Classification of integrable quadratic Hamiltonians on e(3)

Thomas Wolf

Department of Mathematics, Brock University

500 Glenridge Avenue, St.Catharines,

Ontario, Canada L2S 3A1 twolf@brocku.ca

and

Olya V. Efimovskaya

Department of Mechanics and Mathematics,

Moscow State University

Vorob’evy gory, Moscow, 119992, Russia

efvitaly@ttk.ru

January 29, 2003

1 Introduction.

In this work we consider quadratic Hamiltonians of the form

H = (M, AM) + (M, Bγ) + (γ, Cγ) + (P, M) + (Q, γ).  (1)

where M = (M_{1}, M_{2}, M_{3}), γ = (γ_{1}, γ_{2}, γ_{3}) and A, C are constant symmetric matrices, B is a general constant matrix and P, Q are constant vectors. Without loss of generality we choose

A = diag(a_{1}, a_{2}, a_{3}).  (2)

The equations of motion are given by

\frac{dM_{i}}{dt} = \{H, M_{i}\}, \frac{dγ_{i}}{dt} = \{H, γ_{i}\}

where Poisson brackets are defined by

{M_{i}, M_{j}} = ε_{ijk} M_{k},  {M_{i}, γ_{j}} = ε_{ijk} γ_{k},  {γ_{i}, γ_{j}} = 0  (3)

with ε_{ijk} being the totally anti-symmetric tensor. These linear Poisson brackets are related to the Lie algebra e(3).

Because they admit the two Casimir functions

J_{1} = (γ, γ), J_{2} = (γ, M)  (4)

we need only one additional first integral for Liouville integrability [1]. In our paper we restrict ourselves to polynomial first integrals. The relevance of the combination of quadratic Hamiltonians (1) with Poisson brackets (3) arises from the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and from the Kirchhoff model describing the motion of a rigid body in ideal fluid [2, 3, 4].

In section 2 we give a list of known integrable quadratic Hamiltonians with additional polynomial integrals of motion. Results of our search are given in section 3.

It is known from literature that for pairwise non-equal a_{1}, a_{2}, a_{3} in (2) no extra integrable cases besides those mentioned in section 2 exist. In addition all Hamiltonians (1) with linear or quadratic first integral are known too. Hamiltonians with cubic first integral are discussed in [7]. We therefore study first integrals I of fourth degree and consider the case a_{1} = a_{2} ≠ a_{3}.

The motivation for a systematic investigation of this ansatz arose from the recent finding of V. Sokolov of a new integrable Hamiltonian of this form with a fourth degree first integral. Our investigation is in so far more general as we consider quadratic Hamiltonians with linear terms. Our complete classification of this problem resulted in new integrable Hamiltonians with complex coefficients. The simplest example of such a kind is

H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + iM_{1} + M_{2}

which commutes with some fourth degree polynomial under so(3)-brackets.

We also were inspired by the Goryachev-Chaplygin Hamiltonian [5, 6] which admits a fourth degree integral exclusively on the Casimir level J_{2} = 0. Unfortunately our complete classification did not result in new integrable cases.

In section 4 we follow [17, 18] and consider the quantum counterpart

[M_{i}, M_{j}] = ε_{ijk} M_{k},  [M_{i}, γ_{j}] = ε_{ijk} γ_{k},  [γ_{i}, γ_{j}] = 0  (5)

of the Poisson bracket (3). Here M_{i}, γ_{j} are elements of an associative algebra with commutator relations (5). The Hamiltonian is a (non-commutative) polynomial of second degree and first integrals are polynomials which commute with the Hamiltonian. Also here we classify quadratic Hamiltonians with linear terms having a fourth degree first integral and obeying the restriction a_{1} = a_{2} ≠ a_{3}. We find four cases, among them the quantum analogue of the above mentioned integrable Hamiltonian of V. Sokolov.

In all three investigations an ansatz for the Hamiltonian and an ansatz for a fourth degree polynomial were made, the Poisson bracket or commutator were computed and set to zero, resulting in large non-linear over-determined algebraic systems. Although they are linear in the coefficients of H and linear in the coefficients of the first integral, the algebraic systems are still very large (see table 1) and challenging to solve.

We used the computer algebra program Crack which was originally designed to solve over-determined PDE-systems. But its interactive capabilities allowed to solve large algebraic systems at first interactively and in doing so to learn how to take advantage of the bi-linearity. Due to an ongoing effort to generalize gathered experience and to incorporate it into the program it is now able to solve large bi-linear systems, for example the first and third system in table 1 automatically and the second system with only few manual interactions. An overview of essential features of the program and other applications requiring the solution of bi-linear systems is given in [7].

type of problem  e(3)  e(3), J_{2} = 0 e(3) quant.
# of unknowns (H,I,total) 17,200,217 17,176,193 17,200,217
# of equations  451  396  451
total # of terms  5469  5243  9681
average # of terms/equ.  12.1  13.2  21.5
time to solve  18h 53min  ≈ 15h  11h 43min
details in section  3.1  3.2  4

Table 1. An overview of the solved algebraic systems

Times are measured on a 1.7GHz Pentium 4 running a 120 MByte REDUCE session under Linux.

2 Known integrable Hamiltonian on e(3)

In the following we list all known integrable Hamiltonian of type (1) on e(3). First let us consider all cases where the matrices A = {a_{ij}}, B = {b_{ij}} and C = {c_{ij}} are diagonal, i.e. the Hamiltonian takes the form:

\begin{matrix}H = a_{1}M_{1}^{2} + a_{2}M_{2}^{2} + a_{3}M_{3}^{2} + 2b_{1}M_{1}γ_{1} + 2b_{2}M_{2}γ_{2} + 2b_{3}M_{3}γ_{3}+ \\ c_{1}γ_{1}^{2} + c_{2}γ_{2}^{2} + c_{3}γ_{3}^{2} + p_{1}M_{1} + p_{2}M_{2} + p_{3}M_{3} + q_{1}γ_{1} + q_{2}γ_{2} + q_{3}γ_{3}. \end{matrix} (6)

Kirchhoff’s problem of the motion of a rigid body in ideal fluid. In this case the Hamiltonian does not contain linear terms, i.e. we have p_{i} = q_{i} = 0. For this problem there are three known classical integrable cases of Kirchhoff [8], Clebsch [9] and Steklov-Lyapunov [10, 11].

The Kirchhoff case is defined by the identities

a_{1} = a_{2},  b_{1} = b_{2},  c_{1} = c_{2}.

The additional integral I is linear: I = M_{3}.

For the Clebsch case the coefficients a_{i} are arbitrary and the remaining parameters satisfy the following conditions:

b_{1} = b_{2} = b_{3},

\frac{c_{1} − c_{2}}{a_{3}} + \frac{c_{3} − c_{1}}{a_{2}} + \frac{c_{2} − c_{3}}{a_{1}} = 0.

If not all a_{i} are equal then the Hamiltonian can be represented in the form

H = a_{1}M_{1}^{2} + a_{2}M_{2}^{2} + a_{3}M_{3}^{2} + a_{2}a_{3}γ_{1}^{2} + a_{3}a_{1}γ_{2}^{2} + a_{1}a_{2}γ_{3}^{2}.

The additional quadratic integral admitted by H is

I = M_{1}^{2} + M_{2}^{2} + M_{3}^{2} + (a_{2} + a_{3})γ_{1}^{2} + (a_{3} + a_{1})γ_{2}^{2} + (a_{1} + a_{2})γ_{3}^{2}.

If a_{1} = a_{2} = a_{3} then we get the Neumann Hamiltonian

H = M_{1}^{2} + M_{2}^{2} + M_{3}^{2} + c_{1}γ_{1}^{2} + c_{2}γ_{2}^{2} + c_{3}γ_{3}^{2}.

The additional integral in this case coincides with H.

For the Steklov-Lyapunov case the coefficients a_{i} are arbitrary and the remaining parameters satisfy the following conditions:

\frac{b_{1} − b_{2}}{a_{3}} + \frac{b_{3} − b_{1}}{a_{2}} + \frac{b_{2} − b_{3}}{a_{1}} = 0,

c_{1} − \frac{(b_{2} − b_{3})^{2}}{a_{1}} = c_{2} − \frac{(b_{3} − b_{1})^{2}}{a_{2}} = c_{3} − \frac{(b_{1} − b_{2})^{2}}{a_{3}} .

If not all the a_{i} are equal then the Hamiltonian can be represented in the form

\begin{matrix}H = a_{1}M_{1}^{2} + a_{2}M_{2}^{2} + a_{3}M_{3}^{2} + 2a_{2}a_{3}M_{1}γ_{1} + 2a_{3}a_{1}M_{2}γ_{2} + 2a_{1}a_{2}M_{3}γ_{3} \\ +a_{1}(a_{2} − a_{3})^{2}γ_{1}^{2} + a_{2}(a_{3} − a_{1})^{2}γ_{2}^{2} + a_{3}(a_{1} − a_{2})^{2}γ_{3}^{2}. \end{matrix}

The additional integral is quadratic:

\begin{matrix}I = M_{1}^{2} + M_{2}^{2} + M_{3}^{2} + 2(a_{2} + a_{3})M_{1}γ_{1} + 2(a_{3} + a_{1})M_{2}γ_{2} + 2(a_{1} + a_{2})M_{3}γ_{3} \\ +(a_{2} − a_{3})^{2}γ_{1}^{2} + (a_{3} − a_{1})^{2}γ_{2}^{2} + (a_{1} − a_{2})^{2}γ_{3}^{2}. \end{matrix}

If a_{1} = a_{2} = a_{3} we have to interchange the Hamiltonian H and the integral I just as in the Clebsch case.

Recently in the paper [12] by Sokolov a Hamiltonian with non-diagonal matrix B having an integral of fourth degree was presented. One of the possible forms of this Hamiltonian is:

H = M_{1}^{2} + M_{2}^{2} + 2 M_{3}^{2} + 2 (a_{1}γ_{1} + a_{2}γ_{2}) M_{3} − (a_{1}^{2} + a_{2}^{2}) γ_{3}^{2}.  (7)

The integral can be represented as a product I = k_{1} k_{2}, where k_{1} = M_{3} and

k_{2} = (M_{1}^{2} + M_{2}^{2} + M_{2}^{2}) M_{3} + 2 (a_{1}M_{1} + a_{2}M_{2}) (M_{1}γ_{1} + M_{2}γ_{2})
+2 (a_{1}γ_{1} + a_{2}γ_{2}) M_{3}^{2} + (a_{1}γ_{1} + a_{2}γ_{2})^{2} M_{3}
−(a_{1}^{2} + a_{2}^{2}) (2M_{1}γ_{1} + 2M_{2}γ_{2} + M_{3}γ_{3}) γ_{3}.

This case appears to be similar in its properties to the Kowalewski case given below.

The problem of motion of a rigid body around a fixed point. Hamiltonians describing such situations have the form (6), where b_{i} = c_{i} = p_{i} = 0. The following integrable cases are known.

The Lagrange case:

H = M_{1}^{2} + M_{2}^{2} + a_{3}M_{3}^{2} + q_{3}γ_{3}  (8)

where a_{3}, q_{3} are arbitrary. The additional integral I is linear: I = M_{3}.

The Euler case:

H = a_{1}M_{1}^{2} + a_{2}M_{2}^{2} + a_{3}M_{3}^{2}  (9)

with an additional quadratic integral:

I = M_{1}^{2} + M_{2}^{2} + M_{3}^{2}.

The Kowalewski case:

H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + q_{1}γ_{1} + q_{2}γ_{2}  (10)

with arbitrary parameters q_{1}, q_{2}. The additional integral of fourth degree can be written as I = G_{1}^{2} + G_{2}^{2}, where

G_{1} = M_{1}^{2} − M_{2}^{2} − q_{1}γ_{1} + q_{2}γ_{2}, G_{2} = 2M_{1}M_{2} − q_{2}γ_{1} − q_{1}γ_{2}.

We note, that if q_{2} ≠ iq_{1}, then the transformation M → T M, Γ → T Γ, where

T = ( \begin{matrix}cos ϕ & sin ϕ & 0 \\ − sin ϕ & cos ϕ & 0 \\ 0 & 0 & 1 \end{matrix} )  (11)

can be used to make q_{2} to zero.

Generalizations. Terms linear in the moments M_{i} that occur in the Hamiltonian could be interpreted as an action of hydrostatic forces ([4]). We have no comments on the physical meaning of other mixed Hamiltonians (1) (i.e. Hamiltonians having both B ≠ 0 or C ≠ 0, and Q ≠ 0 ).

For example, an obvious hybrid of Lagrange’s and Kirchhoff’s Hamiltonians (with additional hydrostatic member) is

H = M_{1}^{2} + M_{2}^{2} + s_{1}M_{3}^{2} + s_{2}γ_{3}M_{3} + s_{3}γ_{3}^{2} + s_{4}M_{3} + s_{5}γ_{3},  (12)

3 Results

3.1 The classical case.

For the Hamiltonian (1) on e(3) we consider the case of A being diagonal:

A = diag(a_{1}, a_{2}, a_{3}),

where

a_{1} = a_{2} ≠ a_{3},  a_{i} ≠ 0,  i = 1, 2, 3.

Without loss of generality matrix B can be considered as low-triangular, i.e.

b_{12} = b_{13} = b_{23} = 0.

Note, that the addition of Casimir functions (4) to the Hamiltonian does not influence the equation of motion. By subtracting an appropriate linear combination of J_{1} and J_{2} we can make b_{11} = c_{11} = 0 and get

H = M_{1}^{2} + M_{2}^{2} + a_{3}M_{3}^{2} +
b_{21}γ_{1}M_{2} + b_{31}γ_{1}M_{3} + b_{32}γ_{2}M_{3} + b_{22}γ_{2}M_{2} + b_{33}γ_{3}M_{3} +
c_{12}γ_{1}γ_{2} + c_{13}γ_{1}γ_{3} + c_{22}γ_{2}^{2} + c_{23}γ_{2}γ_{3} + c_{33}γ_{3}^{2} +  (13)
p_{1}M_{1} + p_{2}M_{2} + p_{3}M_{3} + q_{1}γ_{1} + q_{2}γ_{2} + q_{3}γ_{3}.

Lemma. For a Hamiltonian of type (13), a canonical transformation can be used to obtain b_{21} = 0.

Thus we will consider only the following Hamiltonian:

H = M_{1}^{2} + M_{2}^{2} + a_{3}M_{3}^{2} + b_{31}γ_{1}M_{3} + b_{32}γ_{2}M_{3} + b_{22}γ_{2}M_{2} + b_{33}γ_{3}M_{3} +
c_{12}γ_{1}γ_{2} + c_{13}γ_{1}γ_{3} + c_{22}γ_{2}^{2} + c_{23}γ_{2}γ_{3} + c_{33}γ_{3}^{2} +  (14)
p_{1}M_{1} + p_{2}M_{2} + p_{3}M_{3} + q_{1}γ_{1} + q_{2}γ_{2} + q_{3}γ_{3}.

Theorem 1.The Hamiltonian of the above form commutes with a polynomial integral of 4^{th} degree iff it coincides with one of the following:

• (12), where s_{i} - are arbitrary, or

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + M_{3}^{2} + 2s_{1}γ_{3}M_{3} − s_{1}^{2}γ_{3}^{2} + s_{2}γ_{1} + s_{3}γ_{2} + s_{4}γ_{3}+ \\ λ (2s_{1}M_{3}^{2} + s_{2}M_{1} + s_{3}M_{2} + s_{4}M_{3} + s_{1}(s_{2}γ_{1} + s_{3}γ_{2})), \end{matrix} (15)

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + s_{1}(iγ_{1} + γ_{2})M_{3}+ \\ s_{2}(−iM_{1} + M_{2}) + s_{3}M_{3} + s_{4}(iγ_{1} + γ_{2}) − s_{1}s_{2}γ_{3}, \end{matrix}  (16)

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + 2(s_{1}γ_{1} + s_{2}γ_{2}) M_{3} − (s_{1}^{2} + s_{2}^{2})γ_{3}^{2}+ \\ s_{3} (M_{3} + s_{1}γ_{1} + s_{2}γ_{2}) + s_{4}γ_{1} + s_{5}γ_{2}, \end{matrix} (17)

where s_{i} are parameters constrained only by the condition:

s_{2}s_{5} + s_{1}s_{4} = 0.

Remarks

1. The case (15) was founded by Rubanovskiyi [13].

2. Under the conditions s_{1} = s_{2} = s_{3} = 0 the formula (17) describes the Kowalewski case. If s_{3} = s_{4} = s_{5} = 0 we have the Sokolov case for which the general formula (17) was presented in [14] and the Lax pair was found by Sokolov and A.V. Tsiganov [15].

3. The Hamiltonian (16) is probably new.

3.2 Classical case with additional condition.

We now consider the case of the area integral J_{2} being equal to zero. In this case the following additional integrable Hamiltonians are known.

The Goryachev-Chaplygin Hamiltonian [5, 6]

H = M_{1}^{2} + M_{2}^{2} + 4M_{3}^{2} − s_{1}γ_{1}  (18)

belongs to the class of Hamiltonians describing the motion of a rigid body around a fixed point. The additional integral is of third degree.

For the Kirchhoff problem is known the Hamiltonian by Chaplygin:

H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + cγ_{1}^{2} − cγ_{2}^{2}.  (19)

Here the additional integral has fourth degree.

Theorem 2. The Hamiltonian of the form (14) commutes with a polynomial integral of fourth degree with additional condition (M, γ) = 0, iff it coincides with one of the Hamiltonian from Theorem 1 or with the following:

•

H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + s_{1}(γ_{1}^{2} − γ_{2}^{2}) + s_{2}γ_{1}γ_{2} + s_{3}M_{3} + s_{4}γ_{1} + s_{5}γ_{2},

(20)

where s_{i} - are arbitrary parameters.

The Hamiltonian (20) is the generalization of Chaplygin’s case [5, 6, 16].

4 Quantum case.

In the papers by Komarov, Sklyanin [17, 18, 19] and others quantum generalizations of classical Hamiltonians on e(3) and so(4) were considered. Similarly, instead of the Poisson bracket (3) we investigate the commutator relations (5) in an associative algebra with generators M_{1}, M_{2}, M_{3}, γ_{1}, γ_{2}, γ_{3}. We can consider M_{i}, γ_{i} to be operators. Due to commutator relations (5) any monomial always could be ordered such, that M_{i} come before any γ_{i} and indices increase (such a presentation is unique). Thus Hamiltonians have the same form (1) as in the classical case, but the multiplication is now noncommutative. In this case, any element I from the associative algebra that satisfies [H, I] = 0 is called an integral.

Theorem 3. The Hamiltonian of the above form commutes with a polynomial integral of 4^{th} degree iff it coincides with one of the following:

•

H = M_{1}^{2} + M_{2}^{2} + s_{1}M_{3}^{2} + s_{2}γ_{3}M_{3} + s_{3}γ_{3}^{2} + s_{4}M_{3} + s_{5}γ_{3},  (21)

where s_{i} - are arbitrary parameters;

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + M_{3}^{2} + 2s_{1}γ_{3}M_{3} − s_{1}^{2}γ_{3}^{2} + s_{2}γ_{1} + s_{3}γ_{2} + s_{4}γ_{3}+ \\ λ (2s_{1}M_{3}^{2} + s_{2}M_{1} + s_{3}M_{2} + s_{4}M_{3} + s_{1}(s_{2}γ_{1} + s_{3}γ_{2})), \end{matrix} (22)

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + s_{1}(iγ_{1} + γ_{2})M_{3}+ \\ s_{2}(−iM_{1} + M_{2}) + s_{3}M_{3} + s_{4}(iγ_{1} + γ_{2}) − s_{1}s_{2}γ_{3}, \end{matrix}  (23)

•

\begin{matrix}H = M_{1}^{2} + M_{2}^{2} + 2M_{3}^{2} + 2(s_{1}γ_{1} + s_{2}γ_{2}) M_{3} − (s_{1}^{2} + s_{2}^{2})γ_{3}^{2}+ \\ s_{3} (M_{3} + s_{1}γ_{1} + s_{2}γ_{2}) + s_{4}γ_{1} + s_{5}γ_{2}, \end{matrix} (24)

where s_{i} are parameters constrained only by the condition:

s_{2}s_{5} + s_{1}s_{4} = 0;

Notice that this list looks exactly as the list of subsection 3.1 although a priori this coincidence is not predictable. The most interesting here is Hamiltonian (24), which is a quantum generalization of the Sokolov Hamiltonian. If s_{1} = s_{2} = s_{3} = 0 we have the quantum Kowalewski Hamiltonian [17, 20].

Acknowledgements

The authors are grateful to V.V. Sokolov for useful discussions. The research was partially supported by RFBR grant 02-01-06670.

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