Solving this non-holonomic system using Dirac-Bergmann theory

[andresB]

I have read in some books and articles that the Dirac-Bergmann procedure to deal with constraints in phase space does not care about holonomic and Non-holonomic constraints, but I've been unable to find a single example. So, I wanted to test that assertion by solving a simple non-holonomic system.

Consider a particle (#m=1#) subject to the non-holonomic constraint $$\phi_{1}=\dot{y}-z\dot{x}=0.$$
The Lagrangian of the system is the standard one
$$L=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-V(\mathbf{r}),$$
and it is non-singular as the momentum can be found to be
$$p_{i}=\frac{\partial L}{\partial\dot{x}_{i}}=\dot{x}_{i}.$$

In phase-space, the dynamic is given by the Hamiltonian

$$H=\frac{1}{2}\left(p_{x}^{2}+p_{z}^{2}+p_{z}^{2}\right)+V(\mathbf{r})$$
constrained to obey $\phi_{1}=\dot{p_{y}}-z\dot{p_{z}}=0$. The time evolution is obtained using the Dirac bracket

$$\dot{F}=\left\{ F,H\right\} _{D}=\left\{ F,H\right\} -\sum_{i,j}\left\{ F,\phi_{i}\right\} \left(M_{ij}\right)^{-1}\left\{ \phi_{j},H\right\}, $$
where the Matrix of constraint has the following entries

$$M_{ij}=\left\{ \phi_{i},\phi_{j}\right\}.$$

Now, with only one constraint, the matrix only has one element, and since $\left\{ \phi_{1},\phi_{1}\right\} =0,$ the matrix is non-invertible and there is no Dirac Bracket.

I tried to remedy this in the usual way of the Dirac-Bergmann theory by introducing a second constraint

$$\phi_{2}=\left\{ \phi_{1},H\right\} \approx0$$
But the equations of motion that come from the Dirac bracket do not coincide with the ones from the standard Lagrangian mechanics+Lagrange multipliers method.

So, given the above Hamiltonian and the constraint, how can the correct equation of motion be found?

 

[PhDeezNutz]

I don't know the answer to your question but non-holonomic constraints have always intrigued me.

Perhaps @vanhees71 or @wrobel can shed insight.

 

[andresB]

Apparently, I made a mistake in the statement. The correct procedure is to use the modified Lagrangian
$$L'=L+\lambda\phi,$$
and then the Momenta are given by
$$p_{i}=\frac{\partial L'}{\partial\dot{x}_{i}}=\dot{x}+\lambda\frac{\partial\phi}{\partial\dot{x}_{i}}.$$
In any case, the usual Dirac-Bergmann procedure seems to lead to nowhere.

 

[vanhees71]

No! That's leading to vakonomic motion, and that's wrong. We have discussed this at length some time ago in the textbook forum, when this occurred in the infamous 3rd edition of the famous mechanics textbook by Goldstein, which was entirely correct in the 1st and 2nd edition, before it was destroyed by some new authors.

The correct treatment is to impose the non-holonomic constraint on the variations (i.e., as in d'Alembert's principle on the "virtual displacements"), i.e., the correct variational ansatz is
$$\int_{t_1}^{t_2} \mathrm{d} t [\delta L+\lambda (\delta y-z \delta x)]=0,$$
where $\lambda$ is a Lagrange parameter. The resulting equations of motion thus are
$$\frac{\partial L}{\partial x}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{x}} -\lambda z=0,$$
$$\frac{\partial L}{\partial y}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{y}} +\lambda=0,$$
$$\frac{\partial L}{\partial z}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{z}}=0,$$
$$\dot{y}-z \dot{x}=0.$$
So you have $3$ coordinates $(x,y,z)$ and $1$ Lagrange multiplier $\lambda$ and $4$ equations, as it should be.

 

[andresB]

Hi Vanhees,

I'm aware of the correct Lagrange treatment for this system. What I don't know is the how to solve it using phase space variables, and, in particular, I wanted to check the statement that the Dirac bracket formalism can be used to find the correct equation of motions*.

*I'm away from my books at the moment, so i can't quote it, but that's what Petter Mann's Lagrangian and Hamiltonian mechanics seems to say.