# Lift Force from Lagrangian Mechanics

1. Sep 20, 2009

### thrillhouse86

I am having a lot of trouble conceptually understanding this issue in Lagrangian mechanics:

I have an airfoil which is immersed in incompressible flow: it has two degrees of freedom: a rotation: alpha and a pitching (up down) motion: h. Now the lift is due to both alpha and the first derivative of h.
i.e. Lift = (1/2)*\rho*b*U^{2}(\alpha + \dot{h}/b)

Whenever the equations of motion for the airfoil are derived the inertial forces and the elastic restoring forces of the airfoil are included in the Lagrangian and the aerodynamic forces are included as generalised forces.

What I really want to know is whether the alpha term of the Lift can be included in the Lagrangian, rather than included as part of the generalised force. My rationale for this is that it is a function of alpha, and therefore is not a dissapative term. Also doing an eigen value analysis shows that if the only term that is neglected from the equations is the \dot{h} term, then the eigenvalues have no real component (as I understand it, indicating no energy gain or loss).

But conceptually this lift term is obviously feeding energy into the airfoil, and as this is only a two degree of freedom system, and I'm not including the (infinite) degrees of freedom of the pressure field around the airfoil the energy of my system should not be conserved and thus I shouldn't be able to include it in my Lagrangian.

Can anyone help me resolve this connundrum ?

On an unrelated note - can someone direct me to a page showing how you insert Latex into these pages ?

Cheers,
Thrillhouse