Re: Hamiltonian for flight?

[Digital ChoreoGraphics <dcg@softcafe.net>]

At 07:14 PM 4/5/99 -0400, you wrote:
>Aeronautical physicists,
>
>	I am looking for the physics formula that demonstrates that
>unpowered aerodynamic flight is characterized by a constant glide
>ratio (as opposed to the parabolic curve of ballistic flight).
>
>	After too many years, I don't remember the detailed derivation
>and have been unable to find it either on the web or in my old
>textbooks.  I vaguelly remember that it involved evaluating a
>Hamiltonian.  I vaguelly remember showing that for some sets of
>constants (air density, mass, drag coefficient, moment of inertia,
>???) the Hamiltonian was stable and minimized for a path that dropped
>a proportionate distance for a given curve length (i.e. constant
>glide ratio).  I vaguelly remember not worrying about the sets of
>constants where the Hamiltonian was unstable.
>
>	The equivalent of this would be to prove that the vertical
>component of lift and drag is constant.  I seem to remember that
>proving the contancy of lift and drag forces was much more
>complicated than evaluating the Hamiltonian.  I further remember
>that completely solving the Hamiltonian was too complicated, but
>that proving a constant glide ratio was straightforward.
>
>	I think my basic problem is that I am missing one of the
>Hamiltonain components.
>
>	o Gravitational Potential (m g z) <= altitude energy
>	o Kinetic Energy (- m v^2 /2) <= energy of motion
>	o Atmospheric compression (Cp p S v^2) <= wing loading
>	o ??? = what am I forgetting?
>
>Thank you,
>
>Bob
>

Bob - 

I am not familiar with the formula in question, but you have the 
Kinetic and Potential Energies, and the drag.  All that is missing is 
an expression for Lift as a function of 'v'. 

However, I cannot find, in my physics texts, such an expression.  Perhaps 
an associate with an aerodynamics background (or textbook).  The following 
is the lift as a function of the 'circulation' about the airfoil:

F = p v L Gamma
p is fluid density
v is velocity of undisturbed fluid flow 
L is length of wing
Gamma is the fluid circulation above & below the wing.

I have seen Hamiltonians used for the equations of motion when there
is no dissipation.  I suppose it will work in the glider scenario if 
you assume negligable drag, in which case you could determine the 
limits of motion. 

Let me know what you work out.
- Don Black


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