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Newton’s force law $\frac{dP}{dt}=F$ is derived from the Schrödinger equation for isolated macroscopic bodies, composite states of e.g. $N\sim 10^{25},10^{51},\dots$ atoms and molecules, at finite body temperatures. We first review three aspects of quantum mechanics (QM) in this context: (i) Heisenberg’s uncertainty relations for their center of mass (CM), (ii) the diffusion of the CM wave packet, and (iii) a finite body temperature which implies a metastable (mixed-) state of the body: photon emissions and self-decoherence. They explain the origin of the classical trajectory for a macroscopic body. The ratio between the range $R_q$ over which the quantum fluctuations of its CM are effective, and the body’s (linear) size $L_0$, $R_q∕L_0\lesssim 1$ or $R_q∕L_0\gg 1$, tells whether the body’s CM behaves classically or quantum mechanically, respectively. In the first case, Newton’s force law for its CM follows from the Ehrenfest theorem. We illustrate this for weak gravitational forces, a harmonic-oscillator potential, and for constant external electromagnetic fields slowly varying in space. The derivation of the canonical Hamilton equations for many-body systems is also discussed. Effects due to the body’s finite size such as the gravitational tidal forces appear in perturbation theory. Our work is consistent with the well-known idea that the emergence of classical physics in QM is due to the environment-induced decoherence, but complements and completes it, by clarifying the conditions under which Newton’s equations follow from QM, and by deriving them explicitly.

In this paper, we show that Hamilton's equations can be recast into the equations of dissipative memristor circuits. In these memristor circuits, the Hamiltonians can be obtained from the principles of conservation of "charge" and "flux", or the principles of conservation of "energy". Furthermore, the dynamics of memristor circuits can be recast into the dynamics of "ideal memristor" circuits. We also show that nonlinear capacitors are transformed into nonideal memristors if an exponential coordinate transformation is applied. Furthermore, we show that the zero-crossing phenomenon does not occur in some memristor circuits because the trajectories do not intersect the i = 0 axis. We next show that nonlinear circuits can be realized with fewer elements if we use memristors. For example, Van der Pol oscillator can be realized by only two elements: an inductor and a memristor. Chua's circuit can be realized by only three elements: an inductor, a capacitor, and a voltage-controlled memristor. Finally, we show an example of two-cell memristor CNNs. In this system, the neuron's activity depends partly on the supplied currents of the memristors.