Jourdain Principle - anybody using it ?

[Oberst Villa]

While reading about Lagrange equations, I found the following remark about the Jourdain principle (I try a quickndirty translation of the german original, hope it makes sense):

"For big systems with very many degrees of freedom - especially in vehicle dynamics and aerospace - the many derivatives in the Lagrange equations result in a considerable effort. There is general agreement that in this cases the Jourdain Principle should be preferred, which is similar to the d'Alambert principle".

Now I'm not looking for an explanation of the principle, but for a statement like "Yep, it's usefull for doing xyz". So that I know whether I should add it to the list in my brain of the 1 trillion things I have to learn someday...

 

[jhc]

If you are already familiar with Lagrange's form of D'Alembert's Principle (the variational statement often referred to as D'Alembert's Principle)

<br /> \sum_{i \in bodies}{(m_i\vec{a}_i-\vec{F}_i) \cdot \delta{\vec{r}_{i}} = 0<br />

and know how to use it to solve multibody problems then for all practical purposes you already know Jourdain's Principle.

D'Alembert's Principal uses virtual displacements (\delta{\vec{r}_{i}}), which expand to

<br /> \delta{\vec{r}_{i}} = \sum_{j \in coords}{\frac{\partial{\vec{r}_i}}{\partial{q}_j}\delta{q_j}<br />

Jourdain's statement is a virtual power law
<br /> \sum_{i \in bodies}{(m_i\vec{a}_i-\vec{F}_i) \cdot \bf{\delta{v_{i}}} = 0<br />

with
<br /> \bf{\delta{v_{i}}} = \sum_{j \in coords}{\frac{\partial{\vec{v}_i}}{\partial{\dot{q}}_j}\delta{\dot{q}_j}<br />

Note that this is NOT the variation of velocity, as there would also have to be a partial with q_j. Even so, Jourdain's Principle is more than the second variation of D'Alembert's Principle because finite variations of the "velocity" are admissible... If one leaves all constraint forces explicit in the statement of D'Alembert's Principle, you can arrive at Jourdain's Principle through direct time differentiation and acknowledging that each of the three terms in the distributed derivative must be zero independently (by Newton's Law).

The advantage of Jourdain (and Gauss) is that the higher order is necessary to represent nonholonomic velocity(acceleration) constraints such as a caster, ice skate, or coin rolling without slip, where the space of displacements has more coordinates than the velocities. It is also often easier to enforce mechanism kinematic looping constraints in this way (four bar linkage has three coordinates but only one Degree of Freedom), and when doing so they are referred to as "simple nonholonomic" because they are integrable (but very ugly) and therefore holonomic.

Everyone using "Kane's Method" or a "generic velocity projection scheme" is using Jourdain's Principle. It is VERY popular.

Enjoy!