The Sutherland and Calogero N-body Models

It describes the dynamics of N-particles lying on a circle, with 1/(r ^2) interaction between them (r being the absolute distance between the particles i.e. chord length).

Sutherland model is described by the Hamiltonian

H = 1/2Underoverscript[∑, j = 1, arg3] p_j^2 + 1/2 (π/L)^2Underoverscript[∑, j, k = 1 (j≠k), arg3] (2α)^2/(sin^2π/L (x_j - x_k) ), (1)

with corresponding equations of motion (L being the length of the circle)

Overscript[x, .] _j = p_j (2)
Overscript[p, .] _j = -∂/∂x_j (π/L)^2Underoverscript[∑, k = 1 (k≠j), arg3] (2α)^2/(sin^2π/L (x_j - x_k) ) (2)

or explicitly

Overscript[p, .] _j = 8 (π/L)^3α^2Underoverscript[∑, k = 1 (k≠j), arg3] (cosπ/L (x_j - x_k))/(sin^3π/L (x_j - x_k) ) (3)

The Calogero N-body model is similar to Sutherland N-body model--Calogero model describes the dynamics of N-particles lying on a straight line, with 1/(r ^2) interaction between them

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