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?



$\boldsymbol x$=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.


1 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.


  1. S. Nguyen & L.A. Turski, Examples of the Dirac approach to dynamics of systems with constraints Phys. A290 (2001) 431; Section 4.
  • $\begingroup$ 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). $\endgroup$ Oct 30, 2015 at 22:00
  • 1
    $\begingroup$ $\uparrow$ Yes. $\endgroup$
    – Qmechanic
    Oct 30, 2015 at 22:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.