Added equations for the recursive constrution of the multipoles.

examples/theory/multipoles.tex

@@ -19,7 +19,7 @@
 
 Bold quantities are vectors. The indices $\alpha,\beta$ run over the directions $x,y,z$.
 
-\section{Construction of multi-poles}
+\section{Construction of multipoles}
 
 For a set of particles $A$ at position $\p{i}=(x_i, y_i, z_i)$ with mass $m_i$, 
 we can construct the total mass of the set 
@@ -30,8 +30,8 @@ $M_A$ and its centre of mass $\muu_A=(\mu_{A,x}, \mu_{A,y}, \mu_{A,z})$:
 m_i\p{i}.
 \end{equation}
 
-The multi-poles can then be computed around the centre of mass $\muu_A$. We use 
-Dehnen's notation.
+For a set of particles $A$, the multipoles can be computed around the 
+centre of mass $\muu_A$. The first LHS term uses Dehnen's notation.\\
 
 Monopole:
 \begin{equation}
@@ -40,24 +40,50 @@ M_{(0,0,0)} = M_A
 
 Dipole:
 \begin{eqnarray}
-M_{(1,0,0)} &=& 0\\
-M_{(0,1,0)} &=& 0\\
-M_{(0,0,1)} &=& 0
+M_{(1,0,0)} &=& P_{A,x}~= 0\\
+M_{(0,1,0)} &=& P_{A,y}~=0\\
+M_{(0,0,1)} &=& P_{A,z}~=0
 \end{eqnarray}
 
 Quadrupole:
 \begin{eqnarray}
-M_{(2,0,0)} &=& I_{xx}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})^2\\
-M_{(0,2,0)} &=& I_{yy}~= \sum_{i\in A}m_i ( p_{i,y}-\mu_{A,y})^2\\
-M_{(0,0,2)} &=& I_{zz}~= \sum_{i\in A}m_i ( p_{i,z}-\mu_{A,z})^2\\
-M_{(1,1,0)} &=& I_{xy}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})( 
+M_{(2,0,0)} &=& I_{A,xx}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})^2\\
+M_{(0,2,0)} &=& I_{A,yy}~= \sum_{i\in A}m_i ( p_{i,y}-\mu_{A,y})^2\\
+M_{(0,0,2)} &=& I_{A,zz}~= \sum_{i\in A}m_i ( p_{i,z}-\mu_{A,z})^2\\
+M_{(1,1,0)} &=& I_{A,xy}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})( 
 p_{i,y}-\mu_{A,y})\\
-M_{(0,1,1)} &=& I_{yz}~= \sum_{i\in A}m_i ( p_{i,y}-\mu_{A,y})( 
+M_{(0,1,1)} &=& I_{A,yz}~= \sum_{i\in A}m_i ( p_{i,y}-\mu_{A,y})( 
 p_{i,z}-\mu_{A,z})\\
-M_{(1,0,1)} &=& I_{xz}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})( 
-p_{i,z}-\mu_{A,z}
+M_{(1,0,1)} &=& I_{A,xz}~= \sum_{i\in A}m_i ( p_{i,x}-\mu_{A,x})( 
+p_{i,z}-\mu_{A,z})
 \end{eqnarray}
 
 \section{Recursive construction of the quadrupoles}
 
+Given a set of multipoles $B$ expressed around their centre of masses 
+$\muu_{B}$, we can construct the total multipoles around the centre of mass
+$\muu_{A}$ of the system consisting of all the particles contained in each 
+individual (disjoint) sub-sets $B$. This allows to construct the multipoles in 
+the tree recursively.\\
+
+Monopole:
+\begin{equation}
+ M_{A} = \sum_{B\in A} M_B
+\end{equation}
+
+Dipole:
+\begin{equation}
+ P_{A,\alpha} = 0
+\end{equation}
+
+Quadrupole:
+\begin{equation}
+ I_{A,\alpha\beta} = \sum_{B\in A}\left( I_{B,\alpha\beta} + 
+M_B\mu_{B,\alpha}\mu_{B,\beta} \right) - \frac{1}{M_A} 
+\mu_{A,\alpha}\mu_{A,\beta}
+\end{equation}
+
+
+\section{Derivatives of the potential}
+
 \end{document}