Solution 5 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r50=0
r51=r519
r52=0
r53=r514
r54=0
r56=0
r57=0
r58=0
r59=0
r510=0
r511=0
r512=2*r519
r513=0
r515=0
r516=0
r517=0
r518=0
r520=0
r521=0
r522=0
1
---*b32*r579
2
r523=--------------
a23
r524=0
1
---*b32*r574
2
r525=--------------
a23
r526=0
r527=0
r528=0
r529=0
r530=0
r531=0
1
---*b32*r579
2
r532=--------------
a23
r533=0
r534=0
r535=0
r536=0
1
a23*r5197 + ---*a33*r579
2
r537=--------------------------
a23
r538=0
2 1 1 2 1 2
a23 *r5199 + ---*a23*a33*r574 + ---*b32 *r5108 - ---*b32 *r5213
2 4 4
r539=-----------------------------------------------------------------
2
a23
r540=0
r541=0
r542=0
r543=0
1
a23*r5197 + ---*a33*r579
2
r544=--------------------------
a23
r545=0
r546=0
r547=0
1 1 1
---*a23*b32*r5223 + ---*a33*b32*r5108 - ---*a33*b32*r5213
2 2 2
r548=-----------------------------------------------------------
2
a23
r549=0
r550=0
r551=0
r552=0
1 2 1 1 2 1 2
---*a23 *r5213 + ---*a23*a33*r5223 + ---*a33 *r5108 - ---*a33 *r5213
2 4 4 4
r553=----------------------------------------------------------------------
2
a23
r554=0
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=r579
r573=0
r575=0
r576=0
r577=0
r578=0
r580=0
r581=0
r582=0
b32*r5108 - b32*r5213
r583=-----------------------
a23
r584=0
r585=0
r586=0
r587=0
a23*r5223 + a33*r5108 - a33*r5213
r588=-----------------------------------
a23
r589=0
r590=0
r591=0
r592=r5197
r593=0
r594=r5199
r595=0
r596=0
r597=0
r598=0
r599=r5197
r5100=0
r5101=0
r5102=0
1
---*b32*r5223
2
r5103=---------------
a23
r5104=0
r5105=0
r5106=0
r5107=0
r5109=0
r5110=0
r5111=0
r5112=0
r5113=0
r5114=0
r5115=0
r5116=0
r5117=0
r5118=r5223
r5119=0
r5120=0
r5121=0
1 1
---*a23*r5213 - ---*a33*r5223
2 4
r5122=-------------------------------
a23
r5123=0
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
r5198=0
r5200=0
r5201=0
r5202=0
r5203=0
r5204=r5197
r5205=0
r5206=0
r5207=0
1
---*b32*r5223
2
r5208=---------------
a23
r5209=0
r5210=0
r5211=0
r5212=0
r5214=0
r5215=0
r5216=0
r5217=0
r5218=0
r5219=0
r5220=0
r5221=0
r5222=0
r5224=0
r5225=0
r5226=0
1
a23*r5213 - ---*a33*r5223
2
r5227=---------------------------
a23
r5228=0
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
1 1
---*a23*r5213 - ---*a33*r5223
2 4
r5247=-------------------------------
a23
r5248=0
r5249=0
r5250=0
r5251=0
c11=0
b33=0
b31=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:
r55, r514, r519, r574, r579, r5108, r5197, r5199, r5213,
r5223, b32, a33, c22, 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.
{b32,a33,c22,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v2*b32 + v2 *c22
has apart from the Hamiltonian and Casimirs the following 10 first integrals:
4 2 2 2 2 2
FI= - u1 *v2*a33 - 2*u1 *u2 *v2*a33 + 4*u1 *u2*u3*v2*a23 + 2*u1 *u3*v2 *b32
4 3 2 2 3
- u2 *v2*a33 + 4*u2 *u3*v2*a23 + 2*u2 *u3*v2 *b32 + 4*u2*u3 *v2*a23
4 3 2
+ u3 *v2*a33 + 2*u3 *v2 *b32
= 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 + 2*u3*v2*b32}
{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 2 2 4 2 2
- 4*u2*u3 *v2*a23*a33 - 4*u2*u3 *v2 *a23*b32 + u3 *v2*(2*a23 - a33 )
3 2 2 3 2
- 2*u3 *v2 *a33*b32 - u3 *v2 *b32
= 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
2 4 2 2 3
- 4*u2*u3 *v2*a23*b32 + u3 *(2*a23 - a33 ) - 2*u3 *v2*a33*b32
2 2 2
- u3 *v2 *b32 }
{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 2 2 2 2 2 2
FI=u1 *v1 *v2 + u1 *v2*v3 + u2 *v1 *v2 + u2 *v2*v3 + u3 *v1 *v2 + u3 *v2*v3
2 2 2 2 2
= a product of the elements of: {v2,v1 + v3 ,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 3 2 2 4 2
FI=4*u2 *u3 *v2*a23 + 4*u2*u3 *v2*a23*a33 + 4*u2*u3 *v2 *a23*b32 + u3 *v2*a33
3 2 2 3 2
+ 2*u3 *v2 *a33*b32 + u3 *v2 *b32
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33 + v2*b32,
2*u2*a23 + u3*a33 + v2*b32,
v2}
{HAM,FI} = 0
2 2 2 2 2 2
FI=2*u2*u3*v1 *v2*a23 + 2*u2*u3*v2*v3 *a23 + u3 *v1 *v2*a33 + u3 *v2*v3 *a33
2 2 2 2
+ u3*v1 *v2 *b32 + u3*v2 *v3 *b32
= a product of the elements of: {v2,
2 2
v1 + v3 ,
u3,
2*u2*a23 + u3*a33 + v2*b32}
{HAM,FI} = 0
3 2 3 4
FI=2*u2*u3*v2 *a23 + u3 *v2 *a33 + u3*v2 *b32
= a product of the elements of: {v2,
v2,
v2,
u3,
2*u2*a23 + u3*a33 + v2*b32}
{HAM,FI} = 0
4 2 2 4
FI=v1 *v2 + 2*v1 *v2*v3 + v2*v3
2 2 2 2
= a product of the elements of: {v1 + v3 ,v1 + v3 ,v2}
{HAM,FI} = 0
2 3 3 2
FI=v1 *v2 + v2 *v3
2 2
= a product of the elements of: {v2,v2,v2,v1 + v3 }
{HAM,FI} = 0
5
FI=v2
= a product of the elements of: {v2,
v2,
v2,
v2,
v2}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v2*b32 + v2**2*c22$
FI= - u1**4*v2*a33 - 2*u1**2*u2**2*v2*a33 + 4*u1**2*u2*u3*v2*a23 + 2*u1**2*u3*v2
**2*b32 - u2**4*v2*a33 + 4*u2**3*u3*v2*a23 + 2*u2**2*u3*v2**2*b32 + 4*u2*u3**3*
v2*a23 + u3**4*v2*a33 + 2*u3**3*v2**2*b32$
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 - 4*u2*u3**2*v2**2*a23*b32 + u3**4*v2*(2*
a23**2 - a33**2) - 2*u3**3*v2**2*a33*b32 - u3**2*v2**3*b32**2$
FI=u1**2*v2**3 + u2**2*v2**3 + u3**2*v2**3$
FI=u1**2*v1**2*v2 + u1**2*v2*v3**2 + u2**2*v1**2*v2 + u2**2*v2*v3**2 + u3**2*v1
**2*v2 + u3**2*v2*v3**2$
FI=4*u2**2*u3**2*v2*a23**2 + 4*u2*u3**3*v2*a23*a33 + 4*u2*u3**2*v2**2*a23*b32 +
u3**4*v2*a33**2 + 2*u3**3*v2**2*a33*b32 + u3**2*v2**3*b32**2$
FI=2*u2*u3*v1**2*v2*a23 + 2*u2*u3*v2*v3**2*a23 + u3**2*v1**2*v2*a33 + u3**2*v2*
v3**2*a33 + u3*v1**2*v2**2*b32 + u3*v2**2*v3**2*b32$
FI=2*u2*u3*v2**3*a23 + u3**2*v2**3*a33 + u3*v2**4*b32$
FI=v1**4*v2 + 2*v1**2*v2*v3**2 + v2*v3**4$
FI=v1**2*v2**3 + v2**3*v3**2$
FI=v2**5$