Angular momentum bounds in particle systems

Abstract

Four expressions involving sums of position and velocity coordinates bounding the total angular momentum of particle systems, and by extension of any continuous or discontinuous material systems, are derived which are tighter for any particle configuration than similar inequalities derived by Sundman (Acta Math 36:105, 1913), Saari (Conference board of the mathematical sciences, No. 104, 2005. http://books.google.ch/books?id=uNXgf2X1GeYC) and Scheeres (Celest Mech Dyn Astron 113:291–320, 2012. https://doi.org/10.1007/s10569-012-9416-0). Eight distinct inequalities can thus be ordered according to their tightness to angular momentum.

Appendices

Appendix I: Hill’s inequations

In Hill’s book (1964, pp. 29–30) is proved an important double inequation that we display here in more current notations. For any pair of real three-vector sets $\{\mathbf {a}_i\}$ and $\{\mathbf {b}_i\}$, $i=1\ldots N$, then

$$\begin{aligned} \sum _i {\mathbf {a}}_i^2\sum _j {\mathbf {b}}_j^2 - \left( \sum _i {\mathbf {a}}_i\cdot {\mathbf {b}}_i\right) ^2\ge & {} \left( \sum _i \left| {\mathbf {a}}_i\wedge {\mathbf {b}}_i \right| \right) ^2 \end{aligned}$$

(52)

$$\begin{aligned}\ge & {} \left( \sum _i {\mathbf {a}}_i\wedge {\mathbf {b}}_i \right) ^2 . \end{aligned}$$

(53)

The demonstration uses the vector relations

$$\begin{aligned} {\mathbf {a}}\cdot {\mathbf {b}} = a b \cos \theta \quad \mathrm {and} \quad \left| {\mathbf {a}}\wedge {\mathbf {b}}\right| = a b \sin \theta . \end{aligned}$$

(54)

where $\theta$ is the angle between the vectors ${\mathbf {a}}$ and ${\mathbf {b}}$, and a, b are the lengths of ${\mathbf {a}}$ and ${\mathbf {b}}$.

Thus, subtracting the right side from the left side in Eq. (52), expanding the squared sums in double sums and using the trigonometric identity $\cos (\theta -\eta ) = \cos (\theta )\cos (\eta ) + \sin (\theta )\sin (\eta )$, we obtain,

$$\begin{aligned}&\sum _{i,j} a_i^2 b_j^2 - a_i b_i a_j b_j \cos \theta _i \cos \theta _j - a_i b_i a_j b_j \sin \theta _i \sin \theta _j \end{aligned}$$

(55)

$$\begin{aligned}&\qquad =\sum _{i,j} a_i^2 b_j^2 - a_i b_i a_j b_j \cos (\theta _i -\theta _j). \end{aligned}$$

(56)

Each diagonal term with $i=j$ vanishes, and each pair of nondiagonal terms $i\ne j$ provides a nonnegative contribution to the sum,

$$\begin{aligned} a_i^2 b_j^2+a_j^2 b_i^2 -2a_i b_ja_jb_i\cos (\theta _i -\theta _j) \ge (a_ib_j-a_j b_i)^2 \ge 0. \end{aligned}$$

(57)

The equality in Eq. (52) is reached when all the ratios $a_i/b_j$ are equal, and all the angles $\theta _i$ are equal.

Finally, Eq. (53) follows from the triangle inequality $|a|+|b|\ge |a+b|$.

Using the triangle inequality for the quadratic norm, $(|a|+|b|)^2 \ge a^2+b^2$, one can also express another inequality

$$\begin{aligned} \sum _i {\mathbf {a}}_i^2\sum _j {\mathbf {b}}_j^2 - \left( \sum _i {\mathbf {a}}_i\cdot {\mathbf {b}}_i\right) ^2\ge & {} \left( \sum _i \left| {\mathbf {a}}_i\wedge {\mathbf {b}}_i \right| \right) ^2 \end{aligned}$$

(58)

$$\begin{aligned}\ge & {} \sum _i \left( {\mathbf {a}}_i\wedge {\mathbf {b}}_i \right) ^2, \end{aligned}$$

(59)

which is distinct from Eq. (53), that is, in different cases either the right-hand side of Eq. (53) or that of Eq. (59) is the largest term.

Appendix II: Weighted Binet–Cauchy identities

For m pairs of real or complex N-dimensional vectors $a_{ki}$ and $b_{ki}$, $k \in [1,\ldots m]$, and the real or complex N-dimensional vector $w_{i}$, $i \in [1,\ldots N]$, we have the following simplifications of antisymmetric double sums.

1.1 1. Odd m

If m is odd,

$$\begin{aligned} \sum _{i=1, j=1}^{N} w_i w_j \prod _{k=1}^m \left( a_{ki} b_{kj} - a_{kj} b_{ki}\right) = 0. \end{aligned}$$

(60)

This follows from the antisymmetry of the global sum, each term occurring twice with opposite signs.

1.2 2. Even m

If m is even,

$$\begin{aligned}&\sum _{i=1, j=1}^{N} w_i w_j\prod _{k=1}^m \left( a_{ki} b_{kj} - a_{kj} b_{ki}\right) = 2\sum _{i=1}^{N-1}\sum _{j=i+1}^{N} w_i w_j \prod _{k=1}^m \left( a_{ki} b_{kj} - a_{kj} b_{ki}\right) . \end{aligned}$$

(61)

This follows from the symmetry of the global sum, each term occurring twice with same signs.

1.3 3. $m=2$

For $m=2$, we have a weighted Binet–Cauchy identity:

$$\begin{aligned}&\sum _{i=1}^{N-1} \sum _{j=i+1}^N w_i w_j \left( a_{1i} b_{1j} - a_{1j} b_{1i}\right) \left( a_{2i} b_{2j} - a_{2j} b_{2i}\right) \nonumber \\&\qquad = \sum _i w_i a_{1i} a_{2i} \sum _j w_j b_{1j} b_{2j} - \sum _i w_i a_{1i} b_{2i} \sum _j w_j a_{2j} b_{1j}. \end{aligned}$$

(62)

If $b_{1i}=b_{2i}$, we have, dropping the index k for the b’s:

$$\begin{aligned}&\sum _{i=1}^{N-1} \sum _{j=i+1}^N w_i w_j \left( a_{1i} b_j - a_{1j} b_i\right) \left( a_{2i} b_j - a_{1j} b_i\right) \nonumber \\&\qquad = \sum _i w_i a_{1i} a_{2i} \sum _j w_j b_j^2 - \sum _i w_i a_{1i} b_i \sum _j w_j a_{2j}b_j. \end{aligned}$$

(63)

If $a_{1i}=\bar{a}_{2i}$ and $b_{1i}=\bar{b}_{2i}$, we have a weighted version of Lagrange’s identity:

$$\begin{aligned}&\sum _{i=1}^{N-1} \sum _{j=i+1}^N w_i w_j \left| a_i b_j - a_j b_i\right| ^ 2 = \sum _i w_i |a_i|^2 \sum _j w_j |b_j|^2 - \left| \sum _i w_i a_i b_i\right| ^2. \end{aligned}$$

(64)

For real nonnegative weights $w_i$, since the left-hand side is nonnegative, a weighted form of Cauchy–Schwarz inequality follows:

$$\begin{aligned} \sum _i w_i |a_i|^2 \sum _j w_j |b_j|^2\ge & {} \left| \sum _i w_i a_i b_i\right| ^2. \end{aligned}$$

(65)

Further, if $b_i=1$, we have

$$\begin{aligned}&\sum _{i=1}^{N-1} \sum _{j=i+1}^N w_iw_j \left| a_i-a_j \right| ^ 2 = \sum _i w_i|a_i|^2\sum _jw_j -\left| \sum _iw_i a_i \right| ^2. \end{aligned}$$

(66)

For larger even m, it is straightforward to derive further similar equalities and inequalities.

Appendix III

We can calculate the inverse of the axial moment of inertia, ${\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}^{-1}$ (or pseudo-inverse ${\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}^{\dagger }$ for degenerate cases) with the following procedure working in all configurations (Pfenniger 1994):

• 1. Calculate the determinant $\bigl |{\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}\bigr |$ with

  $$\begin{aligned} \bigl |{\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}\bigr |= & {} {\mathcal I}_{xx} {\mathcal I}_{yy} {\mathcal I}_{zz} + 2 {\mathcal I}_{xy} {\mathcal I}_{yz} {\mathcal I}_{zx} - \left( {\mathcal I}_{xx}{\mathcal I}_{yz}^2 + {\mathcal I}_{yy}{\mathcal I}_{zx}^2 + {\mathcal I}_{zz}{\mathcal I}_{xy}^2 \right) . \end{aligned}$$

  (67)
• 2.

  • If $\bigl |{\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}\bigr |\ne 0$, the general case, then

    $$\begin{aligned} {\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}^{-1} = \bigl |{\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}\bigr |^{-1} \times \left( \begin{array}{lll} {\mathcal I}_{yy} {\mathcal I}_{zz} \!-\! {\mathcal I}_{yz}^2 &{}\quad {\mathcal I}_{yz} {\mathcal I}_{zx} \!-\! {\mathcal I}_{xy}{\mathcal I}_{zz} &{}\quad {\mathcal I}_{xy} {\mathcal I}_{yz} \!-\! {\mathcal I}_{zx}{\mathcal I}_{yy}\\ {\mathcal I}_{yz}{\mathcal I}_{zx} \!-\! {\mathcal I}_{xy}{\mathcal I}_{zz} &{}\quad {\mathcal I}_{zz} {\mathcal I}_{xx} \!-\! {\mathcal I}_{zx}^2 &{}\quad {\mathcal I}_{zx} {\mathcal I}_{xy} \!-\! {\mathcal I}_{yz}{\mathcal I}_{xx}\\ {\mathcal I}_{xy} {\mathcal I}_{yz} \!-\! {\mathcal I}_{zx}{\mathcal I}_{yy} &{}\quad {\mathcal I}_{zx} {\mathcal I}_{xy} \!-\! {\mathcal I}_{yz}{\mathcal I}_{xx} &{}\quad {\mathcal I}_{xx} {\mathcal I}_{yy} \!-\! {\mathcal I}_{xy}^2 \end{array} \right) \end{aligned}$$

    (68)

  • Else, i.e., $\bigl |{\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}\bigr |= 0$, all particles are located along a line:

    • (a) If $I\ne 0$, then

      $$\begin{aligned} {\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}^{\dagger } = I^{-2} {\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}, \end{aligned}$$

      (69)

    • (b) Else, i.e., $I=0$, all the particles are at the origin,

      $$\begin{aligned} {\mathop {\mathbf {\mathcal I}}\limits ^{\leftrightarrow }}^{\dagger } =0. \end{aligned}$$

      (70)