Extending Hamilton's principle to systems with constraints

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Stuck in Goldstein's Classical Mechanics

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I'm working my way through Goldstein's Classical Mechanics and have followed the arguments until section 2.4 (Extending Hamilton's Principle to Systems with Constraints). In the second paragraph, Goldstein states that "When we derive Lagrange's equations from either Hamilton's or D'Alembert's principle, the holonomic constraints appear in the last step when the variations in the ##q_i## were considered independent of eachother." I believe this refers back to equation 2.17: $$\delta J = \int_1^2\sum_i\bigg(\frac{\partial f}{\partial y_i} - \frac{d}{dx}\frac{\partial f}{\partial \dot{y}_i}\bigg)\delta y_i dx$$ with ##\delta y_i = \big(\frac{\partial y_i}{\partial \alpha}\big)_0 d\alpha##. I understand why deriving the Lagrange equations requires the variations to be independent. However, Goldstein then says "the virtual displacements in the ##\delta q_I##'s may not be consistent with the constraints. If there are ##n## variables and ##m## constraint equations ##f_\alpha## of the form ##f(\mathbf{r}_1, \mathbf{r}_2,...,t) = 0##, the extra virtual displacements are eliminated by the method of Lagrange undetermined multipliers. "

I have more questions about the derivation that follows, but first of all, I don't understand what it means for the virtual displacements in the ##\delta q_I##'s to not be consistent with the constraints. Maybe if I clear this up, the remaining argument will be clear. Thanks!
 
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  • #2
vanhees71
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Be careful with Goldstein. The treatment of anholonomous constraints is plain wrong! It leads to the socalled vakonomic dynamics, which is incorrect:

https://arxiv.org/abs/1204.3387

The correct treatment is as with d'Alembert's principle. The non-holonomic constraints of the form [EDIT: corrected in view of #4]
$$f_k(t,q) \dot{q}_k=0$$
have NOT to be introduced in a naive Lagrange-multiplier way as claimed in Goldstein's book but as a constraint for the variations (i.e., in an equivalent way as in d'Alembert's principle as "virtual discplacements"), i.e., the variations must obey the constraints
$$f_k(t,q) \delta q_k=0.$$
Also see Landau-Lifshits vol. 1, which is anyway a great book on classical point mechanics employing the action-principle-first approach.
 
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  • #3
We have two fixed points in our co-ordinate space and we're looking at all possible smooth curves joining the two points. But owing to the constraints, it may not be possible to choose our virtual displacements as we please.
For example, in 3 Dimensional Space (which is our co-ordinate space), suppose we restrict/constrain motion to take place only in the ##xy## plane. Then that essentially means that we cannot vary the ##z## co-ordinate arbitrarily, it must satisfy ##\delta z = 0##. This is the constraint we impose and in that case it is clear that ##\delta x##, ##\delta y## and ##\delta z## are not linearly independent (there exists a non-trivial linear combination which gives 0). This is what he meant when he said the virtual displacements may not be consistent with the constraints. The given constraints are then taken care of by using the Method of Lagrange Multipliers which he goes on to describe in the next section.
 
  • #4
It is very bad idea to study math by physics textbooks, on the other hand there are a lot of good mathematical textbooks in variational analysis but these books are hard for physics students and for the first reading. Dilemma.
"virtual discplacements"), i.e., the variations must obey the constraints

fk(t,q)δqk+g(t,q)=0.​
variations must obey the following equations
$$f_k(t,q)\delta q_k=0.$$
Also see Landau-Lifshits vol. 1, which is anyway a great book
then why do they make exactly the mistake which is pointed out in https://arxiv.org/abs/1204.3387
 
  • #5
vanhees71
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I thought general linear constraints are allowed. Why do you think ##g=0## is necessary? But it's fine with me to set ##g=0## to discuss one special case, where there's consensus in the literature that the d'Alembert EoM are the correct ones.

Where in landau-Lifshitz vol. 1 is made this mistake? They constrain the variations in Hamilton's principle as in the d'Alembert principle, and thus they get the same equations as with the d'Alembert principle. If I understood the quoted paper right these are the correct equations. An interesting question is, whether there are experimental tests to check them ;-)).

Another nice source, where they consider only the case linear in the ##\dot{v}## as you say, is

http://www.diva-portal.org/smash/get/diva2:528335/FULLTEXT02
 
  • #6
I thought general linear constraints are allowed. Why do you think g=0g=0 is necessary?
I do not think so. I just say that virtual displacements satisfy the corresponding homogeneous equations.

Where in landau-Lifshitz vol. 1 is made this mistake?
They deduce the equations of nonholonomic mechanics from Hamilton's Principle. It is well known (for example, from the article you cited) that in case of nonholonomic constraints, Hamilton's principle implies the vakonomic dynamics equations. And vaconomic dynamics is not the same as nonholonomic one.
 
  • #7
vanhees71
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You are right; ##g=0## doesn't make sense (i corrected my previous posting), but Landau Lifhitz do it right, imposing the contraints on the variations not in the wrong way leading to vakonomic dynamics (see Sect. 38). They write the contraints as
$$\sum_i c_{i \alpha} \dot{q}_{i}=0,$$
but then impose them with Lagrange multipliers in the form
$$\sum_i c_{\alpha i} \delta q_{i}=0,$$
leading to
$$\mathrm{d}_t \frac{\partial L}{\partial \dot{q}_i}-\frac{\partial L}{\partial q_i}=\sum_{\alpha} \lambda_{\alpha} c_{\alpha i}.$$
This are the same equations you get with d'Alembert's principle, which are considered to be the right equations in this case.

The WRONG way leading to the vakonomic dynamics is when imposing the constraints with Lagrange multipliers using the form with the generalized velocities, i.e., minimizing the Lagrangian
$$\tilde{L}=L-\sum_{i,\alpha} \lambda_{\alpha} c_{i \alpha} \dot{q}_{i}), \qquad [\text{WRONG}],$$
leading to
$$\mathrm{d}_t \frac{\partial \tilde{L}}{\partial \dot{q}_i} - \partial{\tilde{L}}{\partial q_i}=0,$$
i.e.,
$$\mathrm{d}_t \frac{\partial {L}}{\partial \dot{q}_i} - \partial{{L}}{\partial q_i}=\mathrm{d}_t \sum_{\alpha} \lambda_{\alpha} c_{i \alpha} - \sum_{\alpha} \lambda_{\alpha} \frac{\partial c_{\alpha j}}{\partial q_i} \dot{q}_j,$$
which is obviously only equivalent to the correct equations, if the constraints are integrable to holonomic contraints and thus lead in the case of nonholonomic constraints to WRONG equations.
 
  • #8
vanhees71
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It's also correct in Sommerfeld, Lectures on Theoretical Physics 1.
 
  • #9
1) Landau Lifhitz write the correct equations of nonholonomic mechanics. But their argument is wrong. Their argument (if only they would do everything right) leads to the vaconomic mechanics.

2) Youcan consider constrains of the form ##f_i(t,q)\dot q^i+g(t,q)=0.## The corresponding equation for virtual displacements is ##f_i(t,q)\delta q^i=0.##
 
  • #10
vanhees71
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Ok about 2.

I still don't see what's wrong with Landau Lifshitz and Sommerfeld though. They get the correct equations using Hamilton's principle and imposing the nonholonomic constraints as constraints for the variations. Then also 2) follows, i.e., you have to read the equation as
$$f_i(t,q) \delta q^i + \delta t g(t,q)=0.$$
Since in the Hamilton principle ##\delta t=0## you get ##f_i \delta q^i=0##.

It's not so clear to me, why these are the correct equations though. In other words, why does d'Alembert's principle hold for nonholonomic constraints? At the end it's just an empirical fact that these are the right equations of course.
 
  • #11
Here is a very nice discussion of all these things: Classical Dynamics by Donald T. Greenwood.
You may find djvu in the internet:)
 
  • #12
vanhees71
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Great book. So the conclusion is that one must assume in Hamilton's principle that the nonholonomic constraints of the discussed kind must be imposed for the variations, i.e., in the above notation by LL
$$c_{\alpha j}(q,t) \delta q_j=0.$$
It's justified, because this leads to the same EoM as d'Alembert's principle of virtual work.

I guess, this is then to be taken as the most general principle of mechanics, and to me it seems to be indeed more general than "naive" Newtonian mechanics a la ##F=m a##.

The only thing, which seems not to be solved is, if you have more general constraints of the form ##f_{\alpha}(q,\dot{q},t)=0##, i.e., which are not linear in the ##\dot{q}##. There it seems that also d'Alembert's principle cannot be used. On the other hand, I'm not aware of any example where such a most general constraint is needed.
 
  • #13
We have two fixed points in our co-ordinate space and we're looking at all possible smooth curves joining the two points. But owing to the constraints, it may not be possible to choose our virtual displacements as we please.
For example, in 3 Dimensional Space (which is our co-ordinate space), suppose we restrict/constrain motion to take place only in the ##xy## plane. Then that essentially means that we cannot vary the ##z## co-ordinate arbitrarily, it must satisfy ##\delta z = 0##.
I think I understand. In this case, the original coordinates ##x,y,z## are not linearly independent as the derivation of Lagrange's equations requires, correct? And that's why we must introduce the Lagrange multipliers?

This is essentially the example he introduces: A smooth solid of hemisphere with a mass sliding down and initial motion in the ##xz## plane.

This is the constraint we impose and in that case it is clear that ##\delta x##, ##\delta y## and ##\delta z## are not linearly independent (there exists a non-trivial linear combination which gives 0). This is what he meant when he said the virtual displacements may not be consistent with the constraints.
So to summarize, the virtual displacements not being consistent with the constraints implies that with the given constraints, the displacements cannot be linearly independent?

The next step in the derivation is modifying the action integral to be:
$$I =\int_1^2\bigg(L+\sum_{\alpha = 1}^m \lambda_{\alpha}f_a\bigg)dt$$
and allowing the ##q_{\alpha}## and the ##\lambda_{\alpha}## to vary independently to obtain ##n+m## equations. The variations of the ##\lambda_{\alpha}##'s give the ##m## constraint equations. The variations of the ##q_i##'s give:
$$\delta I= \int_1^2 dt\bigg(\sum_{i=1}^n \bigg(\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} -\frac{\partial L}{\partial q_i} + \sum_{\alpha = 1}^m \lambda_{\alpha}\frac{\partial f_a}{\partial q_i}\bigg)\delta q_i \bigg) = 0$$

Ahh, I believe it's starting to make sense now. At this point, we have ##m## constraint equations. We can choose the ##\lambda_{\alpha}## such that

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_k} -\frac{\partial L}{\partial q_k} + \sum_{\alpha = 1}^m \lambda_{\alpha}\frac{\partial f_a}{\partial q_k} = 0$$

is true for ##k=1,\ldots,m##, giving ##m## additional equations. Then, we can assume the remaining ##q_i## and therefore the remaining ##\delta q_i##'s for ##m+1,\ldots,n## are independent, giving the final ##n-m## equations in the same form as above.

In short, is the advantage of this method that the coordinates ##q_i## don't need to be independent from the outset, as long as you have the holonomic constraint equations that specify the relationship between the coordinates?
 
  • #14
Be careful with Goldstein. The treatment of anholonomous constraints is plain wrong! It leads to the socalled vakonomic dynamics, which is incorrect:

https://arxiv.org/abs/1204.3387
Thank you for the warning and the resource! In the derivation I'm looking at, he's still considering systems with holonomic constraints. Is Goldstein's treatment with Lagrange undetermined multipliers appropriate here? He only gets to nonholonomic constraints after introducing the holonomic case.
 
  • #15
vanhees71
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Yes, for holonomic constraints everything is fine. I just learnt from @wrobel in this thread that I was myself not very clear about non-holonomic constraints and Hamilton's principle. Of the standard sources I know correct are Landau Lifshitz vol. 1 and Sommerfeld vol. 1.
 

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