Solving this non-holonomic system using Dirac-Bergmann theory

In summary, the Dirac-Bergmann procedure for dealing with constraints in phase space does not take into account holonomic and non-holonomic constraints. However, in a simple non-holonomic system where the Lagrangian is non-singular, the momentum can be found and the Hamiltonian can be constrained to obey the non-holonomic constraint using the Dirac bracket. This leads to the correct equations of motion. However, if there is only one constraint, the matrix of constraints is non-invertible and the Dirac bracket cannot be used. Instead, the correct treatment is to impose the non-holonomic constraint on the variations in the Lagrangian.
  • #1
andresB
626
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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?
 
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  • #2
I don't know the answer to your question but non-holonomic constraints have always intrigued me.

Perhaps @vanhees71 or @wrobel can shed insight.
 
  • #3
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.
 
  • #4
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 occured 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.
 
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  • #5
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.
 

Related to Solving this non-holonomic system using Dirac-Bergmann theory

1. What is a non-holonomic system?

A non-holonomic system is a type of physical system that does not satisfy the holonomic constraints, which are restrictions on the possible motions of the system. This means that the system's degrees of freedom are not independent and cannot be described by a set of generalized coordinates.

2. What is Dirac-Bergmann theory?

Dirac-Bergmann theory is a mathematical framework used to describe and analyze constrained mechanical systems. It was developed by Paul Dirac and Felix Bergmann and is based on Hamiltonian mechanics.

3. How is Dirac-Bergmann theory used to solve non-holonomic systems?

Dirac-Bergmann theory provides a systematic way to incorporate constraints into the equations of motion for a non-holonomic system. This is done by introducing Lagrange multipliers, which act as additional forces that enforce the constraints. The resulting equations of motion can then be solved using standard techniques from Hamiltonian mechanics.

4. What are the advantages of using Dirac-Bergmann theory to solve non-holonomic systems?

One of the main advantages of using Dirac-Bergmann theory is that it allows for a more elegant and systematic treatment of constraints compared to other approaches. It also provides a better understanding of the physical significance of the constraints and their effects on the system's dynamics.

5. Are there any limitations to using Dirac-Bergmann theory for solving non-holonomic systems?

While Dirac-Bergmann theory is a powerful tool for solving non-holonomic systems, it does have some limitations. It can be quite mathematically complex and may not always be applicable to all types of constraints. Additionally, the resulting equations of motion can be difficult to solve in some cases, requiring numerical methods or approximations.

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