An important paper on Lagrangian Mechanics

2010 April 23
by Daniel Lakeland

It's amazing that 250 years after Lagrange came up with his analytical coordinate-independent formulation of Newton's mechanics, we still don't really fully understand what is going on.

I say this because I've been working on a model for a simple system, and it has a non-holonomic constraint, and sure enough the Lagrange formalism doesn't do it for me. The newton's equations are easy to derive for the simple system, but the Euler-Lagrange equations don't give the same dynamics.

Recently a paper was published on why non-holonomic constraints don't work in Lagrangian mechanics. This paper is by professor Flannery at Georgia Tech. There is a class of "non-holonomic" constraints which can be used in Lagrangian mechanics via a "right hand side" but they must be linear in the velocities. In any case, he is hard at work writing the followup article, but for the moment I'm stuck with no good idea of what to do next.

It should be noted that many of the classic textbooks erroneously say that you can plug in lagrange multipliers with the non-holonomic constraints and get the right answer. They are wrong except in special cases where the non-holonomic constraints have an "integrating factor".

To get a brief overview of a generalized approach to non-holonomic constraints there is a decent paper by Leon Bahar from Drexel University where he deals with Lagrange, Jourdain, and Gauss principles for higher and higher order constraints.

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