- #1

andresB

- 626

- 374

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?