Purely Mechanical Memristors: Perfect Massless Memory Resistors, the Missing Perfect Mass-Involving Memristor, and Massive Memristive Systems

We define a mechanical analog to the electrical basic circuit element M = d{\phi}/dQ, namely the ideal mechanical memristance M = dp/dx; p is momentum. We then introduce a mechanical memory resistor which has M(x) independent of velocity v, so it is a perfect (= not-just-memristive) memristor, although its memristance does not crucially involve inert mass. It is practically realizable with a 1cm radius hollow sphere in heavy fuel oil with a temperature gradient. It has a pinched hysteretic loop that collapses at high frequency in the v versus p plot. The mechanical system clarifies the nature of memristor devices that can be hypothesized on grounds of physical symmetries. We hypothesize a missing mechanical perfect memristor, which must be crucially mass-involving (MI) precisely like the 1971 implied EM memristor device needs magnetism. We also construct MI memristive nano systems, which clarifies why perfect MI memristors and EM memristors are still missing and likely impossible.