Dirac bracket for a constrained particle

Question

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.

Answer 1

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

$$\begin{align} {\bf n}({\bf x})~:=~& \frac{{\bf N}({\bf x})}{|| {\bf N}({\bf x})||}, \cr {\bf N}({\bf x})~:=~&\frac{\partial f({\bf x})}{\partial {\bf x}},\cr || {\bf N}({\bf x})||~:=~&\sqrt{{\bf N}({\bf x})\cdot {\bf N}({\bf x})},\end{align}\tag{1}$$

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

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

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

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

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

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

III) At this point we will assume that $f$ and $\chi$ are second class

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

where we have define a Riemann bracket

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

We will then simply postulate that the time evolution is governed by the Dirac bracket

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

Note that the second class constraints are preserved under time evolution, so the proposal (7) is well-defined, and there is no need for tertiary constraints, etc. The Dirac bracket reads

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

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.