Emergence by Smart Integration of Physical Law as Differential Equation

Perfect Harmony of European Parliament: Level curves of political potential generated by an empty spot in the middle.

This is a continuation of previous posts on a new view of Newton's law of gravitation. We here connect to the Fundamental Theorem of Calculus of the previous post, allowing a bypass to compute an integral by tedious laborious summation, using a primitive function of the integrand:

• $\int_0^t f(s)ds = F(t) - F(0)$  if  $\frac{dF}{dt} = f$.

This magical trick of Calculus of computing an integral as a sum without doing the summation,  is commonly viewed to have triggered the scientific revolution shaping the modern world.

The magic is here computing an integral $\int_0^t f(s)ds$ in a smart way, rather than computing a derivative $\frac{dF}{dt}$ in a standard way.

The need of computing integrals comes from the fact that physical laws are usually expressed in terms of derivatives, for example as an initial value problem: Given a function $f(t)$, determine a function $F(t)$ such that

• $DF(t) = f(t)$ for $t\ge 0$ and $F(0) = 0$,

where $DF =\frac{dF}{dt}$ is the derivative of $F$. In other words, given a function $f(t)$, determine a primitive function $F(t)$ to $f(t)$ with $F(0)=0$, that is, determine/compute the integral
by the formula

• $\int_0^t f(s)ds = F(t)$ for $t\ge 0$. 

Using the Fundamental Theorem to compute the integral would then correspond to solving the initial value problem by simply picking a primitive function $F(t)$ satisfying $DF = f$ and $F(0)=0$ from a catalog of primitive functions, allowing to in one leap jump from $t=0$ to any later time $t$. Not very magical perhaps, but certainly smart!

The basic initial value problem of mechanics is expressed in Newton's 2nd Law $f=ma$ where $f$ is force, $m$ mass and $a(t)=\frac{dv}{dt}=\frac{d^2x}{dt^2}$ is acceleration, $v(t)=\frac{dx}{dt}$ velocity and $x(t)$ position, that is,

• $f(t) = m \frac{d^2x}{dt^2}$.           (1)

Note that in the formulation of the 2nd Law, it is natural to view position $x(t)$ with acceleration $\frac{d^2x}{dt^2}$ as given, from which force $f(t)$ is derived by (1) . Why? Because position $x(t)$ and acceleration $\frac{d^2x}{dt^2}$ can be observed, from which the presence of force $f(t)$ can be inferred or derived or concluded, while direct observation of force may not really be possible. In this setting the 2nd Law acts simply to define force in terms of mass and acceleration, rather than to make a connection with some other definition of force.

Writing Newton's 2nd law in the form $f=ma$, thus defining force in terms of mass and acceleration, is the same as writing Newton's Law of Gravitation:

• $\rho = \Delta\phi$,                          (2)

thereby defining mass density $\rho (x)$ in terms of gravitational potential $\phi (x)$ by a differential equation. 

With this view, Newton's both laws (1) and (2) would have the same form as differential equation, and the solutions $x(t)$ and $\phi (x)$ would result from solving differential equations by integration or summation as a form of emergence. 

In particular, this reasoning gives support to an idea of viewing the physics of Newton's Law of Gravitation to express that mass density somehow is "produced from" gravitational potential by the differential equation $\rho =\Delta\phi$. 

To solve the differential equation $\Delta\phi =\rho$ by direct integration or summation in the form

• $\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,

would then in physical terms require instant action at distance, which is difficult to explain. 

On the other hand, if there was a "smart" way of doing the integration by using some form of Fundamental Theorem of Calculus as above, for example by having a catalog of potentials from which to choose a potential satisfying $\Delta\phi =\rho$ for any given $\rho$, then maybe the requirement of instant action at distance could be avoided.

A smart way of solving $\Delta\phi =\rho$ would be to use the knowledge of the solution $\phi (x)$ in the case of a unit point mass at $x=0$ as

• $\phi (x)=\frac{1}{4\pi}\frac{1}{\vert x\vert}$ 

which gives Newton's inverse square law for the force $\nabla\phi$, which is smart in case $\rho$ is a sum of not too many point masses. But the physics would still seem to involve instant action at distance.

In any case, from the analogy with the 2nd Law we have gathered an argument supporting an idea to view the physics of gravitation as being expressed by the differential equation $\rho =\Delta\phi$ with mass density $\rho$ derived from gravitational potential $\phi$. Rather than the opposite standard view with the potential $\phi$ resulting from mass density $\rho$ by integration or summation corresponding to instant action at distance.

The differential equation $\Delta\phi =\rho$ would thus be valid by an interplay "in perfect harmony" in the spirit of Leibniz, where on the one hand "gravitational potential tells matter where to be how to move" and "matter tells gravitational potential what to be".

This would be like a Perfect Parliamentary System where the "Parliament tells People where to be and what to do" and "People tells Parliament what to be".

PS There is a fundamental difference between (1) and (2): (1) is an initial value problem in time while (2) is a formally a static problem in space. It is natural to solve an initial value problem by time stepping which represents integration by summation. A static problem like (2) can be solved iteratively by some form of (pseudo) time stepping towards a stationary solution, which in physical terms could correspond to successive propagation of effects with finite speed of propagation.