Euler-Lagrange equations and partial derivatives


On Thu, 22 Apr 2010, dlakelan wrote:

< On 04/22/2010 12:22 PM, dlakelan wrote:
< 
< > In Euler-Lagrange equations we need to mix these things.
< > 
< > diff(partdiff(foo, bardot),t) - partdiff(foo,bar) = 0
< > 
< > where only the "diff" should take total derivatives, and the
< > non-existent partdiff should do partials only. Unfortunately it is not
< > easily possible (or at least not obvious how) to mix partials and total
< > derivatives in maxima. Does anyone have any suggestions about how to go
< > about it?
< 
< Here is my attempt to solve this problem. I first save the dependencies, then
< remove all dependencies, calculate the partials (via diff) and then replace
< the dependencies and calculate the total derivative with respect to the time
< variable. Is there anything that can go wrong in removing and replacing the
< dependencies? Is this function referentially transparent? (ie. has no
< externally apparent side effects?)
< 
< EuLagEquations(expr,var,vardot,t) :=
< block([deps:copylist(dependencies), term1, term2],
<   print(deps),
<   remove(all,dependency),
<   term1: diff(expr,vardot), term2: -diff(expr,var),
<   for i in deps do (print(i),depends(op(i),args(i))),
<   diff(term1,t)+term2=0);
 
Here is a pretty transparent function which works for multiple
variables. I add a dependency in the function itself, but
otherwise I don't mess with things.

el(L,x,v,[t]):=block([Lv,Lx],
                 t:if t=[] then 't elseif symbolp(part(t,1)) then
part(t,1) else merror("el(L,x,v,[t]): t must be an unbound symbol."),
                 x:if listp(x) then x elseif symbolp(x) then [x] else
merror("el(L,x,v,[t]): x, v must be scalars or lists."),
                 v:if listp(v) then v elseif symbolp(v) then [v] else
merror("el(L,x,v,[t]): x, v must be scalars or lists."),
                 Lx:map(lambda([tau],diff(L,tau,1)),x),
                 Lv:map(lambda([tau],diff(L,tau,1)),v),
                 depends(v,t),
                 flatten([map("=",diff(Lv,t),Lx),map(lambda([xi,eta],eta=diff(xi,t)),x,v)]));
depends(L,[a,b,u,v],[a,b,u,v],t,V,w,[w,z],s);
el(L,[a,b],[u,v]);
W : z^2/2 - V;
EL_W : el(W,w,z,s);
E : W + 2*V;
diff(E,s);
subst(EL_W,%);  <--conservation of energy

Leo


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