# Extendidng Hamilton's Principle to Non-Holonomic sytems

1. Jul 18, 2008

### pardesi

Can someone explain me how to extend Hamilton's principle to non-holonomic system's thru the lagrange undetermined multipliers?
PS:Assume the system is Semi-Holonomic that is $$f_{\alpha}(q_{i},q_{2} \cdots q_{n} ,\dot{q_{1}}, \dot{q_{2}}, \cdots \dot{q_{n}})=0$$ such a equation exists for $$\alpha=1,2,3 \cdots m$$

Last edited: Jul 18, 2008
2. Jul 19, 2008

### gdp

If by "Hamilton's Principle," you mean "the Variational Principle that Hamilton's Action is Extremized," strictly speaking, no such "Nonholonomic Constrained Action Principle" exists, except in the special case that the "nonholonomic constraints" are "integrable" --- i.e., that they are really just "disguised" holonomic constraints that have been written in an apparently non-holonomic form. (One will find a few contrary claims in a few textbooks and papers, but on careful examination, these so-called "nonholonomic variational principles" are all either ill-posed or not self-consistent unless the constraints are integrable.)

IIRC, a mathematically rigorous treatment of constrained variational problems may be found in Rund and Lovelock's https://www.amazon.com/Tensors-Differential-Forms-Variational-Principles/dp/0486658406" but it's fairly heavy going. (Or perhaps I may be thinking of Rund's "Hamilton-Jacobi Theory of the Calculus of Variations," which is sadly now out of print...)

There is a modification of Hamilton's Principle called the "Hamilton-Jacobi-Bellman Principle" (or just the "Bellman Principle" for short), that is used to formulate "Optimal Control Problems" --- including problems with non-holonomic constraints. However, the Bellman Principle is not in general equivalent to Hamilton Principle, and it does in general lead to equations of motion that are "mathematically degenerate" --- i.e., they have a nontrivial "nullspace," implying that their solutions are non-unique.

There is also Dirac's Theory of Constrained Hamilitonian Systems, but Dirac's formalism cannot be derived from a variational principle except in the special case that the constraints are integrable --- i.e., the constraints are equivalent to holonomic constraints.

Last edited by a moderator: May 3, 2017