# Dirac bracket for a constrained particle

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, $$$$H=\frac{\boldsymbol p^2}{2m}+V(\boldsymbol x).\tag{4.4}$$$$ They state that the Dirac brackets for the time evolution of the canonical coordinates are given by, $$$$\boldsymbol{\dot x}=\frac{1}{m}[\boldsymbol p-(\boldsymbol p\cdot\boldsymbol n)\cdot \boldsymbol n]=\frac{\boldsymbol p}{m}$$$$ $$$$\boldsymbol{\dot{p}}=\boldsymbol F-[\boldsymbol F\cdot \boldsymbol n+\frac{1}{m}\boldsymbol p\cdot [(\boldsymbol p\cdot \frac{\partial }{\partial \boldsymbol x})\boldsymbol n]]\boldsymbol n. \tag{4.5}$$$$ There is little information given regarding this particular problem but just above the quoted equations they describe two second class constraints $$\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.

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.
• 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). Commented Oct 30, 2015 at 22:00
• $\uparrow$ Yes. Commented Oct 30, 2015 at 22:03