Kelvin's medium

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Kelvin's one-dimensional medium is a chain consisting of solids interacting via a pair torque potential. In this example, the solids are visualized by rods. The solids interact via the torque potential:

[math] U = C({\bf n}_{1}\cdot{\bf n}_{2}) [/math],

where [math]C[/math] is an interaction constant, [math]\bf{n}_1[/math], [math]\bf{n}_2[/math] are the unit vectors bound to the solids. The interaction torque has the form:

[math] {\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = C({\bf n}_{1}\times{\bf n}_{2}) [/math]

Then the motion equation of the k-th particle is as follows:

[math] J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) [/math]