D'Alembert's principle vs Hamilton's principle

[lriuui0x0]

In the first two chapters of Goldstein mechanics, the Lagrange equations are derived from both D'Alembert's principle and Hamilton's principle. I want to know what're the applicability of these two approaches to Lagrangian mechanics? Is one more powerful than the other or are they completely equivalent?

I noticed there's one assumption that D'Alembert's principle derivation makes, that is the constraint forces do no work. And in the Hamilton's principle, it is assumed the system is monogenic. The elaboration in the book is pretty dense and there doesn't seem to be a summary list on what assumptions each principle makes. So a summary and a comparison on their power and applicability would be really helpful!

 

[anuttarasammyak]

As an old student now I am not sure which is the principle of least action ?

I suspect both are the same, one using Lagrangean, another using Hamiltonian.

 

[Frabjous]

@vanhees71 has some issues with one of Goldstein’s derivations

 

[vanhees71]

Concerning nonholonomous constraints the treatment within d'Alembert's principle is correct in Goldstein but it's incorrect using Hamilton's principle. For classical mechanics the best sources are Landau&Lifshitz volume 1 and Arnold for a more mathematical treatment.

 

[lriuui0x0]

Concerning nonholonomous constraints the treatment within d'Alembert's principle is correct in Goldstein but it's incorrect using Hamilton's principle. For classical mechanics the best sources are Landau&Lifshitz volume 1 and Arnold for a more mathematical treatment.

May I ask how Goldstein got the nonholonomic constraints wrong?

 

[wrobel]

For classical mechanics the best sources are Landau&Lifshitz

they also claim that they deduce the equations of nonholonomic mechanics from the Hamilton principle :(

 

[wrobel]

by the way, I am sorry for self citing but here I tried to write a clear text with explanations about nonholonomic variational principle https://arxiv.org/abs/2104.03913
nothing new just an essence from textbooks

 

[vanhees71]

The nonholonomic constraints (with the usual constraints) can be treated with both the d'Alembert and the Hamilton principle of least action, leading also to the same equations. What Goldstein derives with his version using Hamilton's principle in the wrong way is known as "vakonomic dynamics".

 

[wrobel]

to the same equations

nope
for details see A. M. Bloch, J. Baillieul, P. Crouch, J. Marsden: Nonholonomic Mechanicsand Control (Interdisciplinary Applied Mathematics). Springer, 2000
.

 

[vanhees71]

So what's wrong with the derivation in Landau and Lifshitz? They definitely derive the same equations of motion as you also get from d'Alembert's principle. The key is that in both the d'Alembert principle as well as the action principle the non-holonomic constraints are treated as constraints of the virtual displacements/variations and you introduce the Lagrange multipliers as such but not for the constraints written in terms of linear/affine forms of the generalized velocities.

 

[wrobel]

Landau and Lifshitz put in the basis of mechanics the Hamilton principle. They particularly say: let $x_*(t)$ be a critical point of the functional
$$x(\cdot)\mapsto \int_{t_1}^{t_2}L(x(t),\dot x(t))dt$$ in a class of functions $x(t)$ that satisfy nonholonomic constraints:
$$a_i^j(x)\dot x^i=0.$$ Then $x_*(t)$ is a solution to the following Lagrange equations with multipliers
$$\frac{d}{dt}\frac{\partial L}{\partial\dot x^k}-\frac{\partial L}{\partial x^k}=\lambda_s a^s_k\qquad (*)$$
That is wrong. The explanation why it is so see in the book cited above

 

[vanhees71]

But that's also the result from d'Alembert's principle. So you are saying that this is wrong? Unfortunately I cannot look at this book. What is it claiming to be the right equation and from which principle is it derived?

 

[wrobel]

the solution $x_*(t)$ to the variational problem described in #11 satisfies the equation
$$\frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot x^k}-\frac{\partial \mathcal L}{\partial x^k}=0,\quad \mathcal L=L+\lambda_k(t)a^k_s\dot x^s$$
this equation contains $\dot\lambda$ and it is not the same as (* ) from #11

See also [Greenwood_D.T.]_Classical_dynamics

 

[wrobel]

There is a funny story as well. I think only Russians know it. Very long ago before the famous many valued Landau and Lifshitz textbook appeared, Landau and Pitaevsky wrote a textbook on classical mechanics. This book contained many errors and it was completely smashed by Fok in his article critique published in Uspehi Fizicheskih Nauk (Russian physics journal)
After that in his joint with Lifshitz new book, Landau tried to correct the mistakes pointed out by Fok but failed. First (Russian) version of Arnold's "Mathematical Methods of classical mechanics" contained a lot of ironic footnotes addressed to the Landau Lifshitz first volume. Arnold found another portion of strange things in the Landau Lifshitz book. When Landau died, Arnold removed all his critical comments from further editions of his "Math methods".

 

[vanhees71]

the solution $x_*(t)$ to the variational problem described in #11 satisfies the equation
$$\frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot x^k}-\frac{\partial \mathcal L}{\partial x^k}=0,\quad \mathcal L=L+\lambda_k(t)a^k_s\dot x^s$$
this equation contains $\dot\lambda$ and it is not the same as (* ) from #11

See also [Greenwood_D.T.]_Classical_dynamics

But that's precisely the WRONG vakonomic dynamics, and that's not the standard dynamics derived from d'Alembert's principle or equivalently from Hamilton's least-action principle. In both cases the constraints have to be taken as constraints on the virtual displacements at fixed time, and that's not realized by the term with the Lagrange parameter as given above.

The correct implementation of the Lagrange parameters is thus in the variations, i.e., you have to minimize the original action (Einstein summation implies)
$$S[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)$$
under the constraints on the virtual (!) displacements, i.e.,
$$\delta S[q]+\lambda_k a_s^k \delta q^s=0$$
leading to
$$\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^s}-\frac{\partial L}{\partial q^s}=\lambda_k a_s^k.$$
It may be true that there are some mistakes in Landau Lifshitz vol 1. The most famous one is related to the wrong claim that all systems behave as the integrable systems, and this is rightfully critizized in in Arnold's famous book, but concerning the non-holonomic constraints, LL and Goldstein using d'Alembert's principle is correct.

 

[wrobel]

But that's precisely the WRONG vakonomic dynamics

yes, that is the point

the standard dynamics derived from d'Alembert's principle or equivalently from Hamilton's least-action principle.

that's a key mistake. They are not equivalent. In the case of nonholonomic constraints the Hamilton principle implies vaconomic equations. I can only again refer you to the books cited above.

 

[vanhees71]

No, as I've shown above in #15, it's simply not true. Using the correct implementation of the constraints as constraints on the variations (aka "virtual displacements") d'Alembert and action principles lead to the same equations of motion.