  Hamilton's principle is used to extend for the Toda lattice ODEs to systems
of PDEs called the Toda lattice strand equations (T-Strands). The T-Strands in
the $n$-particle Toda case comprise $4n-2$ quadratically nonlinear PDEs in one
space and one time variable. T-Strands form a symmetric hyperbolic Lie-Poisson
Hamiltonian system of quadratically nonlinear PDEs with constant characteristic
velocities. The travelling wave solutions for the two-particle T-Strand
equations are solved geometrically, and their Lax pair is given to show how
nonlinearity affects the solution. The three-particle T-Strands equations are
also derived from Hamilton's principle. For both the two-particle and
three-particle T-Strand PDEs the determining conditions for the existence of a
quadratic zero-curvature relation (ZCR) exactly cancel the nonlinear terms in
the PDEs. Thus, the two-particle and three-particle T-Strand PDEs do not pass
the ZCR test for integrability.
