Particle on a Torus Knot: A Hamiltonian Analysis

Abstract

We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac’s formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific torus. The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed torus knot.

Appendices

Appendix 1

In the terminology of Dirac constraint analysis [26], the noncommutating constraints are termed as SCC and the commutating constraints, that induces local gauge invariance, are named First Class Constraints (FCC). In a generic Second Class system with n SCCs $\chi _i$, $i=1,2,..n$, the modified symplectic structure (or Dirac brackets) are defined in the following way,

$$\begin{aligned} \{A,B\}^*=\{A,B\}-\{A,\chi _i\}\{\chi ^i,\chi ^j\}^{-1}\{\chi _j,B\}, \end{aligned}$$

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where $\{\chi ^i,\chi ^j\}$ is the invertible constraint matrix. From now on we will use $\{,\}$ notation instead of $\{,\}^*$ for Dirac brackets.

In Cartesian coordinate system, the non-zero constraints matrix element reads

$$\begin{aligned} \{\psi _1(r),\psi _2(p,r)\}=\{\psi _1(r),p.A(r)\}=\frac{\partial \psi _1}{\partial x_i}A_i=A_iA_i=A^2 \end{aligned}$$

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where, $A_i=\frac{\partial \psi _1}{\partial x_i},~A^2=\frac{4a^2}{\phi }+\frac{\alpha ^2}{r^2-x_3^2}.$ Thus the inverse matrix element can be written as,

$$\begin{aligned} \{\psi _1(r),\psi _2(p,r)\}^{-1}=-\frac{1}{A^2} \end{aligned}$$

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The Dirac bracket can be computed in the following way,

$$\begin{aligned} \{x_i,p_j\}_{D.B.}= & {} \delta _{ij}-\{x_i,\psi _2\}\{\psi _2,\psi _1\}^{-1}\{\psi _1,p_j\} \nonumber \\= & {} \delta _{ij}-\{x_i,p.A\}\{\psi _2,\psi _1\}^{-1}\frac{\partial \psi _1}{\partial x_j}=\delta _{ij}-\frac{A_iA_j}{A^2}. \end{aligned}$$

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Appendix 2

In toroidal coordinate system, the non-zero constraints matrix element is

$$\begin{aligned} \{\chi _1,\chi _2\}=\frac{p^2 \sinh ^2\eta +q^2}{\sinh ^2\eta }. \end{aligned}$$

Thus the inverse matrix element can be written as

$$\begin{aligned} \{\chi _1,\chi _2\}^{-1}=-\frac{\sinh ^2\eta }{p^2\sinh ^2\eta +q^2}. \end{aligned}$$

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The Dirac bracket

$$\begin{aligned} \{\phi ,p_\phi \} \end{aligned}$$

can be computed in the following way,

$$\begin{aligned} \{\phi ,p_\phi \}_{D.B.}=\{\phi ,p_\phi \}-\{\phi ,\chi _2\}\{\chi _2,\chi _1\}^{-1}\{\chi _1,p_\phi \}\nonumber \\ =1-\frac{q^2}{p^2\sinh ^2\eta +q^2}=\frac{\sinh ^2\eta }{\alpha ^2+\sinh ^2\eta } \end{aligned}$$

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where $\alpha =-\frac{q}{p}$.

Appendix 3

We provide some useful identities connecting the toroidal coordinate system to Cartesian coordinate system. The unit vectors are related as,

$$\begin{aligned} \hat{i}= & {} \left( \cosh \eta \cos \phi -\frac{\sinh ^2\eta \cos \phi }{\cosh \eta -\cos \theta }\right) ~~\hat{\eta }-\frac{\sinh \eta \sin \theta \cos \phi }{\cosh \eta -\cos \theta }~~\hat{\theta }-\sin \phi ~~\hat{\phi },\\ \hat{j}= & {} \left( \cosh \eta \sin \phi -\frac{\sinh ^2\eta \sin \phi }{\cosh \eta -\cos \theta }\right) ~~\hat{\eta }-\frac{\sinh \eta \sin \theta \sin \phi }{\cosh \eta -\cos \theta }~~\hat{\theta }+\cos \phi ~~\hat{\phi },\\ \hat{k}= & {} -\frac{\sin \theta \sinh \eta }{\cosh \eta -\cos \theta }~~\hat{\eta }+\frac{\cos \theta \cosh \eta -1}{\cosh \eta -\cos \theta }~~\hat{\theta }. \end{aligned}$$

The conjugate momenta are related as,

$$\begin{aligned} p_x=\frac{\cos \phi (1-\cos \theta \cosh \eta )p_\eta }{a}-\frac{(\sinh \eta \cos \phi \sin \theta ) p_\theta }{a}-\frac{\sin \phi (\cosh \eta -\cos \theta )p_\phi }{a\sinh \eta }, \end{aligned}$$

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$$\begin{aligned} p_y=\frac{\sin \phi (1-\cos \theta \cosh \eta )p_\eta }{a}-\frac{(\sinh \eta \sin \phi \sin \theta ) p_\theta }{a}+\frac{\cos \phi (\cosh \eta -\cos \theta )p_\phi }{a\sinh \eta } , \end{aligned}$$

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$$\begin{aligned} p_z=\frac{(\cosh \eta \cos \theta -1)p_\theta }{a}-\frac{\sin \theta \sinh \eta }{a}p_\eta . \end{aligned}$$

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