Is the Author's Approach to Nonholonomic Constraints Physically Correct?

[zwierz]

I would like to discuss here some points from the article Martin Swaczyna Several examples of nonholonomic mechanical systems Communications in Mathematics, Vol. 19 (2011), No. 1, 27--56
This article is available at https://eudml.org/doc/196963
Let us open page 37:

This is a classical problem. Please pay your attention to that the author does not assume the value of dog's velocity to be constant. Below he derives this fact as a consequence from equations of dog's motion.
I find that strange and physically incorrect. Any opinions?

 

[vanhees71]

This is a purely kinematic problem and well known. Why do you think the dog's velocity should be constant? It's trajectory is known as the tractrix.

 

[A.T.]

Why do you think the dog's velocity should be constant?

I think the OP means the speed (magnitude of the dog's velocity).

I also don't quite understand the problem statement. How is the velocity defined by a line? The direction can be given by the line, but what about the magnitude?

 

[zwierz]

It seems I have figured it out. The author uses Lagrange formalism so he implicitly employs hypothesis of ideal constraints. This implies that the reaction force imposed to dog's paws from the ground is perpendicular to the trajectory (the second implicit hypothesis: there are no other active forces! What a smarty dog is it.). Then the energy of the dog is preserved and consequently the absolute value of velocity is preserved. The initial absolute value of velocity should obviously be given in the statement of the problem. The author should explain such things in detail I guess.

 

[vanhees71]

Indeed the problem is formulated a bit vague. Maybe I misunderstood it as the usual tractrix problem. Reading it again it seems to rather mean that the dog is running freely and always heads towards the man. The usual definition of the "dog's curve" (in German "Hundekurve") assumes that the dog runs at a constant speed (of course the velocity is not constant, because it's direction is changing all the time). So in other words we have given the man's trajectory
$$\vec{x}_{m}(t)=(0,v_m t).$$
Now let the dog's trajectory be given by
$$\vec{x}(t)=[x(t),y(t)].$$
The dog's velocity by definition is given by
$$\dot{\vec{x}}=-v_d \frac{\vec{x}-\vec{x}_m}{|\vec{x}-\vec{x}_m|},$$
leading to the system of differential equations
$$\dot{x}=-v_d \frac{x}{\sqrt{x^2+(y-v_m t)^2}},\\
\dot{y}=-v_d \frac{y-v_m t}{\sqrt{x^2+(y-v_m t)^2}}.$$
Damit ist
$$y'(x)=\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\dot{y}}{\dot{x}}=\frac{y-v_m t}{x}$$
or
$$x y'+v_m t=y.$$
Differentiating by $t$ and using $\mathrm{d}_t y'=y'' \dot{x}$ gives
$$\dot{x} y' + x y'' \dot{x} + v_m=\dot{y}.$$
Now
$$|\vec{v}|^2=\dot{x}^2+\dot{y}^2=\dot{x}^2(1+y'^2)=v_h^2.$$
Assuming that the dog starts somewhere at $x>0$, this implies
$$\dot{x}=-\frac{v_h}{\sqrt{1+y'^2}}.$$
Thus we get
$$x y''-\frac{v_m}{v_h \sqrt{1+y'^2}}=0.$$
Substitute $z=y'$ and $v_m/v_h=A$ gives
$$x z'=\frac{A}{\sqrt{1+z^2}}.$$
Separating variables leads to
$$\mathrm{d} z \sqrt{1+z^2}=A \frac{\mathrm{d} x}{x}$$
or integrated
$$\mathrm{arsinh} z=A \ln \left (\frac{x}{x_0} \right).$$
The rest is some algebra ;-)).

 

[zwierz]

These derivations are completely banal and completely irrelevant to the matter discussed by Swaczyna and to this thread

 

[vanhees71]

Well, I don't know what this classical problem has to do with Lagrangeans etc. From the snipped provided in the OP, I couldn't get out more than the guess that it refers to this indeed not very tricky problem.

 

[zwierz]

What does prevent you from downloading the article from the reference I provided?

 

[vanhees71]

I've not seen the link, sorry. The (somewhat overcomplicated) treatment shows that I guessed right, what the author was after. My approach is equivalent to what he does.

 

[A.T.]

The initial absolute value of velocity should obviously be given in the statement of the problem. The author should explain such things in detail I guess.

Yes, it's weird that the constant speed is stated explicity for the man, but not for the dog.

 

[zwierz]

Actually Swaczyna proposed very nice and completely new approach to this old classical problem. He treated it in the dynamical aspect. Moreover, in Swaczyna's formulation this problem provides an example of nonholonomic constraint of a new type. Standard ideal nonholonomic constraints show up when one surface or curve rolls on another surface or curve without slipping. There are also some another known possibilities for nonholonomic constraints but this one is new anyway.
Unfortunately, he committed little sloppiness and it took me some time to comprehend the core of the approach.