812500*b23 *c33 *r6338,
12 2
39375000*b23 *c33 *r6338,
r6140,
r6130,
3
b23 *r6130,
9
78125*b23 *r695,
12
1771875000*b23 *c33*r695,
b23*r684,
r695,
8
7500*b23 *r695,
r684,
11 10
45*b23 *r6361 + 9*b23 *c33*r6130,
10 2 9 3
937500*b23 *c33 *r6338 + 937500*b23 *c33 *r695,
15 2 14 3
106312500*b23 *c33 *r6338 + 1063125000*b23 *c33 *r695,
2 2
b23 *r6140 - 2*b23 *r6336,
9
6*b23 *r615,
r615,
r613,
4
30*b23 *r615,
3
2*b23 *r613,
r60},
a33,
b23,
a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u2*v3*b23 + u3 *a33 + u3*v3*b33 + v3 *c33
has apart from the Hamiltonian and Casimirs the following 10 first integrals:
4 2 2 2 2 2 2 2 4 2 2 2 2 4 2
FI=u1 *v3 + 2*u1 *u2 *v3 + 2*u1 *u3 *v3 + u2 *v3 + 2*u2 *u3 *v3 + u3 *v3
= a product of the elements of: {v3,
v3,
2 2 2
u1 + u2 + u3 ,
2 2 2
u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
FI=u1 *v1 *v3 + u1 *v2 *v3 + u2 *v1 *v3 + u2 *v2 *v3 + u3 *v1 *v3
2 2 2
+ u3 *v2 *v3
= a product of the elements of: {v3,
v3,
2 2
v1 + v2 ,
2 2 2
u1 + u2 + u3 }
{HAM,FI} = 0
4 2 2 2 2 2 2 2 2
FI= - u1 *v3 *a33 - 2*u1 *u2 *v3 *a33 + 4*u1 *u2*u3*v3 *a23*a33
2 3 2 3 2 4 2 4 2 2
+ 2*u1 *u2*v3 *a33*b23 + 2*u1 *u3*v3 *a33*b33 + u1 *v3 *b23 - u2 *v3 *a33
3 2 3 3 2 2 2 2
+ 4*u2 *u3*v3 *a23*a33 + 2*u2 *v3 *a33*b23 - 4*u2 *u3 *v3 *a23
2 3 2 3
+ u2 *u3*v3 *( - 4*a23*b23 + 2*a33*b33) - 4*u2*u3 *v3 *a23*b33
4 2 4 2 2
- 2*u2*u3*v3 *b23*b33 + u3 *v3 *(b23 - b33 )
= a product of the elements of: { - v3,
v3,
4 2 2 2 2 2 2
u1 *a33 + 2*u1 *u2 *a33 - 4*u1 *u2*u3*a23*a33 - 2*u1 *u2*v3*a33*b23
2 2 2 2 4 2 3
- 2*u1 *u3*v3*a33*b33 - u1 *v3 *b23 + u2 *a33 - 4*u2 *u3*a23*a33
3 2 2 2 2
- 2*u2 *v3*a33*b23 + 4*u2 *u3 *a23 + u2 *u3*v3*(4*a23*b23 - 2*a33*b33)
2 2 2 2 2 2
+ 4*u2*u3 *v3*a23*b33 + 2*u2*u3*v3 *b23*b33 + u3 *v3 *( - b23 + b33 )}
{HAM,FI} = 0
4 2 2 2 2 2 2 2 2
FI=u1 *v3 *a33 + 2*u1 *u2 *v3 *a33 - 4*u1 *u2*u3*v3 *a23*a33
2 3 2 3 4 2 2
- 2*u1 *u2*v3 *a33*b23 - 2*u1 *u3*v3 *a33*b33 + u2 *v3 *a33
3 2 3 3 2 2 2 2
- 4*u2 *u3*v3 *a23*a33 - 2*u2 *v3 *a33*b23 + 4*u2 *u3 *v3 *a23
2 3 2 4 2 2 3
+ u2 *u3*v3 *(4*a23*b23 - 2*a33*b33) + u2 *v3 *b23 + 4*u2*u3 *v3 *a23*b33
4 2 4 2
+ 2*u2*u3*v3 *b23*b33 + u3 *v3 *b33
= a product of the elements of: {v3,
v3,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23 + u2*v3*b23 + u3*v3*b33,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23 + u2*v3*b23 + u3*v3*b33}
{HAM,FI} = 0
4 2 2 2 2 2 2 2 3
FI= - u1 *v3 *a33 - 2*u1 *u2 *v3 *a33 + 4*u1 *u2*u3*v3 *a23 + 2*u1 *u2*v3 *b23
2 3 4 2 3 2 3 3
+ 2*u1 *u3*v3 *b33 - u2 *v3 *a33 + 4*u2 *u3*v3 *a23 + 2*u2 *v3 *b23
2 3 3 2 2 3 4 2
+ 2*u2 *u3*v3 *b33 + 4*u2*u3 *v3 *a23 + 2*u2*u3 *v3 *b23 + u3 *v3 *a33
3 3
+ 2*u3 *v3 *b33
= a product of the elements of: {v3,
v3,
2 2 2
u1 + u2 + u3 ,
2 2 2
- u1 *a33 - u2 *a33 + 4*u2*u3*a23 + 2*u2*v3*b23 + u3 *a33 + 2*u3*v3*b33}
{HAM,FI} = 0
2 2 2 2 2 3 2 3
FI=2*u2*u3*v1 *v3 *a23 + 2*u2*u3*v2 *v3 *a23 + u2*v1 *v3 *b23 + u2*v2 *v3 *b23
2 2 2 2 2 2 2 3 2 3
+ u3 *v1 *v3 *a33 + u3 *v2 *v3 *a33 + u3*v1 *v3 *b33 + u3*v2 *v3 *b33
= a product of the elements of: {v3,
v3,
2 2
v1 + v2 ,
2
2*u2*u3*a23 + u2*v3*b23 + u3 *a33 + u3*v3*b33}
{HAM,FI} = 0
4 5 2 4 5
FI=2*u2*u3*v3 *a23 + u2*v3 *b23 + u3 *v3 *a33 + u3*v3 *b33
= a product of the elements of: {v3,
v3,
v3,
v3,
2
2*u2*u3*a23 + u2*v3*b23 + u3 *a33 + u3*v3*b33}
{HAM,FI} = 0
4 2 2 2 2 4 2
FI=v1 *v3 + 2*v1 *v2 *v3 + v2 *v3
2 2 2 2
= a product of the elements of: {v3,v3,v1 + v2 ,v1 + v2 }
{HAM,FI} = 0
2 4 2 4
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,
v3,
v3,
v3,
2 2
v1 + v2 }
{HAM,FI} = 0
6
FI=v3
= a product of the elements of: {v3,
v3,
v3,
v3,
v3,
v3}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u2*v3*b23 + u3**2*a33 + u3*v3*b33 + v3**2*c33$
FI=u1**4*v3**2 + 2*u1**2*u2**2*v3**2 + 2*u1**2*u3**2*v3**2 + u2**4*v3**2 + 2*u2
**2*u3**2*v3**2 + u3**4*v3**2$
FI=u1**2*v1**2*v3**2 + u1**2*v2**2*v3**2 + u2**2*v1**2*v3**2 + u2**2*v2**2*v3**2
+ u3**2*v1**2*v3**2 + u3**2*v2**2*v3**2$
FI= - u1**4*v3**2*a33**2 - 2*u1**2*u2**2*v3**2*a33**2 + 4*u1**2*u2*u3*v3**2*a23*
a33 + 2*u1**2*u2*v3**3*a33*b23 + 2*u1**2*u3*v3**3*a33*b33 + u1**2*v3**4*b23**2 -
u2**4*v3**2*a33**2 + 4*u2**3*u3*v3**2*a23*a33 + 2*u2**3*v3**3*a33*b23 - 4*u2**2
*u3**2*v3**2*a23**2 + u2**2*u3*v3**3*( - 4*a23*b23 + 2*a33*b33) - 4*u2*u3**2*v3
**3*a23*b33 - 2*u2*u3*v3**4*b23*b33 + u3**2*v3**4*(b23**2 - b33**2)$
FI=u1**4*v3**2*a33**2 + 2*u1**2*u2**2*v3**2*a33**2 - 4*u1**2*u2*u3*v3**2*a23*a33
- 2*u1**2*u2*v3**3*a33*b23 - 2*u1**2*u3*v3**3*a33*b33 + u2**4*v3**2*a33**2 - 4*
u2**3*u3*v3**2*a23*a33 - 2*u2**3*v3**3*a33*b23 + 4*u2**2*u3**2*v3**2*a23**2 + u2
**2*u3*v3**3*(4*a23*b23 - 2*a33*b33) + u2**2*v3**4*b23**2 + 4*u2*u3**2*v3**3*a23
*b33 + 2*u2*u3*v3**4*b23*b33 + u3**2*v3**4*b33**2$
FI= - u1**4*v3**2*a33 - 2*u1**2*u2**2*v3**2*a33 + 4*u1**2*u2*u3*v3**2*a23 + 2*u1
**2*u2*v3**3*b23 + 2*u1**2*u3*v3**3*b33 - u2**4*v3**2*a33 + 4*u2**3*u3*v3**2*a23
+ 2*u2**3*v3**3*b23 + 2*u2**2*u3*v3**3*b33 + 4*u2*u3**3*v3**2*a23 + 2*u2*u3**2*
v3**3*b23 + u3**4*v3**2*a33 + 2*u3**3*v3**3*b33$
FI=2*u2*u3*v1**2*v3**2*a23 + 2*u2*u3*v2**2*v3**2*a23 + u2*v1**2*v3**3*b23 + u2*
v2**2*v3**3*b23 + u3**2*v1**2*v3**2*a33 + u3**2*v2**2*v3**2*a33 + u3*v1**2*v3**3
*b33 + u3*v2**2*v3**3*b33$
FI=2*u2*u3*v3**4*a23 + u2*v3**5*b23 + u3**2*v3**4*a33 + u3*v3**5*b33$
FI=v1**4*v3**2 + 2*v1**2*v2**2*v3**2 + v2**4*v3**2$
FI=v1**2*v3**4 + v2**2*v3**4$
FI=v3**6$