lbfgs_nfeval_max SciMax Toolbox lc_l

SciMax Toolbox >> lc2kdt

lc2kdt

Maxima Function

Calling Sequence

lc2kdt (expr)

Description

Simplifies expressions containing the Levi-Civita symbol, converting these to Kronecker-delta expressions when possible. The main difference between this function and simply evaluating the Levi-Civita symbol is that direct evaluation often results in Kronecker expressions containing numerical indices. This is often undesirable as it prevents further simplification. The lc2kdt function avoids this problem, yielding expressions that are more easily simplified with rename or contract.

(%i1) load(itensor);
(%o1)      /share/tensor/itensor.lisp
(%i2) expr:ishow('levi_civita([],[i,j])
                 *'levi_civita([k,l],[])*a([j],[k]))$
                                  i j  k
(%t2)                  levi_civita    a  levi_civita
                                       j            k l
(%i3) ishow(ev(expr,levi_civita))$
                                  i j  k       1 2
(%t3)                       kdelta    a  kdelta
                                  1 2  j       k l
(%i4) ishow(ev(%,kdelta))$
             i       j         j       i   k
(%t4) (kdelta  kdelta  - kdelta  kdelta ) a
             1       2         1       2   j
                               1       2         2       1
                        (kdelta  kdelta  - kdelta  kdelta )
                               k       l         k       l
(%i5) ishow(lc2kdt(expr))$
                     k       i       j    k       j       i
(%t5)               a  kdelta  kdelta  - a  kdelta  kdelta
                     j       k       l    j       k       l
(%i6) ishow(contract(expand(%)))$
                                 i           i
(%t6)                           a  - a kdelta
                                 l           l

The lc2kdt function sometimes makes use of the metric tensor. If the metric tensor was not defined previously with imetric, this results in an error.

(%i7) expr:ishow('levi_civita([],[i,j])
                 *'levi_civita([],[k,l])*a([j,k],[]))$
                                 i j            k l
(%t7)                 levi_civita    levi_civita    a
                                                     j k
(%i8) ishow(lc2kdt(expr))$
Maxima encountered a Lisp error:
 Error in $IMETRIC [or a callee]:
 $IMETRIC [or a callee] requires less than two arguments.
Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i9) imetric(g);
(%o9)                                done
(%i10) ishow(lc2kdt(expr))$
         %3 i       k   %4 j       l     %3 i       l   %4 j
(%t10) (g     kdelta   g     kdelta   - g     kdelta   g
                    %3             %4               %3
              k
        kdelta  ) a
              %4   j k
(%i11) ishow(contract(expand(%)))$
                                  l i      l i
(%t11)                           a    - a g
lbfgs_nfeval_max SciMax Toolbox lc_l