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:

  • gradient, Laplacian
  • 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.

 Describing the World.

Watch

Hint to Solution:

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

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