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The Three Body Toda System and A Parametrization of Plane Cubic Curves
1.A. BARROSO editor, Aspects of Mathematics and its Applications © Elsevier Science Publishers B.V. (1986)
409
THE THREE BODY TODA SYSTEM AND A PARAMETRIZATION OF PLANE CUBIC CURVES M. KAC 1t Department ofMathematics, University of Southern California, Los Angeles, California, U.S.A.
To Leopoldo Nachbin on the occasion of his sixtieth birthday
§1. The three body Toda system is a linear chain of three oscillators with the Hamiltonian given (in conveniently chosen units) by the formula (1.1)
H = 2(pi + Pz +
pD + exp(-(ql -
qz»
+ exp«q2 - q3» + exp(-(q) - qt» . Following Flaschka [1] we introduce the variables (1.2)
(k = 1,2,3),
with the understanding that q4 = qt. In these variables the equations of motion become: (1.3a)
(1.3b) and we shall further set (1.4) 1 The research of the author was supported in part by the N.S.F. under contract MCS 8101739. t Deceased.
M. Kac / Three Body Toda System
410
Because of (1.2) we have the constraint (1.5)
Consider now the matrix
(1.6)
By a general theorem of F1aschka the eigenvalues, and hence the traces, are constants of motion, the first two Tr A 3 and Tr A; are the momentum and the energy. We may as well set the total momentum to be 0, i.e. (1.7a) and write the conservation of energy in the form (1.7b) We also have
=
const ,
where 6 would have been 6X1X2X3 if it were not for the constraint (1.5). It should be added that the constancy (in time) of X 1X2X3 = aia~a; is consistent with the equation of motion (1.3b).
§2.
Since
XI'
x2 and x3 are positive we have
(2.1)
the equality occurring only when
XI
= x 2 = X 3 = 1. Thus
M. Kac I Three Body Toda System
and if E uniquely
= 6, b, = b2 = b3 = O.
Thus if E
=6
411
the matrix A 3 is determined
(2.2)
This is trivial from the point of view of mechanics for it merely states that a system in the state of minimum total energy remains in that state forever.
§3. If E > 6 we can try to extremize Tr A~ under the constraints (1.5), (1.7a) and (1.7b). This leads to a somewhat tedious calculation involving Lagrange multipliers, which for convenience I shall take to be .A (for (1.7a», ~JL (for (1.7b» and 3v for (1.5). Thus we have to extremize
(3.1)
bi + b~ + b~ + 3x t(b t + b2 ) + 3x 2(b2 + b3 ) + 3x 3(b3 + b t )
+ .A (b, + b2 + b3 ) + ~JL (bi + b~ + b; + 2x t + 2x 2 + 2x 3 )
+ 3v(x tx 2x3-1), and I will try to save a skeptical rear some needless labour by sketching the derivation in some detail. By differentiation of (3.1) with respect to the x's we get (3.2a) (3.2b) (3.2c)
so that (using b, + b2 + b3 = 0) we get (3.3a)
(3.3b)
412
M. Kac / Three Body Toda System
(3.3e) and (3.4)
Differentiating (3.1) with respect to the b's we obtain (3.5a) (3.5b) (3.5e) and (again using b, + bz + b3 = 0) we get (3.6)
Substituting the expressions (3.3) for the b's into (3.5a) and (3.6) we obtain (3.7) (p, + ZlX IX3)Z + (Xl + x 3 ) + JL z+ JLZlX IX3
- M(p, + ZlXIX3)Z + (p, + ZlXIXz)Z + (p, + Zlxzx3i + 2(xI + Xz + x 3) ] = 0 . The expression in square brackets in (3.7) is equal to 3JL z + 2Z1JL (XIX3 + XIXZ+ XZx3) + l/(xi x~ + =
6JLz- 2(1- l/)(X 1+ x z + x 3)
xi X~ + x~x~) + 2(x I + Xz + X3)
,
and first three terms in (3.7) add up to
In deriving these formulas repeated use has been made of (3.4), (1.5) and of the identity
which in tum follows from (1.5).
M. Kac I Three Body TOlla System
413
Finally (3.7) becomes
Similarly one obtains (3.8b) and
If ,i ¥ 1 equations (3.8) imply that XI = X 2 = X 3 = 1 and hence (3.3), (3.4) and (1.7a) yield hi = b2 = b3 = O. Thus E = 6 contrary to the assumption E>6.
§4. Having established that Ii = 1 we go back to the formula (3.6) rewriting it in the form bi + b~ + bi + 2(x , + X 2 + x 3) = E and using (3.3) and (3.4) as well as once again the identity
(4.1)
§S.
We can calculate the extreme values of Tr A~. All one has to do is to use formula (1.7c) and substitute in it formulas (3.3) for the b's and use formula (3.4) for u, After remarkable simplifications one obtains
M. Kac I Three Body Toda System
414
and since (5.1) is
}J2
= 1 we have that
}J
= ±1. For
}J
= + 1 the right-hand side of
~(!FE)3 = !EVfE 9V~~ 3 2' and for
}J
=-1
12-~EV~E . The case }J = 1 gives the maximum of Tr A~ and v To summarize we can now say that if (see (1.5))
= -1 the minimum.
(5.2)
and (see (4.1)) (5.3)
then (5Aa)
(5Ab) and (SAc)
Formula (5Ab) is the immediate consequence of E = Ai + A~ + A~ and (5Aa). Formula (SAc) is the same as (5.1). From (5Aa), (5.4b) and (5.4c) it follows that (SAd)
and thus the A'S are roots of the cubic equation (5.5) and it follows at once that the roots are
M. Kac / Three Body Toda System
415
i.e. one of the roots is doubly degenerate!
§6. We must now review briefly the general theory as it applies to the case of three bodies (for the genera) theory for n bodies see [2] and for a more detailed presentation [3]). In addition to the matrix (1.6) we consider the matrix (6.1)
obtained from (1.6) by erasing the first row and the first column of A. Denote the eigenvalues of (6.1) by J.tl and J.tz (as in [2]), consider the function .1 (A) defined by the formula (6.2)
and denote by A ~, A 2' A 3" (in increasing order) the roots of (6.3)
.1(A)=-2.
It can be proved that (6.4)
and furthermore (see [2, formula (3.1) for n (6.5a)
and (6.5b)
= 3])
416
M. Kac I Three Body Toda System
In our special case At = A2 it is clear that ILt = At = A2 and since the A's are constants of motion
This is consistent with ..:i 2(p,t) - 4 = ..:i 2(AJ- 4 = O. Equation (6.5b) is nontrivial and can be written in the form (6.6) Now since
equation (6.6) becomes (6.7)
and hence IL2(t) is an elliptic function of t.
§7. From the general theory it also follows [2, formula (2.5)] that (7.1)
where (7.2)
The choice of the sign in front of the radical can be determined from the initial conditions at t = O. Once determined it is to be changed every time on passes through a turning point, i.e. ..:i 2- 4 = O.
M. Kac / Three Body Toda System
417
To determine x 3 (x 2 is then found by applying (1.5» we observe that
(7.3) and (7.4)
(7.5)
Combining (7.5) with (7.4) we have (7.6)
and finally
It is useful to note that
1
• tr;::;
2
6
= -(2v~E -
II. ) ' " + r-z \1-"2
• tr;::;
1 dJ.L 2 4 dt '
v~E)±-6
with J.L2 being the elliptic function defined by (6.7). Our special solution of the three body Toda system has first been found by Toda [4] (who called it a solution) in an entirely different form. The relation between his form and ours may be of interest to investigate.
References [1] H. F1aschka, The Toda Lattice I, Phys. Rev. 89 (1974) 1924-1925.
418
M Kac I Three Body Toda System
[2] M. Kac and P. van Moerbeke, A complete solution of the periodic Toda problem, Proc. Nat. Ac. Sci. 72 (1975) 2879-2880. [3] M. Kac, Non-linear Dynamic and Inverse Problems, In: Proc. 1977 Internat. School in Statistical Mechanics, eds. E.G.D. Cohen and W. FIszdon (Polish Academy of Sciences) 199-222. [4] M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967) 501-506.
409
THE THREE BODY TODA SYSTEM AND A PARAMETRIZATION OF PLANE CUBIC CURVES M. KAC 1t Department ofMathematics, University of Southern California, Los Angeles, California, U.S.A.
To Leopoldo Nachbin on the occasion of his sixtieth birthday
§1. The three body Toda system is a linear chain of three oscillators with the Hamiltonian given (in conveniently chosen units) by the formula (1.1)
H = 2(pi + Pz +
pD + exp(-(ql -
qz»
+ exp«q2 - q3» + exp(-(q) - qt» . Following Flaschka [1] we introduce the variables (1.2)
(k = 1,2,3),
with the understanding that q4 = qt. In these variables the equations of motion become: (1.3a)
(1.3b) and we shall further set (1.4) 1 The research of the author was supported in part by the N.S.F. under contract MCS 8101739. t Deceased.
M. Kac / Three Body Toda System
410
Because of (1.2) we have the constraint (1.5)
Consider now the matrix
(1.6)
By a general theorem of F1aschka the eigenvalues, and hence the traces, are constants of motion, the first two Tr A 3 and Tr A; are the momentum and the energy. We may as well set the total momentum to be 0, i.e. (1.7a) and write the conservation of energy in the form (1.7b) We also have
=
const ,
where 6 would have been 6X1X2X3 if it were not for the constraint (1.5). It should be added that the constancy (in time) of X 1X2X3 = aia~a; is consistent with the equation of motion (1.3b).
§2.
Since
XI'
x2 and x3 are positive we have
(2.1)
the equality occurring only when
XI
= x 2 = X 3 = 1. Thus
M. Kac I Three Body Toda System
and if E uniquely
= 6, b, = b2 = b3 = O.
Thus if E
=6
411
the matrix A 3 is determined
(2.2)
This is trivial from the point of view of mechanics for it merely states that a system in the state of minimum total energy remains in that state forever.
§3. If E > 6 we can try to extremize Tr A~ under the constraints (1.5), (1.7a) and (1.7b). This leads to a somewhat tedious calculation involving Lagrange multipliers, which for convenience I shall take to be .A (for (1.7a», ~JL (for (1.7b» and 3v for (1.5). Thus we have to extremize
(3.1)
bi + b~ + b~ + 3x t(b t + b2 ) + 3x 2(b2 + b3 ) + 3x 3(b3 + b t )
+ .A (b, + b2 + b3 ) + ~JL (bi + b~ + b; + 2x t + 2x 2 + 2x 3 )
+ 3v(x tx 2x3-1), and I will try to save a skeptical rear some needless labour by sketching the derivation in some detail. By differentiation of (3.1) with respect to the x's we get (3.2a) (3.2b) (3.2c)
so that (using b, + b2 + b3 = 0) we get (3.3a)
(3.3b)
412
M. Kac / Three Body Toda System
(3.3e) and (3.4)
Differentiating (3.1) with respect to the b's we obtain (3.5a) (3.5b) (3.5e) and (again using b, + bz + b3 = 0) we get (3.6)
Substituting the expressions (3.3) for the b's into (3.5a) and (3.6) we obtain (3.7) (p, + ZlX IX3)Z + (Xl + x 3 ) + JL z+ JLZlX IX3
- M(p, + ZlXIX3)Z + (p, + ZlXIXz)Z + (p, + Zlxzx3i + 2(xI + Xz + x 3) ] = 0 . The expression in square brackets in (3.7) is equal to 3JL z + 2Z1JL (XIX3 + XIXZ+ XZx3) + l/(xi x~ + =
6JLz- 2(1- l/)(X 1+ x z + x 3)
xi X~ + x~x~) + 2(x I + Xz + X3)
,
and first three terms in (3.7) add up to
In deriving these formulas repeated use has been made of (3.4), (1.5) and of the identity
which in tum follows from (1.5).
M. Kac I Three Body TOlla System
413
Finally (3.7) becomes
Similarly one obtains (3.8b) and
If ,i ¥ 1 equations (3.8) imply that XI = X 2 = X 3 = 1 and hence (3.3), (3.4) and (1.7a) yield hi = b2 = b3 = O. Thus E = 6 contrary to the assumption E>6.
§4. Having established that Ii = 1 we go back to the formula (3.6) rewriting it in the form bi + b~ + bi + 2(x , + X 2 + x 3) = E and using (3.3) and (3.4) as well as once again the identity
(4.1)
§S.
We can calculate the extreme values of Tr A~. All one has to do is to use formula (1.7c) and substitute in it formulas (3.3) for the b's and use formula (3.4) for u, After remarkable simplifications one obtains
M. Kac I Three Body Toda System
414
and since (5.1) is
}J2
= 1 we have that
}J
= ±1. For
}J
= + 1 the right-hand side of
~(!FE)3 = !EVfE 9V~~ 3 2' and for
}J
=-1
12-~EV~E . The case }J = 1 gives the maximum of Tr A~ and v To summarize we can now say that if (see (1.5))
= -1 the minimum.
(5.2)
and (see (4.1)) (5.3)
then (5Aa)
(5Ab) and (SAc)
Formula (5Ab) is the immediate consequence of E = Ai + A~ + A~ and (5Aa). Formula (SAc) is the same as (5.1). From (5Aa), (5.4b) and (5.4c) it follows that (SAd)
and thus the A'S are roots of the cubic equation (5.5) and it follows at once that the roots are
M. Kac / Three Body Toda System
415
i.e. one of the roots is doubly degenerate!
§6. We must now review briefly the general theory as it applies to the case of three bodies (for the genera) theory for n bodies see [2] and for a more detailed presentation [3]). In addition to the matrix (1.6) we consider the matrix (6.1)
obtained from (1.6) by erasing the first row and the first column of A. Denote the eigenvalues of (6.1) by J.tl and J.tz (as in [2]), consider the function .1 (A) defined by the formula (6.2)
and denote by A ~, A 2' A 3" (in increasing order) the roots of (6.3)
.1(A)=-2.
It can be proved that (6.4)
and furthermore (see [2, formula (3.1) for n (6.5a)
and (6.5b)
= 3])
416
M. Kac I Three Body Toda System
In our special case At = A2 it is clear that ILt = At = A2 and since the A's are constants of motion
This is consistent with ..:i 2(p,t) - 4 = ..:i 2(AJ- 4 = O. Equation (6.5b) is nontrivial and can be written in the form (6.6) Now since
equation (6.6) becomes (6.7)
and hence IL2(t) is an elliptic function of t.
§7. From the general theory it also follows [2, formula (2.5)] that (7.1)
where (7.2)
The choice of the sign in front of the radical can be determined from the initial conditions at t = O. Once determined it is to be changed every time on passes through a turning point, i.e. ..:i 2- 4 = O.
M. Kac / Three Body Toda System
417
To determine x 3 (x 2 is then found by applying (1.5» we observe that
(7.3) and (7.4)
(7.5)
Combining (7.5) with (7.4) we have (7.6)
and finally
It is useful to note that
1
• tr;::;
2
6
= -(2v~E -
II. ) ' " + r-z \1-"2
• tr;::;
1 dJ.L 2 4 dt '
v~E)±-6
with J.L2 being the elliptic function defined by (6.7). Our special solution of the three body Toda system has first been found by Toda [4] (who called it a solution) in an entirely different form. The relation between his form and ours may be of interest to investigate.
References [1] H. F1aschka, The Toda Lattice I, Phys. Rev. 89 (1974) 1924-1925.
418
M Kac I Three Body Toda System
[2] M. Kac and P. van Moerbeke, A complete solution of the periodic Toda problem, Proc. Nat. Ac. Sci. 72 (1975) 2879-2880. [3] M. Kac, Non-linear Dynamic and Inverse Problems, In: Proc. 1977 Internat. School in Statistical Mechanics, eds. E.G.D. Cohen and W. FIszdon (Polish Academy of Sciences) 199-222. [4] M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan 23 (1967) 501-506.