The Secret Agenda

• Everyone has a hidden agenda. Except me! (Michael Crichton)
• Later mathematicians will regard set theory as a disease from which one has recovered. (Henri Poincare)

Here is a document describing a secret agenda that will lead to resolving the mystery of the Navier-Stokes equations and thus the mystery of flight.

1d problem:

• motion of a body $B$ along a 1D straight line with coordinates $x$
• $x(t)$ position of $B$ at time $t$
• velocity $v(t)$ as change $dx$ of position $x(t)$ per unit time step $dt$: $dx=vdt$
• acceleration $a(t)$ as change $dv$ of velocity $v(t)$ per unit time step: $dv=adt$
• Newton’s 2nd Law $F = Ma$ or $a=\frac{F}{M}$ with $F$ force and $M$ mass
• time stepping: With $v^n=v(ndt)$, $x^n=x(ndt)$, do for $n=0,1,2,....,$
• $v^{n+1}= v^n + a^ndt=v^n+\frac{F^n}{M}dt$
• $x^{n+1}= x^n + v^ndt$ .

$B$ moves with zero force: $F=0$:

• $v(t) = v$ is constant
• $x(t) = t v + \bar x$
• $B$ moves with constant force $F$: $a =F/M$ constant:
• $v(t) = t a + \bar v$ straight line
• $x(t) = t^2/2 a + t\bar v +\bar x$ curved line.

Same in 2D or 3D:

• $B$ moves in 2D Euclidean plane with coordinates $x=(x_1,x_2)$ or $x=(x_1,x_2,x_3)$
• $x(t)$ position of $B$ depending on time $t$
• $v(t)$ as change $dx$ of position $x(t)$ per unit time step $dt$: $dx=vdt$
• acceleration $a(t)$ as change $dv$ of velocity $v(t)$ per unit time step: $dv=adt$
• Newton’s 2nd Law $F = Ma$ or $a=\frac{F}{M}$ with $F$ force and $M$ mass.

Time stepping:

• With $v^n=v(ndt)$, $x^n=x(ndt)$, do for $n=0,1,2,....,$
• $v^{n+1}= v^n + a^ndt=v^n+\frac{F^n}{M}dt$
• $x^{n+1}= x^n + v^ndt$
• $B$ moves with $F=0$, $F$ constant, $F(t)$ variable
• connect $B$ to spring with spring force depending on position of $B$
• mass-spring systems of several masses and springs in 1-3D
• mass-spring systems with viscosity.

Derivative with respect to space coordinate: space derivative:

• Euler’s and Navier-Stokes’ equations of fluid mechanics
• Navier’s equations of solid mechanics
• waves in fluids and solids: wave equation
• Fourier’s equation for heat conduction/diffusion
• Maxwell’s equations for electromagnetics
• discover secret of flight.

Two approaches to describing and understanding the World.

Hint to Solution:

Describing the World: Can you identify the equations, and the persons behind the equations?