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5 Flux-Charge Analysis Method of Memristor Circuits
p = mν
where the momentum of f is defined as
p(t) =
t
−∞
f (τ )dτ.
The correct Newton’s Law of Motion for a time-varying mass reads as follows:
p = m(t)ν
where
dp
dt
= f.
This yields
f = m(t)
dν
dt
+
dm(t)
dt
ν(t).
The correct pair of variables for a time-varying mass is then given by (p, ν) =
(f (−1) , x (1) ).
Example 5.3 (Laws of Motion) Section II-E in [4] reports a thorough discussion on
the concept of predicting the motion (i.e., dynamical behaviors) of a physical object
(e.g., stones, material bodies, planets, stars, etc.) by a law relating two observable
(i.e., measurable) attributes (i.e., physical variables).
The first attempts date back at least 2300 years ago, when Greek philosopher
Aristotle proposed the law f = mν. We now realize that Aristotle’s Law of Motion
is not valid because he had chosen an incorrect pair of physical variables (force and
velocity) to characterize a body in motion.
We had to wait almost 2000 years to replace Aristotle’s Law with the Newton’s
Law of Motion
f = m
dν
dt
which is correct at least for mundane velocities ν c, where c is the velocity of
light. It is interesting to note that, actually, Newton’s Law was originally expressed
in the equivalent form using the momentum p(t) =
t
−∞ f (τ )dτ of f as
f =
dp
dt
; p(t) = mν(t).
Unfortunately, as is the case in all models, even the celebrated Newton’s Second
Law is an approximation of reality, and it loses its predictive ability (in the sense