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I am trying to work through a simple example of how to use the Dirac bracket from the following paper. In particular section 4 where the authors consider a constrained particle with the following Hamiltonian, \begin{equation} H=\frac{\boldsymbol p^2}{2m}+V(\boldsymbol x)\tag{4.4} \end{equation} They state that the Dirac brackets for the time evolution of the canonical coordinates are given by, \begin{equation} \boldsymbol{\dot x}=\frac{1}{m}[\boldsymbol p-(\boldsymbol p\cdot\boldsymbol n)\cdot \boldsymbol n]=\frac{\boldsymbol p}{m} \end{equation} \begin{equation} \boldsymbol{\dot{p}}=\boldsymbol F-[\boldsymbol F\cdot \boldsymbol n+\frac{1}{m}\boldsymbol p\cdot [(\boldsymbol p\cdot \frac{\partial }{\partial \boldsymbol x})\boldsymbol n]]\boldsymbol n \tag{4.5} \end{equation} There is little information given regarding this particular problem but just above the quoted equations they describe two second class constraints $$\Theta_1=f(\boldsymbol x)\qquad\text{and}\qquad\Theta _2=\boldsymbol p\cdot \frac{\partial f}{\partial \boldsymbol x,}\tag{4.1}$$ however I do not understand how this leads to the quoted expressions above, are we to take $f=\boldsymbol n$ the unit normal vector?

Legend

$H$=Hamiltonian;

$\boldsymbol x$=coordinate;

$\boldsymbol p$=momentum conjugate to $\boldsymbol x$;

$\boldsymbol n$= unit normal vector to constraint surface;

$\Theta_i$=constraint equation;

$\boldsymbol F$=force;

My attempts are shockingly poor, so are not really worth showing. I am looking for general pointers or tips on how to approach this problem.

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I) It seems OP's main question was spurred by a typo below eq. (4.2) in Ref. 1 in the formula for the unit normal vector

$$\tag{1} {\bf n}({\bf x})~:=~ \frac{{\bf N}({\bf x})}{|| {\bf N}({\bf x})||}, \qquad {\bf N}({\bf x})~:=~\frac{\partial f({\bf x})}{\partial {\bf x}},\qquad || {\bf N}({\bf x})||~:=~\sqrt{{\bf N}({\bf x})\cdot {\bf N}({\bf x})}.\qquad$$

to the constrained surface $\{{\bf x}\in \mathbb{R}^n | f({\bf x})=0 \}$ in the position space $\mathbb{R}^n$.

II) It is interesting to generalize the setting of Ref. 1. Let us consider an $n$-dimensional Riemannian manifold $(M,g)$ endowed with two functions $f, V:M\to\mathbb{R}$, called the constraint and the potential, respectively. The Lagrangian is

$$\tag{2} L~=~L_0 +\lambda f, \qquad L_0~:=~ \frac{1}{2}\dot{x}^i g_{ij}\dot{x}^j-V,$$

where $\lambda$ is a Lagrange multiplier. The extended phase space is the cotangent bundle $T^{\ast}M$ equipped with the canonical Poisson bracket. The bare Hamiltonian is

$$\tag{3} H_0~=~\frac{1}{2}p_i g^{ij}p_j+V. $$

We have a constraint $f \approx 0$. We also have a secondary constraint

$$\tag{4} \chi~:=~\{f,H_0\}_{PB}~=~p_i \nabla^i f .$$

III) At this point we will deviate from the usual Dirac-Bergmann recipe, which instructs us to look for tertiary constraints, etc. In fact, $\{\chi,H_0\}_{PB}$ does in general not vanish! Instead we will look at a modified Hamiltonian system, which generalized the system in Ref. 1, but which in general will be inequivalent to the standard Dirac-Bergmann recipe.

We will assume that $f$ and $\chi$ are second class

$$\tag{5} 0~\neq~ \{f,\chi\}_{PB}~=~ (f,f)_{RB}.$$

where we have define a Riemann bracket

$$\tag{6} (f,f)_{RB}~:=~\partial_if~g^{ij}~\partial_jf. $$

Then we will postulate that the time evolution is governed by the Dirac bracket

$$\tag{7} \dot{x}^i ~=~\{x^i, H_0\}_{DB} , \qquad \dot{p}_j ~=~\{p_j, H_0\}_{DB}. $$

We stress again that this system (7) in general will be inequivalent from the Hamiltonian system derived from the full Dirac-Bergmann recipe. Note that the second class constraints are preserved under time evolution, so the proposal (7) is well-defined. The Dirac bracket reads

$$ \tag{8} \{ a,b\}_{DB}~=~\{ a,b\}_{PB}+\frac{ \{ a,f\}_{PB}~ \{ \chi,b\}_{PB}-\{ a,\chi\}_{PB} ~\{ f,b\}_{PB}}{(f,f)_{RB}} ,$$

where $a,b: T^{\ast}M\to\mathbb{R}$ are two arbitrary functions. Eqs. (4.3) and (4.5) in Ref. 1 are special cases of eqs. (8) and (7), respectively.

References:

  1. S. Nguyen & L.A. Turski, Examples of the Dirac approach to dynamics of systems with constraints Phys. A290 (2001) 431; Section 4.
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  • $\begingroup$ Once again I am very impressed by your ability to simplify a complicated topic, thank you for your time Qmechanic. Can I substitute $(f,f)_{RB}=\{f,\chi\}_{PB}$ and $\chi=\{f,H_0\}_{PB}$ into the final formula? (for secondary constraints). $\endgroup$ – AngusTheMan Oct 30 '15 at 22:00
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    $\begingroup$ $\uparrow$ Yes. $\endgroup$ – Qmechanic Oct 30 '15 at 22:03

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