Why is Hamilton's Principle assumed to be valid for non-holonomic systems?

[jv07cs]

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:​

And the he goes on to use Hamilton's Principle and Lagrange Multipliers to arrive at:

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?

 

[wrobel]

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

we cannot!

for details see p.14
Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics) (Anthony Bloch, et al)

 

[wrobel]

or Greenwood_D.T.]_Classical_dynamics
p. 159

 

[vanhees71]

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:​

View attachment 330565And the he goes on to use Hamilton's Principle and Lagrange Multipliers to arrive at:

View attachment 330564

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:​

View attachment 330563

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?

First of all, that's the correct application, leading to the correct equations of motion, and it's equivalent to the results of D'Alembert's Principle. It's all correct in the 2nd edition of Goldstein's book (I don't know the 1st editition). In the 3rd edition some new authors thought that they had to modernize this great book and by an elementary mistake, they derived wrong dynamics for non-holonomic constraints.

The point is that the non-holnomic constraints are constraints that have to imposed on the variations in Hamilton's principle, and there by definition $\delta t=0$. That's also clear from the alternative ansatz, using D'Alembert's principle, which of course leads to the same equations, i.e., the Euler-Lagrange equations of the 1st kind for non-holonomic constraints.

I can't answer the question about "semi-holonomic constraints", because I don't know the definition of this term.

 

[vanhees71]

or Greenwood_D.T.]_Classical_dynamics
p. 159

But there he uses the variational principle in a too restrictive way. With the correct interpretation of the an-holonomous constraints as constraints for the "allowed variations" in Hamilton's principle you get the correct equations of motion, i.e., the ones you also get from D'Alembert's principle, which is no surprise, because D'Alember's principle and the variational principle are entirely equivalent.

 

[jv07cs]

or Greenwood_D.T.]_Classical_dynamics
p. 159

Thank you very much for the references, they also helped me a lot with the derivation using d'Alembert's Principle.
So Goldstein's approach assuming that δS = 0 for the non-holonomic constrains of the form:

would only be correct if these constraints are actually integrable, that is only if the constraints are actually holonomic?

 

[jv07cs]

First of all, that's the correct application, leading to the correct equations of motion, and it's equivalent to the results of D'Alembert's Principle. It's all correct in the 2nd edition of Goldstein's book (I don't know the 1st editition). In the 3rd edition some new authors thought that they had to modernize this great book and by an elementary mistake, they derived wrong dynamics for non-holonomic constraints.

The point is that the non-holnomic constraints are constraints that have to imposed on the variations in Hamilton's principle, and there by definition $\delta t=0$. That's also clear from the alternative ansatz, using D'Alembert's principle, which of course leads to the same equations, i.e., the Euler-Lagrange equations of the 1st kind for non-holonomic constraints.

I can't answer the question about "semi-holonomic constraints", because I don't know the definition of this term.

Thank you very much for your reply. I believe I understood the derivation of these Lagrange Equations for the Non Holonomic case using d'Alembert's Principle and it would make sense to me if I could say that it is valid to assume that Hamilton's Principle, δS = 0, applies to this case because, if it does, I get the same equation that d'Alembert's Principle gives me. Would this be correct? Would it be a valid justification if I said that it is valid to assume that Hamilton's Principle holds because, in doing so, I get to the right equations as derived from d'Alembert's Principle?

And by "semi-holonomic constraints" I mean integrable constraints.

 

[vanhees71]

Yes, that's correct. It's important to keep in mind, how the various variational principle of mechanics are defined. The Hamilton principle comes in 2 versions: (a) Lagrangian version: here the variation is done for the configuration-space trajectories. For the usual cases the Lagrangian is of the form $L(q,\dot{q},t)$, and the variation is such that $\delta t=0$ and the action is defined over a time interval $(t_1,t_2)$, and the endpoints have to be held fixed, $\delta q(t_1)=\delta q(t_2)$; (b) Hamiltonian version: here the variation is done for the phase-space trajectories, and again $\delta t=0$ and $\delta q(t_1)=\delta q(t_2)=0$, while the canonical momenta are unconstrained, i.e., $\delta p$ is completely free.

There are other variational principles, which however are usually not used anymore in the literature. You can find a nice treatment in A. Sommerfeld, Lectures on Theoretical Physics, vol. 1.

 

[wrobel]

This is a piece of my lecture notes. Hope it will also be of some use.

 

[vanhees71]

But doesn't this lead to the false "vakonomic dynamics", we've discussed pretty often in this forum in connection with this distorted 3rd edition of Goldstein's classic?

The point is that in the non-holonomic case, you have to realize the constraints as constraints for the variations in the sense of Hamilton's principle. Then you get to the same equations of motion as with d'Alembert's principle, which is no surprise since the "virtual-work principle" is just using the same variations.

So the non-holonomic constraints, that have a clear meaning should be of the form
$$a_j^k(q,t) \mathrm{d} q_k + b_j(q,t) \mathrm{d} t=0, \quad j \in \{1,\ldots,r \}.$$
Then the correct implementation of the Lagrange multipliers in Hamilton's principle is to apply the defining constraint $\delta t=0$:
$$\delta L+\lambda_j a_j^{k} \delta q^k=0.$$
Now the $\delta q^k$ ($k \in \{1,\ldots,f \}$) can be taken as independent, thanks to the introduction of the Lagrange multipliers and you get $f$ the Euler-Lagrange equations + the $r$ equations for the $\lambda_j$ from the constraints,
$$a_j^k(q,t) \dot{q}_k + b_j(q,t)=0.$$
That's, how it's correctly explained in the 2nd edition of Goldstein's book, as well as in Landau and Lifshitz and Sommerfeld vol. 1.

 

[wrobel]

But doesn't this lead to the false "vakonomic dynamics", we've discussed pretty often in this forum in connection with this distorted 3rd edition of Goldstein's classic?

We have different views on this theory, so I prefer not to start the discussion again, but merely provide the topic starter with references and proofs. Let him decide by himself.

 

[vanhees71]

At the end you have to look at the equations and finally check them by experiment. AFAIK the latter decided in favor of the non-holonomic dynamics a la d'Alembert's principle, which then is equivalent to the interpretation of the constraints given in Goldstein 2nd edition et al.

https://doi.org/10.1587/nolta.4.482

 

[wrobel]

I am sure that d'Alembert completely agrees with the experiment. That is not the point.

 

[andresB]

This question gets asked frequently here. I understand your confusion because, frankly, the situation is not well explained in many books.

The answer is as follows:

(1) We know the form of the Linear* non-holonomic Lagrange equations from the D'alembert principle (see Greenwod, classical dynamics, p 54) and from the very Newton equations (see https://iopscience.iop.org/article/10.1088/0143-0807/14/5/005/meta).

(2) Hamilton's principle leads to the same equations as before, therefore we say it's good.

(3) In the non-holonomic case, Hamilton's principle is NOT a variational principle (see greenwood again). This is, in the non-holonomic case, there is no action functional $$S=\int L\,dt$$ whose variation $$\delta S=\delta\int L\,dt=0$$ leads to the equations of motion.
Notice that, in the non-Holonomic case, Hamilton's principle must involve $\int\delta L\,dt$ and not $\delta\int L\,dt$. The equality
$$\int\delta L\,dt=\delta\int L\,dt$$
only holds in the Holonomic case (the proof is somewhere in Pars, a treatise on analytical dynamics). Equation 2.2 from your notes is wrong in the non-holonomic case.*Everything I say here is for the linear constraints 2.66, non-linear constraints are entirely different bests.

 

[vanhees71]

No, as in the non-holonomic case the action function is
$$S=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t),$$
but the variations of the $\delta q$ are not independent, but restricted by the non-holonomous constraints
$$a_{jk}(q,t) \delta q^k=0,$$
which have to be implemented using Lagrange parameters, since in the non-holonomous case, these constraings are not integrable to holonomous constraints. Since in the Hamilton variational principle $\delta t=0$ and $\delta q(t_1)=\delta q(t_2)=0$ you have
$$\delta S=\int_{t_1}^{t_2} \mathrm{d} t \delta q^k \left (\frac{\partial L}{\partial q^k} - \mathrm{d}_t \frac{\partial L}{\partial \dot{q}^k} -\lambda_j a_{jk} \right)=0,$$
Now thanks to the introduction of the $\lambda_j$ you can treat the $\delta q^k$ as if they were independently variable and fulfill the constraints (which can be of the general form),
$$a_{jk}(q,t) \dot{q}^k + b_j(q,t)=0.$$
Then you obviously get the same equations as with D'Alembert's principle (see also the quoted EJP paper).

 

[andresB]

Since in the Hamilton variational principle $\delta t=0$ and $\delta q(t_1)=\delta q(t_2)=0$ you have
$$\delta S=\int_{t_1}^{t_2} \mathrm{d} t \delta q^k \left (\frac{\partial L}{\partial q^k} - \mathrm{d}_t \frac{\partial L}{\partial \dot{q}^k} -\lambda_j a_{jk} \right)=0,$$

Please justify this step from

No, as in the non-holonomic case the action function is
$$S=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t),$$
but the variations of the $\delta q$ are not independent, but restricted by the non-holonomous constraints
$$a_{jk}(q,t) \delta q^k=0,$$

 

[wrobel]

May I ask a question first.

vanhees71:​

Please give a definition of a stationary point of the functional $S$ in the nonholonomic case.

 

[vanhees71]

Please justify this step from

You have to introduce the Lagrange parameters for the given constraint, i.e., you write
$$\delta S + \delta q^k \lambda_j a_{jk}=0.$$
Now you can assume the $\delta q^k$ being independent and use the $\lambda_j$ to fulfill the contraint. It's just the usual way Lagrange multipliers are introduced to imply constraints on the variations, if it's not possible (or not desirable) to put the contraints in holonomous form and then eliminate the constraints by expressing the original generalized coordinates, subject to constraints, by a set of independent generalized coordinates.

 

[vanhees71]

May I ask a question first.

vanhees71:​

Please give a definition of a stationary point of the functional $S$ in the nonholonomic case.

It's of course $\delta S=0$, what else?

 

[wrobel]

It's of course $\delta S=0$, what else?

Detail it please. It is the core of the whole of our issues. I gave the definition in #9 now it is important that you explain what you mean.

 

[vanhees71]

I gave it in #15, but here it's again:
$$\Delta S=\int_{t_1}^{t_2} \mathrm{d} t \delta q^k \left (\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k} \right).$$
Here, I've used the definition of variations of Hamiltonian's principle of least action, i.e., $\delta t=0$, $\delta q(t_1)=\delta q(t_2)=0$.

Now since there are non-holonomous constraints on the variations,
$$a_{jk} \mathrm{d} q^k + b_j \mathrm{d} t=0.$$
This imposes the constraints on the variations (taking into account that $\delta t=0$ in Hamilton's principle)
$$a_{jk} \delta q^k=0.$$
Now you use the usual argument with Lagrange multipliers, you add the corresponding term to the variation of the action,
$$\delta S=\int_{t_1}^{t_2} \mathrm{d} t \delta q^k \left (\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k} + \lambda_j a_j \right)=0,$$
and now you treat the variations of the $\delta q^k$ as independent and use the $\lambda_j$ to take care of the constraints, leading to the usual set of equations, which are identical to those derived from d'Alember's principle:
$$\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k} + \lambda_j a_j=0, \quad a_{jk} \dot{q}^k + b_j=0.$$

 

[wrobel]

I gave it in #15, but here it's again:

Let's drop the dependence on $t$ in the function $L$ and in the constraints.

Can you deduce your formulas from such a definition:
$$\delta S:=\frac{d}{d\varepsilon}\Big|_{\varepsilon=0}\int L(q(t,\varepsilon), q_t(t,\varepsilon))dt=0,\quad a_{ij}(q(t,\varepsilon))q_t^i(t,\varepsilon)=0,$$
where $\tilde q(t)=q(t,0)$ is the stationary point?

 

[vanhees71]

No, because this is formulating the problem in terms of a different variational ansatz, leading to vakonomic dynamics, and only experiment can deside what's the right description of the problem in question. I'm not aware of lot of work on this question, but the few I know, always come to the conclusion that the standard d'Alembert principle and the corresponding interpretation of the non-holonomous constraints in the action principle (as constraining the variations/virtual displacements).

 

[wrobel]

No, because this is formulating the problem in terms of a different variational ansatz, leading to vakonomic dynamics

Exactly! You do not have a functional to differentiate. Speaking informally, what you write is an infinite dimensional differential form of $\delta q$ and this differential form can not be presented as a differential of the Action functional for the nonholonomic case.

and only experiment can deside what's the right description of the problem in question

Nobody doubts the d'Alembert principle. The point is in the math mistake. When one says that the nonholonomic d'Alembert principle is deduced from the Least Action principle it is a mistake because the Least Action principle is formulated in #22, and as you know it leads to completely different equations .

 

[vanhees71]

There is no math mistake in either of the two approaches. The question is, how the non-holonomous constraints have to be understood in the variational principle, and for centuries it's not formulated as in #22 but in consistency with d'Alembert's principle, which is due to the success of this principle to derive equations of motion which describe the physics right.

Also the formulation as in #22 is not mathematically wrong, but it derives different equations ("vakonomic dynamics"), by solving a different variational problem. Comparison to experiment in some simple cases rather favors the prediction from d'Alembert's principle and the standard definition of the least-action principle in case of non-holonomous constraints.

 

[wrobel]

Comparison to experiment in some simple cases rather favors the prediction from d'Alembert's principle

There are articles by Alexandre Karapetyan where he shows that the nonholonomic constraints can be considered as a limit case of some field of viscous friction forces.
Alexandre Karapetyan Realization of nonholonomic constraints by viscouse friction and stability of rotation of Celtic stone" J.Appl.Math.Mech. (PMM), 45(1981), 1.