equating space-time curvature G with stress-energy/mass density T. The trouble with Einstein’s equations is that they cannot be understood
, and even more cumbersome, they cannot be solved
. But the differential equations listed above can both be understood (more or less) and solved (more or less) by suitable computation.
Wellposed and Illposed Problems
We understand that the formulation of a problem is very important: A wellposed problem can be solved. Trying to solve and illposed problem is meaningless, because it cannot be solved. An example of a mathematically illposed
problem is the Clay Navier-Stokes Problem
What Is an Equation? What Does an Equality Sign Mean?
Since formulating equations is so fundamental we now scrutinize the concept of equation. The basic equation
has the form A = A expressing identity: A is equal to A. This is a tautology which is true whatever A means and as an always true tautology, it carries no information of interest.
Let us compare with an equation of the form A = B supposing that B is not identically equal to A to avoid the
previous tautology of no interest. If B is not identically equal to A, the equation A = B can only express that there is an aspect of A which is shared with B, and the equality sign expresses this aspect.
We thus have to be very careful with the meaning of the equality sign = which expresses the relevant aspect of equality in each case. As an example let us consider the familiar equation
1 + 1 = 2.
How are we to interprete this equation? Is it a definition of 2 in the form 2 = 1 + 1, like the assignment
statement 2 := 1 + 1 in computer science? Maybe, but if so it is like a tautology or identity which has no
If now 2 = 1 + 1 is not simply an assignment, it can be interpreted as expressing the dynamics of forming a couple or pair by letting two things unite, just as when a married couple is formed by two people getting into
some form of fusion process when forming a couple. This interpretation thus has a real content and thus is of more interest. The equality sign in 1 + 1 = 2 then expresses that there is a certain aspect of 2 which is shared with 1 + 1 which can be described as two-ness: Clearly two-ness is an aspect of a couple (2), and
1 + 1 also has an obvious aspect of two-ness resulting from putting together 1 and 1 through the + sign.
1 frog + 1 frog = couple of frogs = 2
Similarly, a differential equation usually expresses a physical process or balance, which to have a real interest should be beyond definition or assignment. For example, Newton’s 2nd law
F = M x A with F force, M mass and A accelleration
can express the physical process of pulling a spring connected to a body and noting a relation between the spring force, the mass of the body and the accelleration, and thus expresses a balance of force and accelleration.
Alternativey, M = F / A can be used as a definition mass M in terms of spring force F and accelleration A as discussed in Does The Earth Rotate?
F = G x M1 x M2 / R x R
where G is a the gravitational constant, and thus connects force and mass.
We have to specify the relevant interpretation in each specific case: definition or physics? If this distinction is not made, the result is confusion. The master of this form of confusion was Einstein as explained in Many-Minds Relativity
What Does It Mean to Solve an Equation?
Once we have decided on the meaning of a differential equation D(U) = F, which can express a physical
law of balance like Newton’s 2nd law, we have to define what it means to solve the equation and then the interpretation of the equality sign = comes up again. If = does not represent an uninteresting identity, but
an equality of numbers, then the only possible interpretation of D(U) = F is that the residual
R(U) = D(U) – F
is small enough. A solution U of the equation D(U) = F is thus qualified by the size of its residual R(U) and
we can speak about solutions of different residual quality.
To a computational mathematician this makes perfect sense but not so to a pure mathematicians: In formulating a system of a differential equations like the Navier-Stokes equations, pure mathematicians take for granted that the meaning of the equality sign as an identity is completely clear, but this leads into serious difficulties as expressed in the ambigious unfortunate formulation of the Clay Navier-Stokes Problem
. In mathematical terms the size of the residual in some relevant norm has to be specified. To just say that the residual should be zero is mathematically meaningless.
Confronted with a differential equation it is useful to make precise what physical process the equality sign
expresses, and when facing an (approximate) solution one should ask how big is its residual in different norms. Let us consider a basic example:
What Is the Squareroot of Two?
We know that the positive solution X of the equation X x X = 2 is called the squareroot of 2 denoted by √2 .
The Pythagorean School based on natural numbers collapsed when it was understood that √2 is not a rational number as a quotient p / q of two natural numbers. In other words you can not find a decimal number X with a finite number of digits such that X x X = 2, but you can make the residual X x X – 2 as small as you want (but not zero) by using more and more digits:
√2 = 1.41421356237309504880168872420969807856967187537694807317667973799….
Since X x X – Y x Y = (X + Y) (X – Y) it follows that the residual of the listed decimal number has as many zero decimals as the number of listed decimals (65). It is small but not identically zero.
A constructivist mathematician would say that √2 as the positive solution of the equation X x X = 2 does not exist, but approximate solutions with arbitrary small residuals do exist since they can be computed/constructed decimal by decimal. A formalist mathematician would argue that √2 exists as an infinitely long non-periodic decimal expansion, but would probably agree that from any practical point of view only finitely many decimals would be of interest and could be determined. For a detailed analysis of the wonderful/tragic story of √2 see Chapters 14-15 of Body&Soul Vol I.
In short, the digital school of Pythagoras was outpowered by the geometric school of Euclide in which √2 was simply the length of the diagonal of a unit square and thus did not pose a problem. Arithmetic was thus replaced by geometry which delayed the development of science 1500 years until Descartes replaced geometry by numbers in his analytic geometry and thereby initiated the scientific revolution. Control of √2 gives control of the World…
d (MV) /dt = M dV/dt = F with A = dV/dt accelleration.
The action of the force F changes the momentum MV at the rate d (MV) /dt = F. If F = 0, then momentum is
does not change d (MV) /dt = 0.
- what goes in ultimately has to go out if nothing is lost.
Physical laws in the form of constitutive laws
connect physical quantities of different nature:
Constitutive laws and conservation laws with non-zero source generally express real physics by connecting different physical aspects, and thus are not simply definitions.
In the case of zero source (F = 0 in the 2nd law) a conservation law can be viewed to express the tautology in = out, but with nonzero source also a constitutive law is involved (connecting change of momentum to force in the 2nd law).
The basic systems of differential equations listed above all express conservation laws combined with constitutive laws.
More on Solving Differential Equations
A differential equation A(U) = F expresses a local balance because differentiation is a local operation. The differential equation
dS/dt = V
connects the rate of change dS/dt of distance S(t) to momentary or local in time velocity V(t) with t time. To solve the differential equation dS/dt = V for S in terms of V amounts to integration or summation of
S = sum of dS = sum of V dt = integral of V dt
the whole = sum of the pieces.
Whole human body = sum of its pieces
The summation or integration is a global operation, and thus to solve a differential equation involves a global operation of summation starting from local information. This is like summing the prices of the items you have
have accumulated in the grocery store to get a total price to pay. Summation requires work and thus computational solution of a differential equation requires computational work.
Normally you solve for U in A(U) = F with A and F given, by some form of summation or integration. The essence of theoretical science based on differential equations can thus be described as a process of global accumulation from local input.
You can also seek to determine coefficients of the differential operator A such as heat conductivity and viscosity from measurements of F and U, which is called an inverse problem
You can view the formula
the whole = sum of the pieces
- assembly – integration: summing the pieces gives the whole
- dissection – differentiation: taking the whole apart gives the pieces.
Assembly is global, dissection is local.
The principle formulate – solve equation is quite general and can be stretched outside physics:
- Darwin’s theory of evolution is based on solving the equation of life in the form of survival of the fittest by combining genetic variation and natural selection.
- Marx’ equation of a classless stateless society is solved by the method of class struggle.
- Freud’s equation of mental health is solved by physchoanalysis using free association on a sofa.
- The equation of capitalistic economy expressing equilibrium of supply and demand is solved by varying prices.
- The equation of music is solved by combining melody with harmony with rythm.
- The equation of literature is solved by combining plot with style.
- The equation pf politics is solved by combining vision with rethoric.
Freud’s equipment for computing The pieces of human physche.