Real Coupled-Oscillator Ledger

Coupled-Oscillator Energy Really Swaps Masses

Every value in this lesson is rounded to four decimal places from a real physim coupled-oscillators numerical integration (scenario examples/coupled-oscillators-beats.json, integrator dp45), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The real energy hands almost entirely from the first mass to the second

A real physim run starts with the first mass displaced 1.0000 metres and the second mass at rest. By t = 16.0000 seconds the first mass has fallen to 0.0017 metres while the second mass has grown to 0.9441 metres — almost all the energy has crossed the weak coupling spring to the other mass.

xa(0.0000)=1.0000 m → xa(16.0000)=0.0017 mx_a(0.0000)=1.0000\ \mathrm{m}\ \to\ x_a(16.0000)=0.0017\ \mathrm{m}
Real coupled-oscillator energy swapGhost trail traces the real (x1, x2) trajectory; the ledger cites the start and swap rows.t=0.0000 s x1=1.0000 m x2=0.0000 mt=16.0000 s x1=0.0017 m x2=0.9441 menergy=5.2500 J (all rows)

A real integrator brings the energy back again

By t = 39.8000 seconds the energy has swung back — the first mass is at 0.9898 metres, almost its original displacement, while the second mass is back near -0.0087 metres — while the real total energy stays pinned at 5.2500 joules at every sampled row, this is a genuine beat between the two coupled masses, not the fixed single-mode pictures earlier in this book.

xa(39.8000)=0.9898 m,E=5.2500 Jx_a(39.8000)=0.9898\ \mathrm{m},\quad E=5.2500\ \mathrm{J}
Real coupled-oscillator energy returnsThe same trail; the ledger adds the return row and the conserved total energy.t=0.0000 s x1=1.0000 m x2=0.0000 mt=16.0000 s x1=0.0017 m x2=0.9441 mt=39.8000 s x1=0.9898 m x2=-0.0087 menergy=5.2500 J (all rows)