STRUC_CONS:={{U1,U2, U3}, {U2,U3, U1}, {U3,U1, U2},
{U1,V2, V3}, {U2,V3, V1}, {U3,V1, V2},
{U1,V3,-V2}, {U2,V1,-V3}, {U3,V2,-V1} }$
The Hamiltonian:
HAM := n1*U1 + n2*U2 + n3*U3 + m1*V1 + m2*V2 + m3*V3 + a11*U1**2 + a22*U2**2 + a33*U3**2 + 2*a12*U1*U2 + 2*a23*U2*U3 + 2*a13*U1*U3 + b11*U1*V1 + b12*U1*V2 + b13*U1*V3 + b21*U2*V1 + b22*U2*V2 + b23*U2*V3 + b31*U3*V1 + b32*U3*V2 + b33*U3*V3 + c11*V1**2 + c22*V2**2 + c33*V3**2 + 2*c12*V1*V2 + 2*c23*V2*V3 + 2*c13*V1*V3 $Casimirs:
J1 = V1**2 + V2**2 + V3**2 J2 = U1*V1 + U2*V2 + U3*V3We consider all possible cases. In the description of each case we use the following notation. NVT is a list of 3 expressions: the first is replaced by the Hamiltonian, the second by the first Casimir, and the third by the second Casimir. INEQU is a list of inequalities, e.g. {a11} would mean that a11 is non-zero and {a11,a22} would mean that either a11 or a22 is non-zero. We compute the general inhomogeneous solution.
For identifying solutions we use the following Hamiltonians:
C111!_D2!_S1!_={2*m3*a33*(a11-a22) - n3*b11*(a11+a22),
2*m2*a33*(a11-a22) - n2*b11*(a11+a33),
2*m1*a33*(a11-a22) - n1*b11*(a22+a33),
4*c33*a33**2*(a11-a22)**2
- b11**2*(a11**2*a33-a11*a22**2-a11*a33**2+a22**2*a33),
a33**2*c22*(a11-a22)*4- b11**2*(a11*a22-a33**2),
a33*b33*(a11-a22)+a11*b11*(a22-a33),
c23,c13,c12,c11,b32,b31,b23,b22,b21,b13,b12,a23,a13,a12},
WOLF!_EFIMOVSKAYA!_S4!_={a11-a22, a33-2*a11, 4*a11*c33+b31**2+b32**2,
a12, a23, a13, b11, b12, b13, b21, b22, b23, b33,
c11, c22, c12, c23, c13,
b31*(a33*m1-n3*b31)+b32*(a33*m2-n3*b32),
n1, n2, m3},
CHAPLYGIN!_WOLF!_EFIMOVSKAJA!_D4!_={a11-a22, a33-2*a11, c11+c22,
a12, a23, a13, b11, b12, b13, b21, b22, b23, b31, b32, b33,
c33, c23, c13, n1, n2, m3},
CHAPLYGIN!_WOLF!_EFIMOVSKAJA!_D3!_={
a11-a22, a33-4*a11, a11*c33+(b31/4)**2+(b32/4)**2,
a12, a23, a13, b11, b12, b13, b21, b22, b23, b33,
c11, c22, c12, c23, c13, n1, n2, m3},
CLEBSCH!_={b11-b22, b11-b33,
a11*a22*(c11-c22)+a33*a11*(c33-c11)+a22*a33*(c22-c33),
a12, a23, a13, b12, b13, b21, b23, b31, b32, c12, c23, c13,
n1, n2, n3, m1, m2, m3},
STEKLOV!_LYAPUNOV!_={a11*a22*(b11-b22)+a33*a11*(b33-b11)+a22*a33*(b22-b33),
a11*a22*c11-a22*(b22-b33)**2-a11*a22*c22+a11*(b33-b11)**2,
a22*a33*c22-a33*(b33-b11)**2-a22*a33*c33+a22*(b11-b22)**2,
a12,a23,a13,b12,b13,b21,b23,b31,b32,c12,c23,c13,
n1, n2, n3, m1, m2, m3}
The following are special cases of the more general solutions above.
ORIGINAL!_LAGRANGE!_={a11-a22,a12,a23,a13,b11,b12,b13,b21,b22,b23,
b31,b32,b33,c11,c22,c33,c12,c23,c13,n1,n2,n3,m1,m2},
NEW!_LAGRANGE!_={a11-a22,a12,a23,a13,b11,b12,b13,b21,b22,b23,
b31,b32, c11,c22, c12,c23,c13,n1,n2, m1,m2},
EULER!_={a12, a23, a13, b11, b12, b13, b21, b22, b23, b31, b32, b33,
c11, c22, c33, c12, c23, c13, n1, n2, n3, m1, m2, m3},
KIRCHHOFF!_={a11-a22, b11-b22, c11-c22, a33, b33, c33,
a12, a23, a13, b12, b13, b21, b23, b31, b32,
c12, c23, c13, n1, n2, n3, m1, m2, m3},
KOWALEWSKI!_={a11-a22, a33-2*a11, a12, a23, a13,
b11, b12, b13, b21, b22, b23, b31, b32, b33,
c11, c22, c33, c12, c23, c13,
n1, n2, n3, m3},
SOKOLOV!_={a11-a22, a33-2*a11, 4*a11*c33+b31**2+b32**2,
a12, a23, a13, b11, b12, b13, b21, b22, b23, b33,
c11, c22, c12, c23, c13,
n1, n2, n3, m1, m2, m3},
WOLF!_EFIMOVSKAYA!_S1!_={a11-a22, a12, a23, a13, b11, b12, b13, b21, b22, b23,
b31, b32, c11, c22, c33, c12, c23, c13,
n1, n2, m1, m2},
WOLF!_EFIMOVSKAYA!_S2!_={a11-a22, a12, a23, a13, b11, b12, b13,
b21, b22, b23, b31, b32, c11, c22, c12, c23, c13,
a22*c33*4+b33**2, a22*(a22-a33)*m1*2+(a22+a33)*b33*n1,
a22*(a22-a33)*m2*2+(a22+a33)*b33*n2,
m3*(a22-a33)+b33*n3},
WOLF!_EFIMOVSKAYA!_S3!_={a11-a22, a33-2*a11, a12, a23, a13, b31**2+b32**2,
n1*b31+n2*b32, m1*b31+m2*b32, a33*m3+2*b32*n2,
b11, b12, b13, b21, b22, b23, b33,
c11, c22, c33, c12, c23, c13, n1, n2, m3},
C111!_D2!_s2!_={c33*a33*(a11-a22) - c22*a22*(a11-a33),
c23,c13,c12,c11,b33,b32,b31,b23,b22,b21,b13,b12,b11,
a23,a13,a12,m3,m2,m1,n3,n2,n1}
Apart from the above solutions also their versions under permutations of indices are used.
Any extra index 1 at the end, like in EULER_1 stands for a cyclic permuted version
1 --> 2 --> 3 --> 1 and any extra index 2 stands for a flip 1 <--> 2.
Not in use yet are the Hamiltonians from the skew symmetric computation of VS and TW which first have to be transformed into the classification below.
SKEW1={c11,c22,c33,c12,c23,c13, % vanishing c-matrix
b11,b22,b33,b21+b12,b31+b13,b32+b23, % skew symmetric b-matrix
a12,b13,b23, % normalizations for real Hamiltonians
a11-a22,a13,a23,
n1,n2,m1,m2
},
SKEW3={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11**2+a13**2+a23**2+b12**2*kap,a11-a22,a11+a33,
m1,m2
},
SKEW4!.2={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11**2+4*a13**2+4*a23**2+b12**2*kap,a11-a22,a33,
m1,m2,a11*(a13*n1+a23*n2)+2*n3*(a13**2+a23**2)
},
SKEW4!.4={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a22-2*a33,2*a11*a22-a22**2+b12**2*kap,a13,a23,
m1,m2,n1,n3
},
SKEW4!.6={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11-2*a33,2*a11*a22-a11**2+b12**2*kap,a13,a23,
m1,m2,n2,n3
},
SKEW4!.5={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11*a22+b12**2*kap,a13,a23,a33,
m1,m2
},
SKEW6!.1={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11**2+16*a13**2+16*a23**2+b12**2*kap,a11-a22,a11-2*a33,
m1,m2,m3,n3,n1*a13+n2*a23 % general inhom extension
},
SKEW6!.2={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11**2+16*a13**2+16*a23**2+b12**2*kap,a11-a22,a11-2*a33,
m1,m2,n3,a13*n2-a23*n1,a23**2+a13**2 % special inhom extension
},
SKEW6!.3={c11,c22,c33,c12,c23,c13,
b11,b22,b33,b21+b12,b31+b13,b32+b23,
a12,b13,b23,
a11**2+16*a13**2+16*a23**2+b12**2*kap,a11-a22,a11-2*a33,
m1,m2,n3,a13,a23 % even more special inhom extension
}
We use the following notation:
Comments
Sometimes equialent Hamiltonians are not automatically recognised as equivalent because the used procedure preduce does not know about inequalities. --> Check by hand necessary.
Partial Summary
Cases 111,112,121,122,123 for degree 1-4:
Distinguished Hamiltonians:
c111_d2_s1 (inhomogeneous generalization of the Steklov-Lyapunov case),
c111_d2_s2 (Clebsch solution which interestingly does not have an
inhomogeneous generalization)
c111_d4_j2_s1,
c121_d4_s1,
c121_d4_s3,
c121_d3_j2_s1,
c121_d4_j2_s4,
c123_d1_s2,
NVT :={U1**2,V1**2,U2*V2}$
INEQU:={{a11},{a22},{a33},{a11-a22},{a22-a33},{a33-a11}}$
a12:=a13:=a23:=0$ % orthogonal rotation
b12:=b13:=b23:=0$ % U --> U + g*V
b22:=c11:=0$ % adding Casimirs to HAM
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: -
Degree 2:
1(1) [= (J2<>0)],
2(1) [= (J2<>0)]
Degree 3: -
Degree 4: 1(1+)
Degree 5:
NVT :={U1**2,V1**2,U2*V2}$
INEQU:={{a11},{a22},{a11-a22}}$
a12:=a13:=a23:=a33:=0$
b12:=b13:=b23:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: -
Degree 2: 1(1) [= degree 2, J2<>0]
Degree 3: -
Degree 4: 1(4+) [= degree 2, J2<>0],
2(4+) [< c111_d2_j2_s2_, c111_d2_s2_, c111_d2_s1_],
3(1+) [< c111_d4_j2_s1_],
4(3+) [< c112_d2_s1_, c111_d2_s1_]
Degree 5:
NVT :={U1**2,V2**2,U2*V2}$
INEQU:={{a11},{a33},{a33-a11}}$
a12:=a13:=a23:=0$ a22:=a11$
b12:=b13:=b23:=0$ b21:=0$
b11:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1(1) [= c121_d1_s1]
Degree 2: 1(1) [= c121_d2_s1],
2(2) [= c121_d1_s1]
Degree 3: 1(1+),
2(5) [= c121_d1_s1]
Degree 4: 1(1+) [= c121_d4_s1, first integrals
differ only by multiple of J2],
2(1+) [CC of c121_d4_s1, first integrals
differ only by multiple of J2],
3(1+) [= c121_d4_s3_, > c121_d4_s6_],
4(1+) [> c121_d4_s6],
5(4+) [= c121_d2_s1, >c121_d4_s5_],
6(4+) [> c121_d4_s5, < c121_d2_s1,
< c111_d2_s1],
7(8) [= c121_d1_s1],
8(3+) [< c121_d2_s1, < c111_d2_s1],
9(3+) [> c121_d4_s5, < c121_d2_s1, < c111_d2_s1]
Degree 5:
NVT :={U1**2,V2**2,U2*V2}$
INEQU:={{a11}}$
a12:=a13:=a23:=0$ a33:=0$ a22:=a11$
b12:=b13:=b23:=0$ b21:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1(1) [= c122_d1_s1]
Degree 2: 1(1) [= c122_d2_s1, > c122_d4_s3],
2(2) [= c122_d1_s1]
Degree 3: 1(5) [= c122_d1_s1]
Degree 4: 1(4+) [= c122_d2_s1, > c122_d4_s3],
2(8) [= c122_d1_s1],
3(3) [= c122_d4_s3]
Degree 5:
NVT :={U3**2,V1**2,U2*V2}$
INEQU:={{a33}}$
a12:=a13:=a23:=0$ a11:=a22:=0$
b31:=b32:=0$ b21:=-b12$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1(1),
2(1),
3(2)
Degree 2: 1(2),
2(1),
3(1),
4(1),
5(3),
6(1),
7(1),
8(3),
9(2),
10(1),
11(2),
12(2),
13(6),
14(4)
Degree 3: 1(1+),
2(1+),
3(5),
4(1+),
5(1+),
6(1),
7(1),
8(6),
9(6),
10(7),
11(7),
12(5),
13(10,
14(16),
15(3+),
16(3+)
Degree 4: 1(4+),
2(4+),
3(4+),
4(4+),
5(4+),
6(4+),
7(10+),
8(10+),
9(4+),
10(4+),
11(32+),
12(1+),
13(2+),
14(2+),
15(1+),
16(2+),
17(2+),
18(8),
19(10),
20(10),
21(1),
22(4+),
23(4+),
24(15+),
25(15+),
26(4+),
27(4+),
28(4+),
29(8+),
30(3+),
31(16),
32(?)
Degree 5:
NVT :={V3**2,V1**2,U2*V2}$
INEQU:={{c11},{c22},{c11-c22}}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c12:=c13:=c23:=c33:=0$
b22:=0$
NVT :={V3**2,V1**2,U2*V2}$
INEQU:={{c11}}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c12:=c13:=c23:=c22:=c33:=0$
b22:=0$
NVT :={V1**2,U2*V2}$
INEQU:={}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c12:=c13:=c23:=c11:=c22:=c33:=0$
b22:=0$
NVT :={V2**2,V1**2,U2*V2}$
INEQU:={{c22},{c23}}$
EQNS:={(c33-c22)**2+4*c23**2}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c12:=c13:=c11:=0$
b22:=0$
NVT :={V3**2,V1**2,U2*V2}$
INEQU:={{c23}}$
EQNS:={c33**2+4*c23**2}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c12:=c13:=c11:=c22:=0$
b22:=0$
NVT :={V1*V3,V1**2,U2*V2}$
INEQU:={{c13}}$
% c12:=i*c13$
EQNS:={c12**2+c13**2}$
a22:=a33:=a11$
a12:=a23:=a13:=0$
c23:=c11:=c22:=c33:=0$
b22:=0$
NVT :={U1**2,V2**2,U2*V2}$
INEQU:={{a11},{a22},{a23}}$
EQNS:={(a33-a22)**2+4*a23**2}$ % a33:=a22-2*i*a23$
a12:=a13:=0$
b12:=b13:=b23:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1
Degree 2: 1,
2,
3,
4,
5,
6,
7
Degree 3:
Degree 4:
Degree 5:
NVT :={U2**2,V3**2,U2*V2}$
INEQU:={{a22},{a23}}$
% a33:=a22-2*i*a23$
EQNS:={(a33-a22)**2+4*a23**2}$ % a33:=a22-2*i*a23$
a11:=a12:=a13:=0$
b23:=b21:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1,
2,
3
Degree 2: 1,
2,
3,
4,
5,
6,
7,
8,
9
Degree 3:
Degree 4:
Degree 5:
NVT :={U1**2,V2**2,U2*V2}$
INEQU:={{a11},{a23}}$
EQNS:={a33**2+4*a23**2}$ % a33:=-2*i*a23$
a12:=a13:=a22:=0$
b12:=b32:=b13:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: 1
Degree 2: 1,
2,
3,
4
Degree 3:
Degree 4:
Degree 5:
NVT :={U3**2,V2**2,U2*V2}$
INEQU:={{a23}}$
% a33:=-2*i*a23$
EQNS:={a33**2+4*a23**2}$
a11:=a12:=a13:=a22:=0$
b21:=b31:=b32:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: -
Degree 2: 1
Degree 3: 1,
2
Degree 4: 1,
2,
3
Degree 5:
NVT :={U3**2,V2**2,U2*V2}$
INEQU:={{a11},{a13}}$
% a13:=i*a12$
EQNS:={a12**2+a13**2}$
a11:=a22:=a33$ a23:=0$
b21:=b31:=b32:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: -
Degree 2: 1
Degree 3: 1,
2
Degree 4: 1,
2
Degree 5:
NVT :={U1*U3,V2**2,U2*V2}$
INEQU:={{a13}}$
% a13:=i*a12$
EQNS:={a12**2+a13**2}$
a11:=a22:=a33:=a23:=0$
b32:=b23:=b12:=0$
b22:=c11:=0$
J1 <> 0 <> J2
J1 <> 0 = J2
Degree 1: -
Degree 2: 1,
2,
3
Degree 3: 1,
2,
3,
4,
5
Degree 4: 1,
2,
3,
4,
5
Degree 5: