Physics Asked by varantir on January 4, 2021
I am working on this Hamiltonian:
$$ H = frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$
Thank you for the hint that it is a modification of the Toda Lattice Equation.
Let me sketch what I tried until now and why it is not working:
Analogous to the mentioned publications I introduced
$$b_n:= frac{1}{2}Exp{(frac{q_n-q_{n+1}}{2})}
a_n:= -frac{p_n}{2} $$
where it follows directly with $frac{partial H}{partial q_i}=-p_i$ and $frac{partial H}{partial p_i}=q_i$:
$$dot{b_n} = (a_{n+1} – a_n)b_n
dot{a_n} = 2 (b_{n}^2 – b_{n-1}^2)$$
When now using the Lax Pair $L$,$B$:
$$ L f_n = b_n f_{n+1} +b_{n-1} f_{n-1} + a_n f_{n}$$
$$ B f_n = b_n f_{n+1} – b_{n-1} f_{n-1} $$
it can be shown that $partial_t L=[B,L]$. My problem arises in defining the border conditions of my couple $q_1$ and $q_2$ in the 2d lattice above, since one needs to shift to the 3d representation ${b_0,b_1,b_2}$ in order to satisfy the periodic conditions (One mutual coordinate $q_3 = 0$ coupled to the others). Since it can be shown easily that $dot{lambda} = 0$ (where $lambda$ is an eigenvalue $Lv=lambda v$) the constants of motion reduce to the calculation of the eigenvalues. But in this case the eigenvalues of $L$ dont seem to simplify, in fact it doesnt seem to be a solution, which was my inital goal.
In general this approach seems to be at overkill for the 2d problem since it solves the n-dimensional Toda lattice.
The Matrix $L$ seems to yield the wrong solution:
$$ L = begin{pmatrix}
a_0 & b_0 & 0
b_0 & a_1 & b_1
0 & b_1 & a_2
end{pmatrix}$$
The Matrix does neigher solve $partial_t L = [B,L]$ (with $B=L_+ – L_-$) nor are eigenvalues constants of motion. Was has gone wrong?
Since the inverse scattering method can be applied here, I tried to get the scattering data, but actually I was not able to do the task. Any literature?
That's the Toda lattice, check, e.g., here. Also, Berry's paper has a discussion of the three-particle Toda lattice, which is what this is. The paper itself is a really good read, but that may give you a good start.
Answered by stafusa on January 4, 2021
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