Solution 3 to problem over
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Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r50=0
r51=0
r52=0
r53=0
r54=0
r55=0
r57=0
r58=2*r56
r59=0
r511=0
r512=0
r513=0
r514=0
r516=0
r518=0
r519=0
r521=0
r522=0
r523=0
r524=0
r525=0
r526=0
r527=0
r528=0
r529=0
1
---*b31*r575
2
r530=--------------
a23
r531=0
1
---*b31*r575
2
r532=--------------
a23
r533=0
r534=0
1
---*b31*r580
2
r535=--------------
a23
r536=0
r537=0
r538=0
r539=0
1
a23*r5200 + ---*a33*r575
2
r540=--------------------------
a23
r541=0
1
a23*r5200 + ---*a33*r575
2
r542=--------------------------
a23
r543=0
r544=0
2 1 1 2 1 2
a23 *r5205 + ---*a23*a33*r580 + ---*b31 *r5109 - ---*b31 *r5214
2 4 4
r545=-----------------------------------------------------------------
2
a23
r546=0
r547=0
r548=0
r549=0
r550=0
1 1 1
---*a23*b31*r5224 + ---*a33*b31*r5109 - ---*a33*b31*r5214
2 2 2
r551=-----------------------------------------------------------
2
a23
r552=0
r553=0
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
r571=0
r572=0
r573=0
r574=0
r576=0
r578=0
r579=0
r581=0
r582=0
r583=0
r584=0
r585=0
b31*r5109 - b31*r5214
r586=-----------------------
a23
r587=0
r588=0
a23*r5224 + a33*r5109 - a33*r5214
r589=-----------------------------------
a23
r590=0
r591=0
r592=0
r593=0
r594=0
r595=r5200
r596=0
r597=r5200
r598=0
r599=0
r5100=r5205
r5101=0
r5102=0
r5103=0
r5104=0
r5105=0
1
---*b31*r5224
2
r5106=---------------
a23
r5107=0
r5108=0
r5110=0
r5111=0
r5112=0
r5113=0
r5114=0
r5115=0
r5116=0
r5117=0
r5118=0
r5119=r5224
r5120=0
r5121=0
r5122=0
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
r5196=0
r5197=0
r5198=0
r5199=0
r5201=0
r5202=r5200
r5203=0
r5204=0
r5206=0
r5207=0
r5208=0
r5209=0
r5210=0
1
---*b31*r5224
2
r5211=---------------
a23
r5212=0
r5213=0
r5215=0
r5216=0
r5217=0
r5218=0
r5219=0
r5220=0
r5221=0
r5222=0
r5223=0
r5225=0
r5226=0
r5227=0
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
r5246=0
r5247=0
1 1
---*a23*r5214 - ---*a33*r5224
2 4
r5248=-------------------------------
a23
r5249=0
r5250=0
r5251=0
c22=0
b33=0
b32=0
r577=r575
r517=r515
r510=r56
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:
r56, r515, r575, r520, r580, r5109, r5200, r5205, r5214,
r5224, b31, a33, c11, 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,c11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + v1 *c11
has apart from the Hamiltonian and Casimirs the following 10 first integrals:
4 2 2 2 2 2
FI= - u1 *v1*a33 - 2*u1 *u2 *v1*a33 + 4*u1 *u2*u3*v1*a23 + 2*u1 *u3*v1 *b31
4 3 2 2 3
- u2 *v1*a33 + 4*u2 *u3*v1*a23 + 2*u2 *u3*v1 *b31 + 4*u2*u3 *v1*a23
4 3 2
+ u3 *v1*a33 + 2*u3 *v1 *b31
= 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 + 2*u3*v1*b31}
{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 2 2 4 2 2
- 4*u2*u3 *v1*a23*a33 - 4*u2*u3 *v1 *a23*b31 + u3 *v1*(2*a23 - a33 )
3 2 2 3 2
- 2*u3 *v1 *a33*b31 - u3 *v1 *b31
= 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
2 4 2 2 3
- 4*u2*u3 *v1*a23*b31 + u3 *(2*a23 - a33 ) - 2*u3 *v1*a33*b31
2 2 2
- u3 *v1 *b31 }
{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 2 2 2 2 2 2
FI=u1 *v1*v2 + u1 *v1*v3 + u2 *v1*v2 + u2 *v1*v3 + u3 *v1*v2 + u3 *v1*v3
2 2 2 2 2
= a product of the elements of: {v2 + v3 ,v1,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 3 2 2 4 2
FI=4*u2 *u3 *v1*a23 + 4*u2*u3 *v1*a23*a33 + 4*u2*u3 *v1 *a23*b31 + u3 *v1*a33
3 2 2 3 2
+ 2*u3 *v1 *a33*b31 + u3 *v1 *b31
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33 + v1*b31,
2*u2*a23 + u3*a33 + v1*b31,
v1}
{HAM,FI} = 0
3 2 3 4
FI=2*u2*u3*v1 *a23 + u3 *v1 *a33 + u3*v1 *b31
= a product of the elements of: {v1,
v1,
v1,
u3,
2*u2*a23 + u3*a33 + v1*b31}
{HAM,FI} = 0
5
FI=v1
= a product of the elements of: {v1,
v1,
v1,
v1,
v1}
{HAM,FI} = 0
2 2 2 2 2 2
FI=2*u2*u3*v1*v2 *a23 + 2*u2*u3*v1*v3 *a23 + u3 *v1*v2 *a33 + u3 *v1*v3 *a33
2 2 2 2
+ u3*v1 *v2 *b31 + u3*v1 *v3 *b31
2 2
= a product of the elements of: {v2 + v3 ,
v1,
u3,
2*u2*a23 + u3*a33 + v1*b31}
{HAM,FI} = 0
3 2 3 2
FI=v1 *v2 + v1 *v3
2 2
= a product of the elements of: {v1,v1,v1,v2 + v3 }
{HAM,FI} = 0
4 2 2 4
FI=v1*v2 + 2*v1*v2 *v3 + v1*v3
2 2 2 2
= a product of the elements of: {v2 + v3 ,v2 + v3 ,v1}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + v1**2*c11$
FI= - u1**4*v1*a33 - 2*u1**2*u2**2*v1*a33 + 4*u1**2*u2*u3*v1*a23 + 2*u1**2*u3*v1
**2*b31 - u2**4*v1*a33 + 4*u2**3*u3*v1*a23 + 2*u2**2*u3*v1**2*b31 + 4*u2*u3**3*
v1*a23 + u3**4*v1*a33 + 2*u3**3*v1**2*b31$
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 - 4*u2*u3**2*v1**2*a23*b31 + u3**4*v1*(2*
a23**2 - a33**2) - 2*u3**3*v1**2*a33*b31 - u3**2*v1**3*b31**2$
FI=u1**2*v1**3 + u2**2*v1**3 + u3**2*v1**3$
FI=u1**2*v1*v2**2 + u1**2*v1*v3**2 + u2**2*v1*v2**2 + u2**2*v1*v3**2 + u3**2*v1*
v2**2 + u3**2*v1*v3**2$
FI=4*u2**2*u3**2*v1*a23**2 + 4*u2*u3**3*v1*a23*a33 + 4*u2*u3**2*v1**2*a23*b31 +
u3**4*v1*a33**2 + 2*u3**3*v1**2*a33*b31 + u3**2*v1**3*b31**2$
FI=2*u2*u3*v1**3*a23 + u3**2*v1**3*a33 + u3*v1**4*b31$
FI=v1**5$
FI=2*u2*u3*v1*v2**2*a23 + 2*u2*u3*v1*v3**2*a23 + u3**2*v1*v2**2*a33 + u3**2*v1*
v3**2*a33 + u3*v1**2*v2**2*b31 + u3*v1**2*v3**2*b31$
FI=v1**3*v2**2 + v1**3*v3**2$
FI=v1*v2**4 + 2*v1*v2**2*v3**2 + v1*v3**4$