From Abstract to Concrete by Computation

The computer changes practice of science and technology. Let us see if the computer also changes the nature and role of theory as expressed in mathematical models. 

A classical Newtonian paradigm is to formulate a model as a set of differential equations describing laws of physics such as Newton's laws of motion, as  a dynamical system with state $U(t)$ depending on time $t$ satisfying (with the dot signifying differentiation with respect to time):

•  $\dot U(t)=F(U(t)$ for $0<t\le T$ with $U(0)$ given and $T$ a given final time,      (N)

where $F(v)$ is a given function of $v$. We call the function $U(t)$ the trajectory of the system.

Calculus was developed to solve (N) using symbolic mathematics of integrals and derivatives, which worked for a limited set of dynamical systems. When symbolic solution failed or was too complicated numerical solution could always be used in the form of time-stepping

•  $U(t+dt) = U(t)+dtF(U(t))$ with $dt>0$ a small time step,               (C)

allowing successive computation of $U(t)$ for $0<t\le t)$, without the use of symbolic Calculus.

Before the computer (C) could be too time consuming, and so the symbolic Calculus was developed by reformulating (N) into a new paradigm of Lagrangian mechanics, where a specific dynamical system solution $U(t)$ was not described by (N) but instead by a Variational Principle VP  stating that the integral

• $\int_0^TL(u(t))dt$                               (VP)

with Lagrangian $L(v)$ (determined by $f(v)$) depending on an arbitrary trajectory $u(t)$, has a vanishing small variation under small variations of $u(t)=U(t)$. 

The differential equation (N) with one specific solution $U(t)$ was thus replaced by a VP including variation over many trajectories $u(t)$. The generality of VP formulation turned the 18th century into Lagrangian mechanics, thus from (N) to (VP). This was a step towards abstraction from concrete Newtonian mechanics to Lagrangian mechanics based on abstract VP.

We have seen that (N) has a natural computational form as (C), while the computational form of a VP is not direct since comparison over a rich variation is not efficient.

With the computer there is thus today a shift from VP back to (N). From abstract to concrete, because computation is concrete like taking yet another time step forward.

This has important implications because the generalisation to classical mechanics to quantum mechanics has followed the path of Lagrangian abstraction put to an extreme in the Quantum Field Theory by Feynman as "sum over all paths".

In particular the generalisation of Schrödinger's equation from one electron to many electrons took an abstract path into multi-dimensional wave functions, which has haunted Quantum Mechanics from start. 

RealQM offers a concrete generalisation which directly lends itself to computation.

Summary: Computation takes concrete form and so naturally connects to concrete differential equations formulation rather than to some abstract variational principle. 

PS1 Recall the light can be viewed to propagate following a principle of least time (without direct computational realisation), as an alternative to wave propagation (with direct computational form).

PS2 Certain configurations can be characterised as minimising energy, which can be resolved computationally by gradient method as a form of time-stepping.