wnde

  
Purpose 		 

Construct the differential equation for the logarithmic coordinates 
$g_k$ of the Wei-Norman formula, see wne, wner.





Syntax 		 
e:=wnde(x,r,max_bracket_order,B,lbt);




Description 		

Constructs the differential equation for the logarithmic coordinates 
$g_k$ of the Wei-Norman formula.  This function equates the
coefficients of the $X_k$ in the left-hand
side of the Wei-Norman equation 40 to the corresponding
coefficients of $X_k$ in the right-hand side 43,
yielding a set of equations of the form:



$$\begin{eqnarray*}
u_1&=&F_1(g_1,\ldots,g_r)\dot{g}_1\\ 
u_2&=&F_2(g_1,\ldots,g_r)\dot{g}_2\\ 
 &\vdots& \\ 
u_r&=&F_r(g_1,\ldots,g_r)\dot{g}_r
\end{eqnarray*}$$









or equivalently in matrix form as






$$u=F(g)\dot{g}$$ 
(44)





where $u,g,\dot{g}\in\mathbb{R}^r$, and $F(g):\mathbb{R}^r\rightarrow
 \mathbb{R}^{r\times r}$.
This procedure returns in its first argument the matrix $F(g)$ and 
the set of equations of the form $u_k=F_k(g_1,\ldots,g_r)\dot{g}_k$, 
$k=1,\ldots,r$ for as a second output element.  In order to equate 
the coefficients on each side of the Wei-Norman equation, this 
function requires the right-hand side as calculated with the 
procedure wne.





Arguments 		 
 $x$  $\parbox{0.64\textwidth}{The right-hand side of
 the Wei-Norman equat...
...omputed with \texttt{wne},
 \texttt{wner} (described on page~\pageref{ss:wne}.}$


 		 $r$ $\parbox{0.64\textwidth}{Dimension of the Lie
 algebra basis. In part...
...\ corresponds to the number of Lie brackets in the PHB
 which are independent.}$


 		 max_bracket_order


 		 		 $\parbox{0.64\textwidth}{The
 maximum bracket order in the exponentia...
...exttt{wner},
 instead of passing the result of \texttt{wner} as argument $x$.}}$


 		 $B$		 $\parbox{0.64\textwidth}{A Philip Hall basis.}$


 		 $lbt$		 $\parbox{0.64\textwidth}{A Lie bracket t...
...n empty list or set if no
 dependencies between the brackets in the PHB exist.}$




Examples 		

Consider the P. Hall basis, B, given in the example for the
function phb on page .  Additionally, suppose
that $B_6=[f_1,f_2]=0$, then the Lie algebra can be expressed in terms
of the following 14-dimensional basis of independent Lie brackets, in
terms of which w5r is expressed (see example for the function
wne, wner on page ): 
  BB := [f0~, f1~, f0~ &* f1~, f0~ &* (f0~ &* f1~),

         f0~ &* (f0~ &* (f0~ &* f1~)), f2~, f0~ &* f2~,

         f0~ &* (f0~ &* f2~), f0~ &* (f0~ &* (f0~ &* f2~)),

         f2~ &* (f0~ &* f1~), f2~ &* (f2~ &* (f0~ &* f1~)),

         f1~ &* (f0~ &* f1~), f1~ &* (f1~ &* (f0~ &* f1~)),

         f1~ &* (f0~ &* (f0~ &* f1~))]

The Wei-Norman equations can now be calculated using wnde, note
that max_bracket_order is an empty list, since
the current version of wnde does not of this argument.
Notice that lbt is also an empty list, since in this case we are
passing directly the basis of the Lie algebra BB in the
argument for the PHB.    
> lc:=wnde(w5r,14,{},BB,{});

                                3
  lq := Fg, {u[9] = 1/6 dg3~ g1~ , u[10] = dg4~ g3~, u[1] = dg1~,

                                                                 2
        u[2] = dg2~, u[3] = dg2~ g1~ + dg4~, u[11] = 1/2 dg4~ g3~ ,

                                             2
        u[12] = dg4~ g2~, u[4] = 1/2 dg2~ g1~  + dg4~ g1~,

                            2
        u[13] = 1/2 dg4~ g2~ , u[6] = dg3~,

                           3               2
        u[5] = 1/6 dg2~ g1~  + 1/2 dg4~ g1~ , u[7] = dg3~ g1~,

                                                 2
        u[14] = dg4~ g1~ g2~, u[8] = 1/2 dg3~ g1~ }



		

The matrix $F(g)$ of logarithmic coordinates $g$ can be obtained from
the first element returned by wnde. 
> eval(lc[1]);
    [1     0       0       0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]  
    [0     1       0       0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [0    g1~      0       1     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [          2                                             ]
    [0  1/2 g1~    0      g1~    0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [          3              2                              ]
    [0  1/6 g1~    0   1/2 g1~   0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [0     0       1       0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [0     0      g1~      0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [                  2                                     ]
    [0     0    1/2 g1~    0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [                  3                                     ]
    [0     0    1/6 g1~    0     0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [0     0       0      g3~    0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [                         2                              ]
    [0     0       0   1/2 g3~   0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [                                                        ]
    [0     0       0      g2~    0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [                         2                              ]
    [0     0       0   1/2 g2~   0  0  0  0  0  0  0  0  0  0]
    [                                                        ]
    [0     0       0   g1~ g2~   0  0  0  0  0  0  0  0  0  0]

Practical application examples involving the derivation of the
logarithmic coordinates for an underactuated rigid body in space can
be found in the directory ../ltp/dev, in the files
weinorman_rb1.ws and weinorman_rbfull.mws. 





Notes 		 

Care should be put when passing the P. Hall basis, the Lie bracket
table to this function, and the right-hand side of the Wei-Norman
equation.  It could be the case that the number of brackets in each
argument are not consistent.  This function attempts to construct the
list containing the actual basis for the Lie algebra from the P. Hall
basis and the Lie bracket table.  It also tries to detect situation of
argument inconsistency, however the checking procedure is rather simple
still and it may not correctly identify all possible errors.

To avoid problems perhaps the best way is to provide the function with
the Lie algebra basis and pass it in the argument for the P. Hall
basis and pass an empty list or set as argument for the Lie bracket
table.  The disadvantage of this approach is the possible calculation
involved in the determination of a suitable Lie algebra basis prior to
invoking wnde.





See Also 		

wne, wner.





References 		 

See [32] and references therein for an explanation on the
derivation of the Wei-Norman equations.