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I am working on a variational problem involving elastic stability of a beam.

The deformation of the beam is given by six functions of the material coordinate along the beams longitudinal axis. The governing functional looks like $$ \int_{s_1}^{s_2} F \left( u(s),v(s),w(s),\phi(s),\chi(s),\psi(s) \right) ds. $$ (It represents the beam's strain energy.)

Now I want to impose the condition that transverse shear deformation and elongation are zero. The consequence is that the 6 degrees of freedom of the beam are reduced to essentially 3. It can be shown that the three relations $$ \frac{du}{ds} = \sin(\chi), \quad \frac{dv}{ds} = -\sin(\phi)\cos(\chi), \quad \frac{dw}{ds} = \cos(\phi)\cos(\chi) -1 \qquad (*) $$ between the coordinates are valid in case transverse shear deformation and elongation are neglected.

My understanding is that there are now two ways to incorporate this constraint condition:

  1. eliminate variables in the original functional by substitution
  2. include equations (*) by means of Lagrange multipliers

I would like to use the latter method. The book The variational principles of mechanics by Lanczos explains this for both holonomic and non-holonomic constraints (and isoperimetric ones as well). Though I can see that equations (*) are clearly not some algebraic relation $$ f_i(u,v,w,\phi,\chi,\psi)=0, \quad i=1\dots3 $$ between the coordinates, does this means the constraint qualifies as non-holonomic? I am confused because it seems integrable. Or does it then become an isoperimetric constraint?


I am getting more confused; can we only call constraints non-holonomic if derivatives with respect to time are involved? (Meaning that all equations (*) above just represent holonomic constraints, since it is all static?)

In The enigma of nonholonomic constraints by Flannery (2005), it is explained that for the typical action $$ S = \int_{t_1}^{t_2} L({q},\dot{q},t) dt $$ the Lagrange multiplier rule can only be used for holonomic and semi-holonomic (exact linear) conditions. Here, the independent variable clearly is time, but in my case it is some spatial coordinate.

If we assume that the independent variables space and time are interchangeable, my (*) equations appear non-holonomic (so, not to be incorporated with Lagrange multipliers), as they cannot be expressed as exact differential.. But, is this correct reasoning?

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