Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r521=0
r522=0
r523=0
r524=0
r525=0
r526=0
r527=0
r528=0
r529=0
r530=0
r531=0
r532=0
r533=0
r534=0
r535=0
1
a23*r5196 + ---*a33*r571
2
r536=--------------------------
a23
1
a23*r5197 + ---*a33*r572
2
r537=--------------------------
a23
1
a23*r5199 + ---*a33*r574
2
r539=--------------------------
a23
1
a23*r5200 + ---*a33*r575
2
r540=--------------------------
a23
1
a23*r5201 + ---*a33*r576
2
r541=--------------------------
a23
1
a23*r5205 + ---*a33*r580
2
r545=--------------------------
a23
r546=0
r547=0
r548=0
r549=0
r550=0
r551=0
1 2 1 1 2 1 2
---*a23 *r5212 + ---*a23*a33*r5222 + ---*a33 *r5107 - ---*a33 *r5212
2 4 4 4
r552=----------------------------------------------------------------------
2
a23
1 2 1 1 2 1 2
---*a23 *r5213 + ---*a23*a33*r5223 + ---*a33 *r5108 - ---*a33 *r5213
2 4 4 4
r553=----------------------------------------------------------------------
2
a23
1 2 1 1 2 1 2
---*a23 *r5214 + ---*a23*a33*r5224 + ---*a33 *r5109 - ---*a33 *r5214
2 4 4 4
r554=----------------------------------------------------------------------
2
a23
r555=0
r556=0
r557=0
r558=0
r559=0
r560=0
r561=0
r562=0
r563=0
r564=0
r565=0
r566=0
r567=0
r568=0
r569=0
r570=0
r581=0
r582=0
r583=0
r584=0
r585=0
r586=0
a23*r5222 + a33*r5107 - a33*r5212
r587=-----------------------------------
a23
a23*r5223 + a33*r5108 - a33*r5213
r588=-----------------------------------
a23
a23*r5224 + a33*r5109 - a33*r5214
r589=-----------------------------------
a23
r590=0
r591=r5196
r592=r5197
1
a23*r538 - ---*a33*r573
2
r593=-------------------------
a23
r594=r5199
r595=r5200
r596=r5201
r597=r5202
1
a23*r543 - ---*a33*r578
2
r598=-------------------------
a23
r599=r5204
r5100=r5205
r5101=0
r5102=0
r5103=0
r5104=0
r5105=0
r5106=0
r5110=0
r5111=0
r5112=0
r5113=0
r5114=0
r5115=0
r5116=0
r5117=r5222
r5118=r5223
r5119=r5224
r5120=0
1 1
---*a23*r5212 - ---*a33*r5222
2 4
r5121=-------------------------------
a23
1 1
---*a23*r5213 - ---*a33*r5223
2 4
r5122=-------------------------------
a23
1 1
---*a23*r5214 - ---*a33*r5224
2 4
r5123=-------------------------------
a23
r5124=0
r5125=0
r5126=0
r5127=0
r5128=0
r5129=0
r5130=0
r5131=0
r5132=0
r5133=0
r5134=0
r5135=0
r5136=0
r5137=0
r5138=0
r5139=0
r5140=0
r5141=0
r5142=0
r5143=0
r5144=0
r5145=0
r5146=0
r5147=0
r5148=0
r5149=0
r5150=0
r5151=0
r5152=0
r5153=0
r5154=0
r5155=0
r5156=0
r5157=0
r5158=0
r5159=0
r5160=0
r5161=0
r5162=0
r5163=0
r5164=0
r5165=0
r5166=0
r5167=0
r5168=0
r5169=0
r5170=0
r5171=0
r5172=0
r5173=0
r5174=0
r5175=0
r5176=0
r5177=0
r5178=0
r5179=0
r5180=0
r5181=0
r5182=0
r5183=0
r5184=0
r5185=0
r5186=0
r5187=0
r5188=0
r5189=0
r5190=0
r5191=0
r5192=0
r5193=0
r5194=0
r5195=0
1
a23*r538 - ---*a33*r573
2
r5198=-------------------------
a23
1
a23*r543 - ---*a33*r578
2
r5203=-------------------------
a23
r5206=0
r5207=0
r5208=0
r5209=0
r5210=0
r5211=0
r5215=0
r5216=0
r5217=0
r5218=0
r5219=0
r5220=0
r5221=0
r5225=0
1
a23*r5212 - ---*a33*r5222
2
r5226=---------------------------
a23
1
a23*r5213 - ---*a33*r5223
2
r5227=---------------------------
a23
1
a23*r5214 - ---*a33*r5224
2
r5228=---------------------------
a23
r5229=0
r5230=0
r5231=0
r5232=0
r5233=0
r5234=0
r5235=0
r5236=0
r5237=0
r5238=0
r5239=0
r5240=0
r5241=0
r5242=0
r5243=0
r5244=0
r5245=0
1 1
---*a23*r5212 - ---*a33*r5222
2 4
r5246=-------------------------------
a23
1 1
---*a23*r5213 - ---*a33*r5223
2 4
r5247=-------------------------------
a23
1 1
---*a23*r5214 - ---*a33*r5224
2 4
r5248=-------------------------------
a23
r5249=0
r5250=0
r5251=0
c22=0
c11=0
b33=0
b32=0
b31=0
1
a23*r5204 + ---*a33*r579
2
r544=--------------------------
a23
1
a23*r5202 + ---*a33*r577
2
r542=--------------------------
a23
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:
r5202, r5204, r50, r51, r52, r53, r54, r55, r56, r57,
r58, r59, r510, r511, r512, r513, r514, r515, r516, r517,
r518, r519, r520, r538, r543, r571, r572, r573, r574,
r575, r576, r577, r578, r579, r580, r5107, r5108, r5109,
r5196, r5197, r5199, r5200, r5205, r5212, r5213, r5214,
r5222, r5223, r5224, r5201, a33, 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.
{a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2
HAM=2*u2*u3*a23 + u3 *a33
has apart from the Hamiltonian and Casimirs the following 50 first integrals:
2 2 2
FI=u1 *v1*v2*v3 + u2 *v1*v2*v3 + u3 *v1*v2*v3
2 2 2
= a product of the elements of: {v3,v2,v1,u1 + u2 + u3 }
{HAM,FI} = 0
4 2 2 2 4
FI= - u1 *v1*a33 - 2*u1 *u2 *v1*a33 + 4*u1 *u2*u3*v1*a23 - u2 *v1*a33
3 3 4
+ 4*u2 *u3*v1*a23 + 4*u2*u3 *v1*a23 + u3 *v1*a33
= a product of the elements of: {v1,
2 2 2
u1 + u2 + u3 ,
2 2 2
- u1 *a33 - u2 *a33 + 4*u2*u3*a23 + u3 *a33}
{HAM,FI} = 0
4 2 2 2 4
FI= - u1 *v2*a33 - 2*u1 *u2 *v2*a33 + 4*u1 *u2*u3*v2*a23 - u2 *v2*a33
3 3 4
+ 4*u2 *u3*v2*a23 + 4*u2*u3 *v2*a23 + u3 *v2*a33
= a product of the elements of: {v2,
2 2 2
u1 + u2 + u3 ,
2 2 2
- u1 *a33 - u2 *a33 + 4*u2*u3*a23 + u3 *a33}
{HAM,FI} = 0
4 2 2 2 4
FI= - u1 *v3*a33 - 2*u1 *u2 *v3*a33 + 4*u1 *u2*u3*v3*a23 - u2 *v3*a33
3 3 4
+ 4*u2 *u3*v3*a23 + 4*u2*u3 *v3*a23 + u3 *v3*a33
= a product of the elements of: {v3,
2 2 2
u1 + u2 + u3 ,
2 2 2
- u1 *a33 - u2 *a33 + 4*u2*u3*a23 + u3 *a33}
{HAM,FI} = 0
4 2 2 2 2 2 2 2 4 2
FI=2*u1 *v1*a23 + 4*u1 *u2 *v1*a23 + 4*u1 *u3 *v1*a23 + 2*u2 *v1*a23
3 4 2 2
- 4*u2*u3 *v1*a23*a33 + u3 *v1*(2*a23 - a33 )
= a product of the elements of: {v1,
4 2 2 2 2 2 2 2 4 2 3
2*u1 *a23 + 4*u1 *u2 *a23 + 4*u1 *u3 *a23 + 2*u2 *a23 - 4*u2*u3 *a23*a33
4 2 2
+ u3 *(2*a23 - a33 )}
{HAM,FI} = 0
4 2 2 2 2 2 2 2 4 2
FI=2*u1 *v2*a23 + 4*u1 *u2 *v2*a23 + 4*u1 *u3 *v2*a23 + 2*u2 *v2*a23
3 4 2 2
- 4*u2*u3 *v2*a23*a33 + u3 *v2*(2*a23 - a33 )
= a product of the elements of: {v2,
4 2 2 2 2 2 2 2 4 2 3
2*u1 *a23 + 4*u1 *u2 *a23 + 4*u1 *u3 *a23 + 2*u2 *a23 - 4*u2*u3 *a23*a33
4 2 2
+ u3 *(2*a23 - a33 )}
{HAM,FI} = 0
4 2 2 2 2 2 2 2 4 2
FI=2*u1 *v3*a23 + 4*u1 *u2 *v3*a23 + 4*u1 *u3 *v3*a23 + 2*u2 *v3*a23
3 4 2 2
- 4*u2*u3 *v3*a23*a33 + u3 *v3*(2*a23 - a33 )
= a product of the elements of: {v3,
4 2 2 2 2 2 2 2 4 2 3
2*u1 *a23 + 4*u1 *u2 *a23 + 4*u1 *u3 *a23 + 2*u2 *a23 - 4*u2*u3 *a23*a33
4 2 2
+ u3 *(2*a23 - a33 )}
{HAM,FI} = 0
2 3 2 3 2 3
FI=u1 *v1 + u2 *v1 + u3 *v1
2 2 2
= a product of the elements of: {v1,v1,v1,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v1*v3 + u2 *v1*v3 + u3 *v1*v3
2 2 2
= a product of the elements of: {v3,v3,v1,u1 + u2 + u3 }
{HAM,FI} = 0
2 3 2 3 2 3
FI=u1 *v2 + u2 *v2 + u3 *v2
2 2 2
= a product of the elements of: {v2,v2,v2,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v2*v3 + u2 *v2*v3 + u3 *v2*v3
2 2 2
= a product of the elements of: {v3,v3,v2,u1 + u2 + u3 }
{HAM,FI} = 0
2 3 2 3 2 3
FI=u1 *v3 + u2 *v3 + u3 *v3
2 2 2
= a product of the elements of: {v3,v3,v3,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 3 4 2
FI=4*u2 *u3 *v1*a23 + 4*u2*u3 *v1*a23*a33 + u3 *v1*a33
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33,
2*u2*a23 + u3*a33,
v1}
{HAM,FI} = 0
2 2 2 3 4 2
FI=4*u2 *u3 *v2*a23 + 4*u2*u3 *v2*a23*a33 + u3 *v2*a33
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33,
2*u2*a23 + u3*a33,
v2}
{HAM,FI} = 0
2 2 2 3 4 2
FI=4*u2 *u3 *v3*a23 + 4*u2*u3 *v3*a23*a33 + u3 *v3*a33
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33,
2*u2*a23 + u3*a33,
v3}
{HAM,FI} = 0
3 2 3
FI=2*u2*u3*v1 *a23 + u3 *v1 *a33
= a product of the elements of: {v1,
v1,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2
FI=2*u2*u3*v1 *v2*a23 + u3 *v1 *v2*a33
= a product of the elements of: {v1,
v1,
v2,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2 2 2
FI= - u1 *v1 *v3*a33 - u2 *v1 *v3*a33 + 2*u2*u3*v1 *v3*a23
= a product of the elements of: {v1,
v1,
v3,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23}
{HAM,FI} = 0
2 2 2
FI=2*u2*u3*v1*v2 *a23 + u3 *v1*v2 *a33
= a product of the elements of: {v2,
v2,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2
FI=2*u2*u3*v1*v2*v3*a23 + u3 *v1*v2*v3*a33
= a product of the elements of: {v3,
v2,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2
FI=2*u2*u3*v1*v3 *a23 + u3 *v1*v3 *a33
= a product of the elements of: {v3,
v3,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
3 2 3
FI=2*u2*u3*v2 *a23 + u3 *v2 *a33
= a product of the elements of: {v2,
v2,
v2,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2 2 2
FI= - u1 *v2 *v3*a33 - u2 *v2 *v3*a33 + 2*u2*u3*v2 *v3*a23
= a product of the elements of: {v2,
v2,
v3,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23}
{HAM,FI} = 0
2 2 2
FI=2*u2*u3*v2*v3 *a23 + u3 *v2*v3 *a33
= a product of the elements of: {v3,
v3,
v2,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
3 2 3
FI=2*u2*u3*v3 *a23 + u3 *v3 *a33
= a product of the elements of: {v3,
v3,
v3,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v1 *v3 + u2 *v1 *v3 + u3 *v1 *v3
2 2 2
= a product of the elements of: {v1,v1,v3,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v2 *v3 + u2 *v2 *v3 + u3 *v2 *v3
2 2 2
= a product of the elements of: {v2,v2,v3,u1 + u2 + u3 }
{HAM,FI} = 0
5
FI=v1
= a product of the elements of: {v1,
v1,
v1,
v1,
v1}
{HAM,FI} = 0
4
FI=v1 *v2
= a product of the elements of: {v1,
v1,
v1,
v1,
v2}
{HAM,FI} = 0
4
FI=v1 *v3
= a product of the elements of: {v1,
v1,
v1,
v1,
v3}
{HAM,FI} = 0
3 2
FI=v1 *v2
= a product of the elements of: {v1,
v1,
v1,
v2,
v2}
{HAM,FI} = 0
3
FI=v1 *v2*v3
= a product of the elements of: {v1,
v1,
v1,
v3,
v2}
{HAM,FI} = 0
3 2
FI=v1 *v3
= a product of the elements of: {v1,
v1,
v1,
v3,
v3}
{HAM,FI} = 0
2 3
FI=v1 *v2
= a product of the elements of: {v2,
v2,
v2,
v1,
v1}
{HAM,FI} = 0
2 2
FI=v1 *v2 *v3
= a product of the elements of: {v2,
v2,
v1,
v1,
v3}
{HAM,FI} = 0
2 2
FI=v1 *v2*v3
= a product of the elements of: {v3,
v3,
v1,
v1,
v2}
{HAM,FI} = 0
2 3
FI=v1 *v3
= a product of the elements of: {v3,
v3,
v3,
v1,
v1}
{HAM,FI} = 0
4
FI=v1*v2
= a product of the elements of: {v2,
v2,
v2,
v2,
v1}
{HAM,FI} = 0
3
FI=v1*v2 *v3
= a product of the elements of: {v2,
v2,
v2,
v3,
v1}
{HAM,FI} = 0
2 2
FI=v1*v2 *v3
= a product of the elements of: {v3,
v3,
v2,
v2,
v1}
{HAM,FI} = 0
3
FI=v1*v2*v3
= a product of the elements of: {v3,
v3,
v3,
v2,
v1}
{HAM,FI} = 0
4
FI=v1*v3
= a product of the elements of: {v3,
v3,
v3,
v3,
v1}
{HAM,FI} = 0
5
FI=v2
= a product of the elements of: {v2,
v2,
v2,
v2,
v2}
{HAM,FI} = 0
4
FI=v2 *v3
= a product of the elements of: {v2,
v2,
v2,
v2,
v3}
{HAM,FI} = 0
3 2
FI=v2 *v3
= a product of the elements of: {v2,
v2,
v2,
v3,
v3}
{HAM,FI} = 0
2 3
FI=v2 *v3
= a product of the elements of: {v3,
v3,
v3,
v2,
v2}
{HAM,FI} = 0
4
FI=v2*v3
= a product of the elements of: {v3,
v3,
v3,
v3,
v2}
{HAM,FI} = 0
5
FI=v3
= a product of the elements of: {v3,
v3,
v3,
v3,
v3}
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v1 *v2 + u2 *v1 *v2 + u3 *v1 *v2
2 2 2
= a product of the elements of: {v1,v1,v2,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v1*v2 + u2 *v1*v2 + u3 *v1*v2
2 2 2
= a product of the elements of: {v2,v2,v1,u1 + u2 + u3 }
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33$
FI=u1**2*v1*v2*v3 + u2**2*v1*v2*v3 + u3**2*v1*v2*v3$
FI= - u1**4*v1*a33 - 2*u1**2*u2**2*v1*a33 + 4*u1**2*u2*u3*v1*a23 - u2**4*v1*a33
+ 4*u2**3*u3*v1*a23 + 4*u2*u3**3*v1*a23 + u3**4*v1*a33$
FI= - u1**4*v2*a33 - 2*u1**2*u2**2*v2*a33 + 4*u1**2*u2*u3*v2*a23 - u2**4*v2*a33
+ 4*u2**3*u3*v2*a23 + 4*u2*u3**3*v2*a23 + u3**4*v2*a33$
FI= - u1**4*v3*a33 - 2*u1**2*u2**2*v3*a33 + 4*u1**2*u2*u3*v3*a23 - u2**4*v3*a33
+ 4*u2**3*u3*v3*a23 + 4*u2*u3**3*v3*a23 + u3**4*v3*a33$
FI=2*u1**4*v1*a23**2 + 4*u1**2*u2**2*v1*a23**2 + 4*u1**2*u3**2*v1*a23**2 + 2*u2
**4*v1*a23**2 - 4*u2*u3**3*v1*a23*a33 + u3**4*v1*(2*a23**2 - a33**2)$
FI=2*u1**4*v2*a23**2 + 4*u1**2*u2**2*v2*a23**2 + 4*u1**2*u3**2*v2*a23**2 + 2*u2
**4*v2*a23**2 - 4*u2*u3**3*v2*a23*a33 + u3**4*v2*(2*a23**2 - a33**2)$
FI=2*u1**4*v3*a23**2 + 4*u1**2*u2**2*v3*a23**2 + 4*u1**2*u3**2*v3*a23**2 + 2*u2
**4*v3*a23**2 - 4*u2*u3**3*v3*a23*a33 + u3**4*v3*(2*a23**2 - a33**2)$
FI=u1**2*v1**3 + u2**2*v1**3 + u3**2*v1**3$
FI=u1**2*v1*v3**2 + u2**2*v1*v3**2 + u3**2*v1*v3**2$
FI=u1**2*v2**3 + u2**2*v2**3 + u3**2*v2**3$
FI=u1**2*v2*v3**2 + u2**2*v2*v3**2 + u3**2*v2*v3**2$
FI=u1**2*v3**3 + u2**2*v3**3 + u3**2*v3**3$
FI=4*u2**2*u3**2*v1*a23**2 + 4*u2*u3**3*v1*a23*a33 + u3**4*v1*a33**2$
FI=4*u2**2*u3**2*v2*a23**2 + 4*u2*u3**3*v2*a23*a33 + u3**4*v2*a33**2$
FI=4*u2**2*u3**2*v3*a23**2 + 4*u2*u3**3*v3*a23*a33 + u3**4*v3*a33**2$
FI=2*u2*u3*v1**3*a23 + u3**2*v1**3*a33$
FI=2*u2*u3*v1**2*v2*a23 + u3**2*v1**2*v2*a33$
FI= - u1**2*v1**2*v3*a33 - u2**2*v1**2*v3*a33 + 2*u2*u3*v1**2*v3*a23$
FI=2*u2*u3*v1*v2**2*a23 + u3**2*v1*v2**2*a33$
FI=2*u2*u3*v1*v2*v3*a23 + u3**2*v1*v2*v3*a33$
FI=2*u2*u3*v1*v3**2*a23 + u3**2*v1*v3**2*a33$
FI=2*u2*u3*v2**3*a23 + u3**2*v2**3*a33$
FI= - u1**2*v2**2*v3*a33 - u2**2*v2**2*v3*a33 + 2*u2*u3*v2**2*v3*a23$
FI=2*u2*u3*v2*v3**2*a23 + u3**2*v2*v3**2*a33$
FI=2*u2*u3*v3**3*a23 + u3**2*v3**3*a33$
FI=u1**2*v1**2*v3 + u2**2*v1**2*v3 + u3**2*v1**2*v3$
FI=u1**2*v2**2*v3 + u2**2*v2**2*v3 + u3**2*v2**2*v3$
FI=v1**5$
FI=v1**4*v2$
FI=v1**4*v3$
FI=v1**3*v2**2$
FI=v1**3*v2*v3$
FI=v1**3*v3**2$
FI=v1**2*v2**3$
FI=v1**2*v2**2*v3$
FI=v1**2*v2*v3**2$
FI=v1**2*v3**3$
FI=v1*v2**4$
FI=v1*v2**3*v3$
FI=v1*v2**2*v3**2$
FI=v1*v2*v3**3$
FI=v1*v3**4$
FI=v2**5$
FI=v2**4*v3$
FI=v2**3*v3**2$
FI=v2**2*v3**3$
FI=v2*v3**4$
FI=v3**5$
FI=u1**2*v1**2*v2 + u2**2*v1**2*v2 + u3**2*v1**2*v2$
FI=u1**2*v1*v2**2 + u2**2*v1*v2**2 + u3**2*v1*v2**2$