Nonholonomic mechanics in Landau and Lifshitz

[vanhees71]

<<Mentor's note: this is spin-off this thread>>

"Mechanics" by LL contains errors. For example, it is well known that the equations of nonholonomic mechanics do not follow from the variational principle of Hamilton. This fact has serious physical and geometric reasons. For details see for example
[Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics), Anthony Bloch, et al]. Nevertheless LL claim that they obtain equations of nonholonomic mechanics from the variational principle.

One error I'm aware of in LL vol. I is the claim on integrability. But what's wrong with LL's treatment of anholonomous constraints (in sectin 38 in my German edition)? It just leads to the usual equations with Lagrange parameters you also get from d'Alembert's principle ("virtual displacements"). There are wrong statements about this in the literature, e.g., in Goldstein, where he uses (38.2) instead of (38.4), leading to the socalle "vaconomic mechanics", which is considered wrong. On this issue, see

Zampieri, Gaetano. "Nonholonomic versus vakonomic dynamics." Journal of Differential Equations 163.2 (2000): 335-347.
https://www.researchgate.net/profile/Gaetano_Zampieri/publication/224039915_Nonholonomic_versus_Vakonomic_Dynamics/links/544fb9a50cf201441e934bcd.pdf

 

[wrobel]

But what's wrong with LL's treatment of anholonomous constraints (in sectin 38 in my German edition)? It just leads to the usual equations with Lagrange parameters you also get from d'Alembert's principle ("virtual displacements").

They write correct equations but they obtain these equations from significantly incorrect argument.

Furthermore Their definition of canonical transformation is also not equivalent to the standard one. In the sequel pages this generates another error. etc

 

[vanhees71]

Hm, you must have read another book ;-)).

BTW also in the by you recommended book [Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics), Anthony Bloch, et al] the make come to the same conclusion as Landau/Lifhitz and other standard textbooks, i.e., that the usual equations derived from d'Alembert's principle of virtual displacement leads to the right equations of motion, i.e., you have to take the contraints (holonomic or anholonomic) as constraints on the variations with ##\delta t=0##. Then you can use the Lagrange-multiplier method also in the action principle, i.e., you make
$$A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)$$
stationary under the constraints ##\delta q(t_1)=\delta q(t_2)=0## and under the (perhaps anholonomous) contraints
$$c_{\alpha,j} \delta q^j=0$$
Then you get
$$\delta A=\int_{t_1}^{t_2} \mathrm{d} t \delta q^j \left [\frac{\partial L}{\partial q^j} -\frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial L}{\partial \dot{q}^j}-\lambda^{\alpha} c_{\alpha,j} \right ] \stackrel{!}{=} 0,$$
leading to the same equation as with the d'Alembert principle
$$\frac{\partial L}{\partial q^j} -\frac{\mathrm{d}}{\mathrm{d} t}\frac{\partial L}{\partial \dot{q}^j}=\lambda^{\alpha} c_{\alpha,j},$$
Together with the contraints these is a set of differential equations for the ##q^j## and ##\lambda^{\alpha}##.

I don't know to what section you are referring to when claiming that there are errors concerning canonical transformations. What's in Sect. 45 (again in my German edition of LL vol. 1) is very much standard and a very economic way to derive canonical transformations via generating functions. Also the Poisson brackets are right (note that LL uses a convention where the sign of the brackets differs from the more usual one, but that you cannot claim to be an error, it's just a different convention, which may be unfortunate but it's not wrong).

 

[wrobel]

you make

A[q]=∫t2t1dtL(q,˙q,t)​

A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)
stationary under the constraints δq(t1)=δq(t2)=0\delta q(t_1)=\delta q(t_2)=0 and under the (perhaps anholonomous) contraints

cα,jδqj=0​

this is exactly vaconomic variational principle and it leads to vaconomic equations, you can not deduce formula:

δA=∫t2t1dtδqj[∂L∂qj−ddt∂L∂˙qj−λαcα,j]!=0,​

from this principle

 

[vanhees71]

No! I just have derived it for you in the previsou posting (indeed that's also what LL does in his book, and I guess you find it in any textbook on analytical mechanics, except in Goldstein who derives the errorensou vakonomic equations without a clear discussion of its (in)validity; of course you find also in this textbook the right equations, derived from d'Alembert's principles).

The vakonomic equations follow if you take
$$c_{\alpha j} \dot{q}^j=0 \qquad (*)$$
as constraints, which is NOT what I did in my previous posting. To get the vakonomic equations, you use the Lagrange-multiplier method for these constraints, but these are different from the constraints on the variations that I (and any reasonable textbook does). It's also all the same in chapter 1 of the textbook by Bloch et al! It's only amazing that he discusses vakonomic dynamics at length in a later chapter, but I have to check what's his motivation for this in more detail to understand why he is doing that.

The point is that the right way of anholonomic constraints have to be applied to the variations themselves. Then you derive the equations of motion for the generalized coordinates and Lagrange multiplyers and then, i.e., "on shell" you can use the constraints in terms of (*), and that's how it is done in LL using Hamilton's principle. LL doesn't discuss the vakonomic dynamics at all, but also doesn't put a warning against their use. Admittedly, I've only recently seen the vakonomic dynamics via a discussion in this forum.

 

[wrobel]

which is NOT what I did in my previous posting.

o, now I have read your posting carefully. But then another question arises. What is the statement of variational problem you are solving? You have the functional ##q(\cdot)\mapsto A[q(\cdot)]##, ok. In which class of functions ##q(t)## you differentiate this functional ##A##?

 

[wrobel]

upd in which class of functions you are looking for extremums of the functional?

 

[vanhees71]

Well, I use the usual physicist's sloppyness (which is also applied in d'Alembert's principle and all other principles of mechanics). I just assume the the functions are "sufficiently smooth". Then you apply the usual (sloppy) way to vary the action but take into account the nonholonomous constraints on this variation using Lagrange multipliers.

Here's a paragraph on the difference between the correct d'Alembert and vaskonomic problem from Bloch's book (which is marvelous, as far as I can say from the little I read in it):

Perhaps it is surprising, at least at first, that these two procedures give
different equations. What, exactly, is the difference in the two procedures?
The distinction is one of whether the constraints are imposed before or
after taking variations. These two operations do not, in general, commute.
We shall see this explicitly with the vertical rolling disk in the next section.
With the dynamic Lagrange–d’Alembert equations, we impose constraints
only on the variations, whereas in the variational problem we impose the
constraints on the velocity vectors of the class of allowable curves.

 

[wrobel]

Well, I use the usual physicist's sloppyness

I did not mean to nitpick. I will also hold the sloppiness. General statement of the conditional extremums finding problem is as follows. Let ##f:X\to\mathbb{R}## stand for a functional of a vector space ##X##. The space ##X## may be infinite-dimensional. And there is a manifold ##M\subset X##. We shall say that a point ##x_0\in M## is a conditional extremum of ##f## iff ##d g(x_0)=0##, where ##g=f\mid_M:M\to\mathbb{R}.##

If you apply this concept to our case: $$f(x)=\int_{t_0}^{t_1}L(x,\dot x)dt,\quad M=\{x(t)\mid \langle c(x),\dot x\rangle=0,\quad x(t_i) --\mathrm{fixed}\}\qquad (**)$$
then you will have the vaconomic statement.

The statement
$$\int_{t_0}^{t_1}\Big(\frac{d}{dt}\frac{\partial L}{\partial \dot x^k}-\frac{\partial L}{\partial x^k}\Big)\delta x^kdt=0,\quad \langle c(x),\delta x\rangle=0\qquad (*)$$ gives the equations of classical mechanics, but this is not a problem on finding conditional extremums, it is just d'Alembert's principle in the integral form. So it is called in Bloch's book and it is correct. In general the solution ##x(t)## to problem (*) is not an exteremum of problem (**). It is an extremum to problem (**) if only the constraint ##\langle c(x),\dot x\rangle=0## is holonomic. LL sound like that they still apply the Hamilton variational principle just with several additional constraints that is incorreect: equation (*) is the D'Alambert principle it does not reduce to Hamilton's one.

To summarize this, roughly speaking, (**) does not imply (*)

 

[vanhees71]

Ok, then it's semantics. You can call it d'Alembert's principle in integral form. The only thing I wonder about is, how do we know, in the case of nonholonomic constraints, which principle to apply. Both principles are equivalent for holonomic constraints, and obviously every source claims that d'Alembert's principle is the correct description in the case of nonholonomic constraints. In the paper of Zampieri it's shown that the vakonomic treatment of the problem of a skater on an inclined plane leads to quite strange solutions, which is not the case for d'Alembert's principle and thus it is argued that the d'Alembert principle of virtual displacement is the correct treatment. On the other hand, in the book of Bloch there are whole chapters about the vakonomic dynamics. So I wonder, why one should consider them at all, if it's known to be the wrong description? I guess, one should only warn students against the use of vakonomic mechanics and then stay with good old d'Alembert.

 

[wrobel]

In the paper of Zampieri it's shown that the vakonomic treatment of the problem of a skater on an inclined plane leads to quite strange solutions

moreover, vaconomic problems do not have uniqueness of solutions in some sense. I am not an expert in vaconomic, I have just heard that vaconomic problems appear in the control theory.

I guess, one should only warn students against the use of vakonomic mechanics and then stay with good old d'Alembert.

Sure students should be aware about these stumbling blocks. But I believe that these things a very beautiful and fundamental and it is good to discuss them in lectures. It is also good to discuss the Frobenius' theorem in this context I guess.
Variational principles are useful for example to prove that the double pendulum has very many periodic trajectories. Anothe interesting thing is that the problems (*), (**) help to define a weak solution to the dynamical equations, this leads to pretty things in the theory of collisions

 

[PhDeezNutz]

Sorry for the mess of a post earlier.

I've seen two different versions of formulas for non-holonomic constraints in two different versions of Goldstein. And now I hear LL has a different formula. So I'd like to know which one is correct.

My version of Goldstein says for a system with ##n## coordinates and ##m## holonomic constraints of the form

##f_j \left( \dot{q}_1 , ..., \dot{q}_n \right) = c_j## where of course ##j = 1,..,m##

The Euler-Lagrange Equations are as follows

## \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i }\right) - \left(\frac{\partial L }{\partial q_i} \right) = \sum \limits_{j=1}^{m} \mu_j \left( t \right) \frac{\partial f_j}{\partial \dot{q}_i}##

Something never sat right with me about this. It was always "too analogous" to the formula for non-holonomic constraints.

I have some ideas on what I think it should be but don't know for sure. Also I will refrain from posting what I think they are because to avoid breaking the forum rules by being overly speculative.

 

[vanhees71]

The joke is that Goldstein has two different formulae within his book. The one derived from D'Alembert's principle is correct and is identical with the correct treatment in Landau&Lifshitz vol. 1 using the least-action principle. The formula Goldstein gives for nonholonomic constraints derived in a wrong way within the least-action principle is wrong.

 

[PhDeezNutz]

The joke is that Goldstein has two different formulae within his book. The one derived from D'Alembert's principle is correct and is identical with the correct treatment in Landau&Lifshitz vol. 1 using the least-action principle. The formula Goldstein gives for nonholonomic constraints derived in a wrong way within the least-action principle is wrong.

I’m really interested in this topic but have very little experience in it. Can I PM you some questions? This is in reference to a particular problem that I don’t want to post here because it’s a test question from several semesters ago and I don’t want to enable the next generation of students to cheat. I already finished the class with a B+ (I can provide proof if necessary).