MaplePrimes - answers and comments on Question, Euler-Lagrange equation

frontend

Joe Riel — Fri, 10 Sep 2010 18:48:06 Z

One approach is to use ?frontend. For example,

(**) z := y^2*diff(y(x),x)^3;         
                                    2 /d      \3
                              z := y  |-- y(x)|
                                      \dx     /

(**) frontend(diff, [z,diff(y(x),x)]);
                                   2 /d      \2
                                3 y  |-- y(x)|
                                     \dx     /

Here is a post describing some of the subtleties of using frontend.

difff from Aladjev's library

djc — Fri, 10 Sep 2010 21:07:45 Z

You can use difff from the free Aladjev's library:

http://www.download.com/Aladjev-s-Library-for-Maple/3000-2070_4-10716687.html

F:=y(x)^2;

difff(F,y(x));

2*y(x)

z := y^2*diff(y(x),x)^3;

difff(z,diff(y(x),x));

3*y^2*(diff(y(x), x))^2

Look at recent question

hirnyk — Fri, 10 Sep 2010 23:15:05 Z

Look at that . PS. Found by the "differentiation" search in MaplePrimes.

Ahh thanks.... the infamous frontend ! I've

longrob — Fri, 10 Sep 2010 21:04:36 Z

Ahh thanks.... the infamous frontend ! I've always had trouble getting my head around it ! The help pages are a bit terse - thanks for the link - I think I've managed it now. At least it gives me the result I expected. Here's an example of the kind of functional I'm dealing with

z := exp(-x)*sqrt(y(x)-exp(x)*diff(y(x),x));

z1:=frontend(diff, [z,diff(y(x),x)],[{`*`, `+`, radical}]);

z2:=frontend(diff, [z,y(x)],[{`*`, `+`, radical}]);

z3:=diff(z1,x);

EL:=simplify(z3-z2);

VC:=VariationalCalculus[EulerLagrange](z,x,y(x));

is(VC[1]=EL);

is(VC[1]=-EL);

#...which was the whole point of doing this.

# Hooray !!

For completeness, here's how I did it with subs

F := z;

F1:=subs(diff(y(x),x)=yp,F);
F2:=subs(y(x)=y,F1);
F3:=diff(F2,y);
F4:=subs(y=y(x),F3);
F5:=subs(yp=diff(y(x),x),F4); # This gives the partial wrt to y(x)

# Then...
FF1:=diff(F1,yp);
FF2:=subs(yp=diff(y(x),x),FF1); # This gives the partial wrt to diff(y(x),x)

# So now we can formulate the Euler-Lagrange equation:
FF3:=diff(FF2,x); #This gives the derivative with respect to x of the partial derivative with respect to diff(y(x),x)
ELL:=simplify(FF3-F5);

is(EL=ELL);

# Hooray again !

Are there any other ways of doing this ?

Just in case anyone happens to come to this

longrob — Mon, 04 Oct 2010 23:28:36 Z

Just in case anyone happens to come to this thread with the same question/problem I had, the best solution is actually in the post linked by hirnyk above, but in case that gets overlooked, the answer is to use the diff command in the Physics package:

eg:

z := exp(-x)*sqrt(y(x)-exp(x)*diff(y(x),x)):

Physics:-diff(z,diff(y(x),x));
Physics:-diff(z,y(x));

Calculation of the Euler-Lagrange (or Lagrange-Eul...

lemelinm — Wed, 08 Jul 2020 21:06:59 Z

@longrob

Just to mention that 10 years later, I come accross that simple way to dol it in Maple 2020. So I was able to do manually the calculation and check with Maple if I was right. Thank you.