Nikolai Varankine - Proton PoC Research Project

Angular momentum

Angular momentum is an important conserved physical quantity. It has both a direction and a magnitude, and both are conserved. It is classically represented as a pseudovector in the three-dimensions. For a dot object, computation performs cross product of the object's position vector (relative to some origin) and its linear momentum vector at this position.

This model uses so called grid method of computing instead of calculation of the integral analythically over the volume of torus body. The grid is uneven: it uses points of visual granulation that are determined by crossing of granulated angles and radius in polar coordinates. Number of these dots determines accuracy versus ideal calculation. Finally, this method provides satisfactory result suitable for the purpose of the project and capabilities of visualizatoin.

Dots inside the volume of torus body are computed using dot torus model, using full granulation crossing. In addition to the torus surface granulation, enhanced model granulates poloidal radius too. As always, Cartesian coordinates are computed with formulae of the smooth torus model.

{\begin{matrix} θ_{m} (t, f) = 2π ∙ (t ∙ k_{p} + f) \\ φ_{m} (t, f) = 2π ∙ (t ∙ k_{t} + f) \\ R_{m} (f) = R_{p} ∙ f \end{matrix}

{\begin{matrix} P_{x} (t, f_{t}, f_{p}, f_{r}) = x_{u} (θ_{m} (t, f_{p}), φ_{m} (t, f_{t}), R_{t}, R_{m} (f_{r})) \\ P_{y} (t, f_{t}, f_{p}, f_{r}) = y_{u} (θ_{m} (t, f_{p}), φ_{m} (t, f_{t}), R_{t}, R_{m} (f_{r})) \\ P_{z} (t, f_{t}, f_{p}, f_{r}) = z_{u} (θ_{m} (t, f_{p}), φ_{m} (t, f_{t}), R_{t}, R_{m} (f_{r})) \end{matrix}

\begin{matrix} P_{d} (t, f_{t}, f_{p}, f_{r}) = (P_{x} (t, f_{t}, f_{p}, f_{r}), P_{y} (t, f_{t}, f_{p}, f_{r}), P_{z} (t, f_{t}, f_{p}, f_{r})) \end{matrix}

Three kinds of sets are needed next: origin on the toroidal axis, dots in the torus volume, dots on the torus surface. Sets differ only in their crossings of granulation parameters. Two sets of dot angles amend definition of the model grid (granulation).

\begin{matrix} {\begin{matrix} ⭕ P_{c} = P_{d} (t, f_{t}, f_{p}, f_{r}) \\ ✖️ [t = T] \times [f_{t} = G_{t}] \times [f_{p} = G_{p}] \times [f_{r} = repeat (0, g_{r})] \end{matrix} \\ {\begin{matrix} ⭕ P_{b} = P_{d} (t, f_{t}, f_{p}, f_{r}) \\ ✖️ [t = T] \times [f_{t} = G_{t}] \times [f_{p} = G_{p}] \times [f_{r} = G_{r}] \end{matrix} \\ {\begin{matrix} ⭕ P_{s1} = P_{d} (t, f_{t}, f_{p}, f_{r}) \\ ✖️ [t = T] \times [f_{t} = G_{t}] \times [f_{p} = G_{p}] \times [f_{r} = G_{r1}] \end{matrix} \\ {\begin{matrix} ⭕ θ_{b} = θ_{m} (t, f_{p}) \\ ⭕ φ_{b} = φ_{m} (t, f_{t}) \\ ✖️ [t = T] \times [f_{t} = G_{t}] \times [f_{p} = G_{p}] \times [f_{r} = G_{r}] \end{matrix} \end{matrix}

Momentum, to be computed at every dot, requires its own local Cartesian coordinate system. Both normal and two tangent vectors are computed for this purpose. All vectors have unit length. Functions $f_{110}$ (z-rotation) and $f_{cp}$ (cross product) help in calculations.

\begin{matrix} N_{s} = \frac{P_{s1} - P_{c}}{R_{p}} \\ N_{b} = N_{s} [i], [i = 1 \dots count (N_{s})] \times [f_{r} = G_{r}] \\ T_{t} = f_{110} ((0, 1, 0), φ_{b}) \\ T_{p} = f_{cp} (T_{t}, N_{b}) \end{matrix}

Values of velocity are determined by which circle dots are belonging to. Amplitudes of poloidal and toroidal velocities depend on frequency of rotaton, given in full knot rotations per second. Result is determined by multiplying them by the knot coefficients. Vector of velocity of the dot is computed as geometric sum of vectors for poloidal and toroidal velocities.

\begin{matrix} V_{t} = 2π ∙ (R_{t} + (R_{p} ∙ G_{rb}) ∙ \sin (θ_{b})) ∙ f ∙ k_{t} \\ V_{p} = 2π ∙ (R_{p} ∙ G_{rb}) ∙ f ∙ k_{p} \\ V = T_{t} ∙ V_{t} + T_{p} ∙ V_{p} \end{matrix}

Provided with vectors shown above, computation of linear and angular momentums, as well as kinetic energy, is simple.

\begin{matrix} P = M_{1} ∙ V \\ L = P \times P_{b} = f_{cp} (P, P_{b}) \\ E = M_{1} ∙ V^{2} = M_{1} ∙ f_{2} (V) \\ f_{2} (N) = {N.x}^{2} + {N.y}^{2} + {N.z}^{2} \end{matrix}

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