M089 · Discrete mechanics

FPUT Recurrence Laboratory

Release one sine mode of a fixed-end nonlinear spring chain. Compare linear, quadratic-force and cubic-force experiments while tracking every harmonic mode, the nonlinear energy remainder, modal entropy and the first sampled return after energy leaves the initial mode. Drag the release profile to change its amplitude.

Acquired normalized time $0$

Total mechanical energy $0$

Kinetic energy $0$

Bond potential energy $0$

Signed energy defect $0$

Chain momentum $0$

Independent net wall impulse $0$

Momentum minus initial and wall impulse $0$

Half-step displacement difference · first ten time units $0$

Actual normalized integration step $0$

Maximum integration energy defect · full record $0$

Record / model domain $0$

Initial-mode share of harmonic energy $0$

Normalized modal entropy $0$

Sum of harmonic mode energies $0$

Signed nonlinear bond-energy contribution $0$

Lowest acquired initial-mode share $0$

First sampled return · fundamental periods $0$

Departure and return status $0$

Linear fundamental period $0$

This is a normalized, isolated, fixed-end chain of mathematical point masses with nearest-neighbour polynomial springs, no contact collision or damping. The time unit uses unit reference mass and stiffness. Harmonic modal energies exclude nonlinear bond energy and are normalized by their own sum; their sum is not the full conserved energy. Return means a sampled initial-mode share below four fifths followed by at least nine tenths; linear confinement without departure is not recurrence. The first linear fundamental period sets the time scale for every release mode. Entropy describes a finite harmonic distribution, not proof of thermalization. Sampling can shift or miss a threshold return. A quadratic-force-only bond is a local polynomial model; recording stops before its nonconfining exterior. A conservative rate cap and a 500000-step budget may shorten extreme records. Half-step comparison covers only the first ten time units.

Start with the essentials

Focus question: Does the observed return survive departure, a linear control and a smaller integration step?
One-sentence intuition: Track the full Hamiltonian separately from harmonic projections. Require departure before calling a later threshold crossing a return.

Core mathematical model

Polynomial bond potential

V(r)=\tfrac12r^2+\tfrac\alpha3r^3+\tfrac\beta4r^4,\qquad r_i=u_i-u_{i-1}

Quadratic and cubic force terms share one Hamiltonian.

Fixed-end chain dynamics

\ddot u_i=V^{\prime}(u_{i+1}-u_i)-V^{\prime}(u_i-u_{i-1}),\qquad u_0=u_{N+1}=0

There is no external driving, damping or contact solver.

Linear harmonic projection

Q_j=\sqrt{\frac2{N+1}}\sum_{i=1}^N u_i\sin\frac{ij\pi}{N+1},\qquad E_j=\tfrac12(\dot Q_j^2+\omega_j^2Q_j^2)

Harmonic modes remain a diagnostic basis when the bonds are nonlinear.

Full energy and the nonlinear remainder

H=\sum_jE_j+\sum_i\left(\tfrac\alpha3r_i^3+\tfrac\beta4r_i^4\right)

Only the complete Hamiltonian is the conserved mechanical energy.

Normalized harmonic modal entropy

p_j=E_j/\sum_kE_k,\qquad S=-\frac{\sum_jp_j\log p_j}{\log N}

A concentrated harmonic distribution has low entropy; the nonlinear remainder is excluded.

Common difficulties

Confinement is not recurrence

Typical misconceptionA linear mode returning to the same displacement proves nonlinear recurrence.

Better mental modelThe linear control never loses its modal energy; this protocol first requires a share below four fifths.

Harmonic energy is not the Hamiltonian

Typical misconceptionThe sum of projected mode energies must be exactly constant.

Better mental modelNonlinear bond energy is a signed remainder and exchanges with the harmonic part.

Entropy does not prove thermalization

Typical misconceptionA broad modal spectrum proves equipartition forever.

Better mental modelThe finite record can return; entropy alone does not establish asymptotic equilibrium.

Run the experiment

01
Establish the linear control
Acquire the linear preset and inspect the modal history.
What to observe: One mode retains its energy, with no departure event.
02
Acquire a nonlinear return
Use the default quadratic-force preset and acquire the full record.
What to observe: Energy spreads, the initial share falls, and a sampled threshold return appears.
03
Change the mixing strength
Compare cubic-force and stronger releases, then inspect the entropy.
What to observe: Different finite histories need not return within the same observation window.
04
Audit the numerical experiment
Reduce the step and compare the full energy defect and early refinement readout.
What to observe: Early convergence and full-record energy monitoring provide different evidence.