Extendidng Hamilton's Principle to Non-Holonomic sytems

[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 $$\alpha=1,2,3 \cdots m$$

 

[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/dp/0486658406/?tag=pfamazon01-20 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.

 

[charng]

Extending Hamilton's principle to non-holonomic systems involves using Lagrange undetermined multipliers to account for the non-holonomic constraints in the system. These constraints can be represented by the equations f_{\alpha}(q_{i},q_{2} \cdots q_{n} ,\dot{q_{1}}, \dot{q_{2}}, \cdots \dot{q_{n}})=0, where \alpha=1,2,3 \cdots m.

To extend Hamilton's principle, we first need to understand the concept of holonomic and non-holonomic systems. In a holonomic system, the constraints can be expressed in terms of the generalized coordinates and their derivatives. In contrast, in a non-holonomic system, the constraints cannot be expressed in this way.

Hamilton's principle states that the true path of a system is the one that minimizes the action integral, which is the difference between the kinetic and potential energy of the system. However, this principle only applies to holonomic systems. To extend it to non-holonomic systems, we need to introduce additional terms in the action integral to account for the non-holonomic constraints.

This is where Lagrange undetermined multipliers come in. These multipliers are introduced into the action integral to incorporate the non-holonomic constraints. They act as additional variables that are determined by the equations of motion of the system.

In the case of a semi-holonomic system, where the constraints can be expressed as f_{\alpha}(q_{i},q_{2} \cdots q_{n} ,\dot{q_{1}}, \dot{q_{2}}, \cdots \dot{q_{n}})=0, we can use Lagrange undetermined multipliers to modify the action integral by adding terms involving these multipliers. This modified action integral can then be minimized to obtain the equations of motion for the system.

In summary, extending Hamilton's principle to non-holonomic systems involves using Lagrange undetermined multipliers to incorporate the non-holonomic constraints into the action integral. This allows us to find the true path of the system and determine its equations of motion.