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jv07cs

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I am using Nivaldo Lemos' "Analytical Mechanics" textbook and on section 2.4 (Hamilton's Principle in the Non-Holonomic Case) he uses Hamilton's Principle and Lagrange Multipliers to arrive at the Lagrange Equations for the non-holonomic case.

I don't understand why it is assumed that the principle of least action is valid for a non holonomic system. On chapter 1, section 1.4, Lagrange Equations are derived from D'Alembert's Principle assuming holonomic constraints and on chapter 2, section 2.3, it is shown that, for a holonomic system, defining the action S as a time integral of the Lagrangian and making it stationary leads us to the same Lagrange Equation, which leads us to state Hamilton's Principle. To arrive at these results, it was needed to assume holonomic systems. However, on section 2.4, the book states that: It is possible to deduce the equations of motion from Hamilton’s principle in the special case in which the non-holonomic constraints are differential equations of the form:

I don't understand why it is assumed that the principle of least action is valid for a non holonomic system. On chapter 1, section 1.4, Lagrange Equations are derived from D'Alembert's Principle assuming holonomic constraints and on chapter 2, section 2.3, it is shown that, for a holonomic system, defining the action S as a time integral of the Lagrangian and making it stationary leads us to the same Lagrange Equation, which leads us to state Hamilton's Principle. To arrive at these results, it was needed to assume holonomic systems. However, on section 2.4, the book states that: It is possible to deduce the equations of motion from Hamilton’s principle in the special case in which the non-holonomic constraints are differential equations of the form:

This textbook does kind of "copy" Goldstein's in many sections and I've seen people saying that Goldstein's third editon onward gives incorrect arguments when using Hamilton's Principle for the non-holonomic case, but on the second edition, which I've seen people saying that is correct, on section 2.4 "Extension of Hamilton's Principle To Nonholonomic Systems", in order to arrive at the Lagrange Equations for Non Holonomic Systems he also assumes that Hamilton's Principle holds for Non Holonomic Systems:

I have actually 3 questions:

1. Can someone please explain to me why we can assume that Hamilton's Principle holds for non holonomic systems?

2. Lemos' books doesn't say it, but do the constraints in eq. 2.66 have to be semi-holonomic?

3. Is there a way to arrive at eq. 2.74 using only D'Alembert's Principle and not Hamilton's Principle?