Nonlinear feedback selects an action only while its energy source supplies power
An oscillator with a finite energy depot relaxes to zero action in the model below. Replenishing the depot above a threshold produces a positive attracting action, independent of the initial seed but dependent on the supplied power and oscillator frequency. The exact construction separates attraction from universality and gives a concrete test of a self-excitation mechanism.
Status: exploratory written derivation, Q03; independent proof/literature review is required before accepted-ledger promotion. This is an autonomous dissipative effective model, not a derivation from a closed Hamiltonian field.
1. Construction and physical choice
Let an oscillator have angular frequency \(\omega>0\), action \(J\ge0\) and angle \(\theta\). Its measured mechanical energy is \(H=\omega J\). Let \(e\ge0\) be stored depot energy. Choose loss rates \(\gamma,\kappa>0\), conversion coefficient \(g>0\) with units \((\text{energy}\,\text{time})^{-1}\), and supplied power \(P\ge0\). Define
\[\dot\theta=\omega,\qquad \dot J=2(ge-\gamma)J,\qquad \dot e=P-\kappa e-2g e\omega J. \tag{1}\]
The nonlinear transfer \(2ge\omega J\) enters oscillator energy and leaves the depot with equal magnitude. The premise is stimulated gain proportional to existing oscillation energy: a charged or active reservoir amplifies a seed, while conversion depletes its stored energy. This is a deliberately simple feedback law to test, rather than a universal mechanical law.
A real-quadrature realization removes the angle singularity at rest. With coordinates \(Q,R\) of units square root of action and \(J=(Q^2+R^2)/2\), put
\[\dot Q=(ge-\gamma)Q-\omega R,\qquad \dot R=\omega Q+(ge-\gamma)R.\]
These equations induce (1), including the invariant state \(Q=R=0\). For a harmonic mechanical readout take \(x=Q/\sqrt{m\omega}\) and \(p=\sqrt{m\omega}R\); then \(p^2/(2m)+m\omega^2x^2/2=\omega J\). Gain acts on both quadratures, so this realization is an effective amplitude model rather than Newton’s unmodified relation \(\dot x=p/m\).
The physical idea has precedent: Schweitzer, Ebeling and Tilch describe particles that store environmental energy and convert it into motion, with limit cycles above an uptake threshold. Their model motivates the comparison; equation (1) is the explicit model tested here.
2. Finite fuel: an exact energy inequality
The positive quadrant is invariant. For \(E=\omega J+e\), cancellation of the internal transfer gives
\[\dot E=P-2\gamma\omega J-\kappa e. \tag{2}\]
Put \(a=\min(2\gamma,\kappa)>0\). At \(P=0\),
\[0\le E(t)\le E(0)e^{-at},\qquad 0\le J(t)\le E(0)e^{-at}/\omega. \tag{3}\]
Thus even an initially amplifying depot, with \(ge(0)>\gamma\), eventually exhausts its energy and sends \(J(t)\) to zero. For arbitrary fixed \(P\), the bound \(\dot E\le P-aE\) also gives global bounded solutions. The conclusion depends explicitly on positive losses. Lossless closed dynamics is a different candidate, whose conserved energy can retain preparation dependence.
3. Replenishment: an explicit positive attractor
The quiet equilibrium is \((J,e)=(0,P/\kappa)\). The linear action growth rate there is \(2(gP/\kappa-\gamma)\). Define
\[P_c=\frac{\kappa\gamma}{g}.\]
For \(P>P_c\) a positive equilibrium exists:
\[e_* = \frac{\gamma}{g},\qquad J_* = \frac{P-P_c}{2\gamma\omega}. \tag{4}\]
It attracts every initial state with \(J(0)>0\) and \(e(0)>0\). To prove this, use the nonnegative function
\[V=\omega\left[J-J_*-J_*\log(J/J_*)\right] +e-e_*-e_*\log(e/e_*).\]
Writing \(\delta J=J-J_*\) and \(\delta e=e-e_*\), the first term contributes \(2g\omega\delta J\delta e\). The depot equation is
\[\dot e=-(\kappa+2g\omega J_*)\delta e-2g\omega e\delta J.\]
Multiplication by \(1-e_*/e\) cancels the cross term exactly, giving
\[\dot V=-(\kappa+2g\omega J_*)\frac{(e-e_*)^2}{e}\le0. \tag{5}\]
Sublevel sets of \(V\) are compact within the positive quadrant. On \(\dot V=0\) we have \(e=e_*\); remaining there requires \(J=J_*\). The invariant-set argument therefore proves convergence to (4). An initially empty depot becomes positive immediately when \(P>0\). An exactly zero oscillator seed stays zero even above threshold; the attracting positive branch has the basin \(J(0)>0\).
The result supplies an actual physical-time action attractor in a classical effective model. It also identifies precisely what powers it.
4. Selection tests and consequence
At the same \(g,\gamma,\kappa,P\), probes of different frequency acquire the same energy \(\omega J_*=(P-P_c)/(2\gamma)\), rather than the same action. Demanding a common action \(K>0\) instead requires the power prescription
\[P(\omega)=P_c+2\gamma\omega K. \tag{6}\]
This prescription inserts the desired scale in the source. It is the physical step a further theory must explain, rather than a consequence of attraction. Moreover \(J_*\downarrow0\) continuously as \(P\downarrow P_c\): instability of the quiet state above threshold supplies no positive lower action bound across the admitted source parameters. Exact rest remains a trajectory throughout.
The promising ingredient is feedback that erases initial amplitude; the failed ingredient is identifying its source-dependent attractor with a universal constant. Together with Q02’s proposed frequency-selective radiation response, this suggests a sharper question: can a self-consistent reservoir dynamically set a common spectral amplitude, with its energy budget included? Merely connecting this depot to a linear radiation bath would still supply \(P\).
Park further pumped-oscillator variants. A different route worth testing is a conservative field sector with a fixed topological constraint: its quiet-state exclusion would be a geometric restriction rather than continuous fueling. The decisive test is whether its minimum action survives dilation, not merely whether a nontrivial stationary configuration exists. This is a proposed next test, not evidence that topology already fixes an action constant.
Source and review boundary
F. Schweitzer, W. Ebeling and B. Tilch, Complex Motion of Brownian Particles with Energy Depots, Physical Review Letters 80, 5044 (1998), publisher record and abstract. Coverage: one discovery query and the publisher abstract; no full-text equation match or novelty assessment. Its energy-uptake/limit-cycle mechanism prompted the finite-fuel versus replenished comparison. Equations (1)–(6) are the present exploratory calculation, with energy accounting and a written Lyapunov check; no numerical or symbolic scripts were used.