Three equal point masses lie on a frictionless line

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Simulating classical dynamical systems

Computational Physics, Ch Stepping the equations of motion: four integrators measured

Problem

Three equal point masses mm lie on a frictionless line, connected in series by two identical springs of constant kk; the two outer masses are otherwise free (not attached to any wall). Writing x1,x2,x3x_1,x_2,x_3 for the displacements of the three masses from their equilibrium positions, Newton's second law for each mass is mx¨1=k(x2−x1),mx¨2=k(x3−x2)−k(x2−x1),mx¨3=−k(x3−x2).\begin{aligned}m\ddot x_1 &= k(x_2-x_1),\\m\ddot x_2 &= k(x_3-x_2)-k(x_2-x_1),\\m\ddot x_3 &= -k(x_3-x_2).\end{aligned} (a) Write this three-body system as a first-order system: give the six-component state vector y\mathbf{y} and f(y)\mathbf{f}(\mathbf{y}). (b) Take k=50 N m−1k = 50\,\mathrm{N\,m^{-1}} and m=0.20 kgm = 0.20\,\mathrm{kg}. Evaluate f\mathbf{f} numerically at x1=0.02 mx_1 = 0.02\,\mathrm{m}, x2=0x_2 = 0, x3=−0.01 mx_3 = -0.01\,\mathrm{m}, with all three masses momentarily at rest.

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