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cliffordan.mac
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/*******************************
Clifford algebra
a lightweight package for component-wise Geometric Algebra calculations
Clifford analysis
@version 2.0 Date 8 Jul 2023
- gradbasis
- assume_pos option
1.9 Date 18 Dec 2021
- constrained Lagrangians
1.8.9 Date 31 Jul 2021
- cinv changed to cinvinblade
- bugfix in dependsv
1.8.8 Date 25 May 2021
- potentiategr - integrates a k-vector
1.8.7 Date 25 Apr 2020
- change in cdirdiff
- cdirdiffn - directional derivative of order n-1
- cdirdiffi -
1.8.6 Date 26 Jan 2020
- climoivre integrated
1.8.5 Date 17 Jan 2020
- pintegrate added
- partcoord added
- grank added
- potentiate added
1.8.4 Date 11 Jan 2020
- cndirdiff sign change
- obsolete parts removed (old convderiv)
1.8.3 1 Sept 2019
- convderiv added
- experimental functions moved
1.8.2 Date 04 Feb 2018
- code refactoring
1.8.1 Date 20 Sept 2017
- diffsubst bugfixes
1.8 Date 23 Aug 2017
- change of variables fixed
- bug fixes
1.7 Date 12 Dec 2016
- projection derivatives
1.6.5 Date 13 Jun 2016
1.6 Date 10 Apr 2016
1.5 Date 20 Feb 2016
1.2 Date 22 Jan 2016
1.1 Date 16 Dec 2015
@depends 'clifford
**********************************
* @license This library is free software you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation either
* version 2.1 of the License, or (at your option) any later version.
*
* This library is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
stringdisp:true;
declare(trigsimp, evfun);
if get('clifford,'version)=false then (
err:errcatch(
load("clifford.mac"),
load("climoivre.mac")
),
if emptyp(err) then (
infix ("|", 130, 129),
infix ("&", 130, 129)
)
);
/* move to clifford ?*/
if get('cliffordan,'version)=false then (
remove("@", operator),
"inner product",
infix ("@", 130, 129),
"@"(a, b):= scprod(a,b),
print("warning: redefining @")
);
%clverbose:false;
%divsimp:true;
matchdeclare([%ff, %xx], symbolp, %nn, integerp, %bb, true);
defmatch(%diffpar, %bb*'diff(%ff, %xx, %nn));
%sclardecl:true;
/*
declaring dependencies
*/
dependsv(F, var):= block([ ee, u, var2, _k],
if listp(F) then
map (lambda([u], dependsv(u, var)), F)
else (
if %sclardecl= true then(
ee:buildq([F], declare(F, scalar) ),
ev(ee, nouns)
),
if subvarp(F) then F:op(F),
var2:var,
var: if subvarp(var) then op(var) else var,
if listp(var) then
var:map(lambda([u], if subvarp(u) then op(u) else u), var),
ee:buildq([var], declare(var, scalar) ),
ev(ee, nouns),
var:var2,
/* display(var),*/
ee:buildq([var], if listp(var) then
makelist(var[_k], _k, length(var))
else
var ),
display(ev(ee, nouns)),
ev(depends(F,ee), nouns)
)
);
/*
Calculus functions
*/
/* total differentiation;
no assumptions */
totdiff(f, x, [lv]):=block( [ret:0, u],
if mapatom(x) then
ret: diff( f, x )
else (
if emptyp(lv) then
lv:listofvars(x)
else
lv:lv[1],
/*display(lv),*/
for u in lv do
if symbolp(u) then
ret: ret + diff(f, u) / diff(x, u)
),
ret
);
/* fix */
ctotdiffs(f, x):=block( [ret:0, lv, ia],
if mapatom(x) then
ret: diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
for u in lv do
if symbolp(u) then
if symbolp(f) then
ret: ret + cinvblade(diff(x, u)). subst(".", "*", diff( f, u ))
else (
ia:inargs(f),
if length(ia)=1 then
ret: ret + cinvblade(diff(x, u)). subst(".", "*", diff(inop(f), u)* diff(first(ia), u))
else
ret: ret + cinvblade(diff(x, u)). subst(".", "*", diff( f, u ))
)
),
ret
);
diffp(expr):= inop(expr)=nounify('derivative);
/* total differentiation;
subsitutions*/
dsymb:nounify('derivative);
diffsubst(deq, dvar, eq):=block([nn, zz, u, sop, sargs, ret, l, r, dsymb:nounify('derivative) ],
if numberp(deq) then return(deq),
/*disp(deq),*/
sop:inop(deq),
if op(eq)="=" then zz:rhs(eq) else error("provide explicit equation"),
if inop(deq)="+" then
ret:map(lambda([u], diffsubst(u, dvar, eq)), deq)
else (
ret:deq,
if diffp(deq) then (
sargs:inargs(deq),
nn:sargs[3],
if not listp (dvar) then dvar:[dvar],
/*display(lhs(eq), sargs[2]),*/
if lhs(eq)=sargs[2] then (
ret:ev(deq, diff),
ret: block([ ff, xx, nn, fu],
subst(dsymb=lambda([ff, xx, nn],
if not mapatom(ff) then (
fu:subst(eq, ff),
'totdiff(fu, zz, dvar)) else 'totdiff(ff, zz, dvar)
), ret)
),
/*ret:subst(dsymb=lambda([ff, xx, nn], disp(ff, zz), totdiff(ff, zz, [dvar])), ret), */
/*ret:subst(dsymb=lambda([ff, xx, nn], totdiff(ff, zz, [dvar])), ret), */
ret:ev(ret, nouns),
if nn>1 then
for k:1 thru nn-1 do ret:totdiff(ret, zz, dvar),
/* display(ret),*/
ret:subst(eq, ret)
)
) else (
[l, r]: oppart(deq, lambda([u], freeof(dsymb, u))),
/*display(l, r),*/
if r#nil then
r: diffsubst(r, dvar, eq)
else r:1,
l: subst(eq, l),
ret:l*r
)
),
ret
);
/*
Clifford-valued
total differentiation;
cinv->cinvblade
*/
ctotdiff(f, x):=block( [ret:0, lv, u, z],
if mapatom(x) then
ret: diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
for u in lv do
ret: ret + cinvblade(diff(x, u)). subst(".", "*", diff( f, u ))
),
if assume_pos=true then
ret:subst(abs=lambda([u], u),ret),
ret
);
/*
connection-type of derivatives
*/
conmderivl(f, x):=block([lv, sq],
sq:makelist( asymbol[i] @ asymbol[i], i , ndim),
lv:makelist( asymbol[i]@x *sq[i], i , ndim),
genmatrix( lambda([i,j], sq[i]* dotsimpc(expand(asymbol[i]. ctotdiff(f, lv[j])))), ndim)
);
conmderivr(f, x):=block([lv, sq],
sq:makelist( asymbol[i] @ asymbol[i], i , ndim),
lv:makelist( asymbol[i]@x *sq[i], i , ndim),
genmatrix( lambda([i,j], sq[i]* dotsimpc(expand(ctotdiff(f, lv[j]). asymbol[i]))), ndim)
);
/*
Metrics table */
metrics(x, lst):=block([n, ek],
ek: maplist (lambda([u], diff(x, u)), lst),
/*disp(ek),*/
n:length(lst),
genmatrix( lambda([i,j], trigsimp( (ek[i] @ ek[j] ))), n)
);
/*
Clifford-valued
directional derivative
*/
cdirdiff(f, v, x):=block( [ret:0, u, lv, m, z, uu:0, qq],
if mapatom(x) then
ret: scalarpart(v)*diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
for u in lv do (
qq: diff(x, u),
m: qq @ qq,
if m#0 then m:1/m,
uu:m*(qq @ v),
ret: ret + uu * subst(".", "*", diff( f, u ))
)
),
ret:dotsimpc(ret),
if assume_pos=true then
ret:subst(abs=lambda([u], u),ret),
ret:clifact(ret)
);
cdirdiffn(f, v, x, n):=block([ret],
if not numberp(n) then return ('cdirdiffn(f, v, x, n)),
ret:cdirdiff(f, v, x),
for i:1 thru n-1 do
ret: cdirdiff(ret, v, x),
ret
);
/**
Cartan decomposition
*/
cdirdiffa(f, v, x):=block( [ret:0, qq, pp, ee, kv, %k:gensym("k"), ipdecomp:true, inprotype:sym, lv],
if mapatom(x) then
ret: scalarpart(v)*diff( f, x )
else (
kv:cvect(%k),
lv: invautom( clv(v)),
ret:mvectdiff( kv | f, x),
display(ret),
for ee in lv do (
qq: ee @ v,
pp: ee @ kv,
ret:subst(pp=qq, ret)
),
ret: v | mvectdiff (f, x) + ret
),
ret:clifact(ret)
);
/*
Clifford-valued
meet of the directional derivative
*/
cdirdiffi(f, v, x):=block ([ret:0, m, z, p],
if mapatom(x) then
ret: scalarpart(v)*diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
for u in lv do (
qq: diff(x, u),
m: qq @ qq,
if m#0 then m:1/m,
uu: (qq | v)*m,
ret: ret + uu * subst(".", "*", diff( f, u ))
)
),
ret:dotsimpc(ret),
ret:clifact(ret)
);
/*
dual of the directional derivative
*/
cdirdualdiff(f, v, x):=block( [ret:0],
ret: cdirdiff(f, v, x),
clihdual(ret)
);
/*
@deprecated
*/
cndirdiff(f, v, x):=block( [ret:0, lv, uu:0, m, qq],
if mapatom(x) then
ret: scalarpart(v)*diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
for u in lv do (
qq: diff(x, u),
m: qq @ qq,
if m#0 then m:1/m,
uu: ( qq. v)*m,
ret: ret + uu. subst(".", "*", diff( f, u ))
)
),
ret:expand(ret),
ret:dotsimpc(ret),
ret:clifact(ret)
);
ctotprojdiff(f, v, x):=block( [ret:0, lv, uu:0, qq, gn, ss:0, ipdecomp:true],
if mapatom(x) then
ret: scalarpart(v)*diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol, z))),
gn:maxgrade(v),
for u in lv do (
qq: cinvblade(diff(x, u)),
uu: dotsimpc(expand(v | qq)),
ret: ret + uu. subst(".", "*", diff( f, u ))
)
),
ret:expand(ret),
ret:dotsimpc(ret),
ret:clifact(ret)
);
ctotintdiff(f, x):=block( [ret:0, lv],
if mapatom(x) then
ret: diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol,z))),
for u in lv do
ret: ret + cinvblade(diff(x, u)) | subst(".", "*", diff( f, u ))
),
ret
);
ctotextdiff(f, x):=block( [ret:0, lv],
if mapatom(x) then
ret: diff( f, x )
else (
lv: sublist( listofvars(x), lambda ([z], freeof(asymbol,z))),
for u in lv do
ret: ret + cinvblade(diff(x, u)) & subst(".", "*", diff( f, u ))
),
ret
);
/*
multi vector derivative
*/
mvectdiff(f, x, [k] ):=block ([s:0, f1:copy(f), p],
if emptyp(k) then k:1 else k: k[1],
x:expand(x),
for p:1 thru k do (
s:ctotdiff(f1, x),
s:cliffsimpall(s),
f1:s
),
if %divsimp=true then (
s:trigsimp(s),
s:clifact(s)
),
s
);
mvectdiffs(f, x, [k] ):=block ([s:0, f1:copy(f), p],
if emptyp(k) then k:1 else k: k[1],
x:expand(x),
for p:1 thru k do (
s:ctotdiffs(f1, x),
s:cliffsimpall(s),
f1:s
),
if %divsimp=true then (
s:trigsimp(s),
s:clifact(s)
),
s
);
/*
partial derivative
*/
pardiff(f, x, [k] ):=block( [ret:0, u:gensym() ],
if emptyp(k) then k:1 else k: k[1],
if mapatom(x) then
ret:diff(f, x, k )
else (
ret:diff(ratsubst( u, x, f), u, k),
ret:subst(u=x,ret)
),
ret
);
/*
vector derivative
reference implementation
*/
vectdiff(f, x, [k] ):=block ([s, n:1, m, i ],
if listp(x) then n:length(x) else x:[x],
if n>ndim then error("size mismatch"),
if emptyp(k) then k:1 else k: k[1],
for p:1 thru k do (
i:1, s:0,
for m in x do (
if not emptyp(m) then
s: s+ cinvblade(asymbol[i]). subst(".", "*", pardiff( f, m )),
i:i+1
),
s:cliffsimpall(s),
f:s
),
s
);
/*
multi vector partial derivative
*/
parmvectdiff(f, x, [k] ):=block ([s, z, m, i, ee, xx, df, qq, es ],
[ee, xx]:clicoeff2(x, 'list),
if emptyp(k) then k:1 else k: k[1],
es: sublist( listofvars(x), lambda ([z], not freeof(asymbol, z))),
if %clverbose=true then
display(es),
for p:1 thru k do (
i:1, s:0,
for m in xx do (
df: expand(pardiff( f, m )),
qq: cinvblade(ee[i]),
s: s + cliffsimpall(qq . subst(".", "*", df )),
i:i+1
),
if %divsimp=true then (
if not emptyp(es) then
s:simpfact(s, es)
),
f:s
),
clifact(s)
);
parmvectdirdiff(f, v, x, [k] ):=block ([s, ee, xx, df, qq, uu ],
[ee, xx]:clicoeff2(x, 'list),
if %clverbose=true then
display(xx),
if emptyp(k) then k:1 else k: k[1],
for p:1 thru k do (
s:0,
for i:1 thru length(xx) do (
df: expand(pardiff( f, xx[i] )),
qq: cinvblade(ee[i]),
uu: qq @ v,
/*display(qq, uu, df),*/
s: s + cliffsimpall(uu*subst(".", "*", df ))
),
f:s
),
s
);
parmvectextdiff(f, x, [k] ):=block ([s, n:1, z, m, i, ee, xx, df, qq, es, ipdecomp:true],
[ee, xx]:clicoeff2(x, 'list),
if %clverbose=true then
display(xx),
if emptyp(k) then k:1 else k: k[1],
es: sublist( listofvars(x), lambda ([z], not freeof(asymbol, z))),
"display(es)",
for p:1 thru k do (
i:1, s:0,
for m in xx do (
df: expand(pardiff( f, m )),
qq: cinvblade(ee[i]),
s: s + cliffsimpall(qq & subst(".", "*", df )),
i:i+1
),
if %divsimp=true then (
/*s:cliffsimpall(s),*/
if not emptyp(es) then
s:simpfact(s, es)
),
f:s
),
clifact(s)
);
parmvectintdiff(f, x, [k] ):=block ([s, n:1, z, m, i, ee, xx, df, qq, es, ipdecomp:true],
[ee, xx]:clicoeff2(x, 'list),
if %clverbose=true then
display(xx),
if emptyp(k) then k:1 else k: k[1],
es: sublist( listofvars(x), lambda ([z], not freeof(asymbol, z))),
"display(es)",
for p:1 thru k do (
i:1, s:0,
for m in xx do (
df: expand(pardiff( f, m )),
qq: cinvblade(ee[i]),
s: s + cliffsimpall(qq | subst(".", "*", df )),
i:i+1
),
if %divsimp=true then (
/*s:cliffsimpall(s),*/
if not emptyp(es) then
s:simpfact(s, es)
),
f:s
),
clifact(s)
);
/*
interior vector derivative
*/
intmvectdiff(f, x, [k] ):=block ([s:0, n:1, m, esm, ipdecomp:true],
if emptyp(k) then k:1 else k: k[1],
x:expand(x),
for p:1 thru k do (
s:ctotintdiff(f, x),
s:cliffsimpall(s),
f:s
),
if %divsimp=true then (
es: sublist( listofvars(x), lambda ([z], not freeof(asymbol, z))),
"display(es)",
s:trigsimp(s),
if not emptyp(es) then
s:simpfact(s, es),
s:clifact(s)
),
s
);
/*
exterior vector derivative
*/
extvectdiff(f, x, [k] ):=block ([s:0, n:1, m, es],
if emptyp(k) then k:1 else k: k[1],
x:expand(x),
for p:1 thru k do (
s:ctotextdiff(f, x),
s:cliffsimpall(s),
f:s
),
if %divsimp=true then (
"display(es)",
s:trigsimp(s),
s:clifact(s)
),
s
);
alias (extmvectdiff, extvectdiff);
svectdiff(f, x, [k] ):=block ([ df],
if emptyp(k) then k:1 else k: k[1],
for p:1 thru k do (
df:mvectdiff(f, x, p),
df:scalarpart(df)
),
df
);
vvectdiff(f, x, [k] ):=block ([ df],
if emptyp(k) then k:1 else k: k[1],
for p:1 thru k do (
df:mvectdiff(f, x, p),
df:nscalarpart(df)
),
if %divsimp=true then
df:clifact(df),
df
);
/*
convective derivative
*/
convderiv(f, t, xx, [vs]):=block([ez, es, lv, tt:gensym("t"), ret:0, pv,
aa:[], inprotype:ic, u, opsubst:true, derivsubst:true],
if not emptyp(vs) then (
ret:parmvectdirdiff(f, vs[1], xx )
) else (
xx:expand(xx),
lv:listofvars(xx),
ez:sublist( lv, lambda ([u], freeof(asymbol, u))),
es:sublist( lv, lambda ([u], not freeof(asymbol, u))),
depends (ez, t),
pv:diff(xx,t),
apply ('remove, [ez, 'dependency]),
ret:parmvectdirdiff(f, pv, xx )
),
ret:diff(f,t)+ret
);
/*
Euler-Lagrange equations
*/
/*
EL field Lagrangians
actually, a functional derivative
*/
ciELfdiff(_F, rr, pvars):=block(
[eq:0, p, pv, ee, qq, ss:[], zz:[], vv, lv ],
[p, pv]:pvars,
if %clverbose=true then
display(p, pv),
[ee, vv]:clicoeff(pv, 'list),
lv: sublist( listofvars(rr), lambda ([z], freeof(asymbol, z) )),
if not mapatom(vv) then vv: substinpart("[",vv, 0)
else vv:[vv],
for j:1 thru length(vv) do (
for i:1 thru length(lv) do (
if inop(vv[j])#"+" then (
push(vv[j]=vv[j](lv[i]), ss),
push(vv[j](lv[i])=vv[j], zz)
)
)
),
/*display(ss, zz),*/
qq:parmvectdiff(_F, pv),
if %clverbose=true then
display(qq),
qq:psubst(ss, qq),
eq:parmvectdiff(_F, p)-psubst(zz, mvectdiff(qq , rr)),
clifact(eq)
);
/**/
alias (EuLagEq2, ciELfdiff);
/*Energy-momentum equation*/
ciELfdiff3(_F, rr, pvars) :=block(
[eq:0, p, pv, ee, qq, ss:[], zz:[], vv, lv, dq],
[p, pv]:pvars,
if %clverbose=true then
display(p, pv),
[ee, vv]:clicoeff(pv, 'list),
lv: sublist( listofvars(rr), lambda ([z], freeof(asymbol, z) )),
if not mapatom(vv) then vv: substinpart("[",vv, 0)
else vv:[vv],
for j:1 thru length(vv) do (
for i:1 thru length(lv) do (
if inop(vv[j])#"+" then (
push(vv[j]=vv[j](lv[i]), ss),
push(vv[j](lv[i])=vv[j], zz)
)
)
),
/*display(ss, zz),*/
qq:parmvectdiff(_F, pv),
if %clverbose=true then
display(vv, qq),
qq: psubst(ss, qq),
eq: psubst(zz, pv @ qq),
eq -_F
);
/* constrained EL equation,
anholonomic constraints
*/
ciELfdiff1(_F, rr, pvars, cvars):=block(
[eq:0, p, pv, ee, qq, ss:[], zz:[], vv, lv, cm, _cfr, qm:0 ],
[p, pv]:pvars,
[cm, _cfr, lv]:cvars,
if %clverbose1=true then
display( _cfr, cm ,lv),
eq:ciELfdiff(_F, rr, pvars),
qm:parmvectdiff(_cfr, p),
/* getting the anholonomic constraints*/
qq: parmvectdiff(_cfr, cm),
if %clverbose1=true then
display(qm, qq),
/* solving for the constraints*/
ss:clisolve(qq, lv),
eq:eq + qm,
/* subsituting the constraints*/
eq:subst(ss, eq),
eq:ev(eq, diff),
[eq, ss]
);
/*
EL particle lagrangians
*/
/*
Euler - Lagrange equation derivation
*/
ciEL1(_F, tt, pvars, smat):=block([eq, p, pv, qq, ee, aa:[], bb:[], es:[], ez:[], qp, qq2, cc:[], dd:[], z ],
if not listp(pvars) then error("list required"),
[p, pv]:pvars,
eq:parmvectdiff(_F, p),
es: sublist( listofvars(p), lambda ([z], freeof(asymbol, z))),
ez: sublist( listofvars(pv), lambda ([z], freeof(asymbol, z))),
/*display(es, ez), */
if %clverbose1=true then
display(pv, eq),
for j:1 thru length(ez) do (
push(ez[j]='diff(es[j], tt, 1), aa),
push('diff(es[j], tt, 1)=ez[j], bb),
push(ez[j]=ez[j](tt) , cc),
push(ez[j](tt)=ez[j] , dd)
),
qq2: parmvectdiff(_F, pv),
qq2: subst(cc, qq2),
/*qp: convderiv(qq2, tt, p, pv),*/
qp: convderiv(qq2, tt, p, pv),
if %clverbose1=true then
display(es, qq2, qp),
eq:eq - qp,
eq: subst(dd, eq),
/*eq: subst(bb, eq),*/
if smat='list then (
/*eq:subst(aa, eq),*/
eq: clifact(eq)
)else if smat='mat then (
eq:bdecompose(eq),
eq:factorby(eq, ez)
),
eq
);
ciEL(_F, tt, pvars, smat):=block([eq, p, pv, qq, ee, aa:[], bb:[], es:[], ez:[], ad1, ad2 ],
if not listp(pvars) then error("list required"),
[p, pv]:pvars,
eq:parmvectdiff(_F, p),
es: sublist( listofvars(p), lambda ([z], freeof(asymbol, z))),
ez: sublist( listofvars(pv), lambda ([z], freeof(asymbol, z))),
/*display(es, ez), */
if %clverbose1=true then
display(pv, eq, ez, es),
for j:1 thru length(ez) do (
if 'diff(es[j], tt, 1)#1 then (
push(ez[j]='diff(es[j], tt, 1), aa),
push('diff(es[j], tt, 1)=ez[j] , bb)
)
),
aa:reverse(aa),
bb:reverse(bb),
if %clverbose1=true then
display (aa, bb),
/*depends(es, tt),*/
qq: parmvectdiff(_F, pv),
qq: diff(qq, tt),
eq:eq - qq,
if %clverbose1=true then
display(qq,eq),
eq: subst(bb, eq),
if %clverbose1=true then
display(qq),
if smat='list then (
eq: clifact(eq)
)else if smat='mat then (
eq:bdecompose(eq),
eq:factorby(eq, ez)
),
eq
);
ciEL2(_F, tt, pvars, cvars):=block(
[eq:0, p, pv, ee, qq, ss:[], zz:[], vv, lv, cm, _cfr, qm:0 ],
[p, pv]:pvars,
[cm, _cfr, lv]:cvars,
display( _cfr, cm ,lv),
eq:ciEL(_F, tt, pvars, 'list),
display(eq),
qm:parmvectdiff(_cfr, p),
/* getting the anholonomic constraints*/
qq: parmvectdiff(_cfr, cm),
display(qm, qq),
/* solving for the constraints*/
ss:clisolve(qq, lv),
eq:eq + qm,
/* subsituting the constraints*/
eq:subst(ss, eq),
eq:ev(eq, diff),
[eq, ss]
);
EL(_F, q, pvars):=block ([eq:0, p, pv, qq, rr, len, ret:[]],
if not listp(pvars) then error("list required"),
len:length(pvars),
p:pvars[1],
eq:pardiff(_F, p),
if len>1 then (
for i:2 thru len do (
pv:pvars[i],
qq: subst('pv(q), pv, pardiff(_F, pv)),
display(qq),
for j:1 thru i-1 do (
rr:- convderiv(qq, q, p),
rr: subst('diff(p, q)='pv(q), rr),
display(q, p, pv, rr),
eq:eq + subst( pv, 'pv(q), rr ),
/*eq:eq+ rr,*/
qq:rr
),
push(pv='diff(pvars[1], q, i-1), ret)
),
ret:reverse(ret)
),
[ ret, eq]
);
/*
EuLagEq2(_F, tt, pvars):=block ([eq:0, p, pv, ret:[]],
[p, pv]:pvars,
display(p, pv),
if listp(p) then
for i:1 thru length (p) do
push(pv[i]='diff(p[i], tt), ret)
else
push(pv='diff(p, tt), ret),
depends(pv, tt),
eq:mvectdiff(_F, p)-diff(parmvectdiff(_F, pv), tt),
eq:subst(ret, eq=0),
remove (pv, dependency),
eq
);*/
/*
coordinate substitution
*/
coordsubst(rr, eqs):=block( [lv, ret:rr, qq, z],
lv: sublist( listofvars(rr), lambda ([z], freeof(asymbol,z))),
if not listp(eqs) then eqs:[eqs],
for i:1 thru length(eqs) do (
if op(eqs[i])="=" then (
qq:rhs(eqs[i]),
ret:ratsubst(qq, lhs(eqs[i]), ret)
) else (
qq:eqs[i],
ret:ratsubst(qq, lv[i], ret)
)
),
ret:trigsimp(ret),
ret:ratsimp(ret),
ret
);
gradbasis(rr, lv):=block([ret],
if not listp(lv) then error(lv, "not a list"),
ret : maplist(lambda([u], diff(rr, u)), lv),
ret:trigsimp(ret),
ret:ratsimp(ret),
ret
);
/*
volume element
coordinate substitution
*/
volel(eqs, [lv]):=block( [ ret, qq:[] ],
if not listp(eqs) then eqs:[eqs],
if not emptyp(lv) then (
lv:lv[1],
for i:1 thru length(eqs) do
push(rhs(eqs[i]), qq)
) else (
for i:1 thru length(eqs) do (
lv: append( sublist( listofvars(rhs(eqs[i])), lambda ([z], freeof(asymbol,z))), lv),
push(rhs(eqs[i]), qq)
),
lv:unique (lv)
),
qq:reverse(qq),
/*disp(qq, lv),*/
/* jacobian computation*/
ret: outermap (lambda([u,z], diff(qq[z], u)), lv, makelist(i,i, length(lv))),
ret: substinpart('matrix, ret,0),
if %clverbose=true then display(lv, qq, ret),
ret:determinant(ret),
if %divsimp=true then (
ret:trigsimp(ret),
ret:ratsimp(ret)
),
%iv.ret
);
clidet(vlist):=block([dd, n ],
n:length(vlist),
dd:vlist[1],
for i:2 thru n do (
dd:dotsimpc(dd & vlist[i]),
if dd=0 then return(dd)
),
factorby(dd, %iv)
);
clivolel(rr, eqs, [lv]):=block([ ret, qq:[], rq, z ],
if not listp(eqs) then eqs:[eqs],
if not emptyp(lv) then (
lv:lv[1],
for i:1 thru length(eqs) do
push(rhs(eqs[i]), qq)
) else (
for i:1 thru length(eqs) do (
lv: append ( sublist( listofvars(rhs(eqs[i])), lambda ([z], freeof(asymbol,z))), lv),
push(rhs(eqs[i]), qq)
),
lv:unique (lv)
),
rq:subst(eqs,rr),
ret:map (lambda([z], diff(rq, z)), lv ),
if %clverbose=true then display(lv, qq, ret),
ret:clidet(ret),
if %divsimp=true then (
ret:trigsimp(ret),