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Question: Laplacian and Euler Lagrange equation

Question:Laplacian and Euler Lagrange equation

maple2015 140 Maple
vectorcalculus laplacian euler-lagrange
February 15 2019
1

Hi

Please download the attachment.

 

I try to find a relation between EL and Lap(EL) in polar coordinate for one variable function w(r), where Lap is laplacian and EL is Euler Lagrange equation. Please check the Maple code and help me to do some manipulations to find a general relation (if any relation exists!).

In fact I need the inverse of Euler Lagrange equation to obtain f(r) for an arbitrary function g(r) in equation below

EL(f) = Lap(EL(g))

Or f=inverseEL(Lap(EL(g)))

Thank you for taking your time

 

 

 

restart; s := proc (f) subs(d[0] = w(r), seq(d[n] = diff(w(r), `$`(r, n)), n = 1 .. 10), f) end proc; ss := proc (f) subs(seq(diff(w(r), `$`(r, 11-n)) = d[11-n], n = 1 .. 10), w(r) = d[0], f) end proc; EL := proc (eq) s(diff(ss(eq), d[0]))+add((diff(s(diff(ss(eq), d[n])), `$`(r, n)))*(-1)^n, n = 1 .. 10) end proc

f := (diff(w(r), r, r))^2*r^4+4*r^6*(diff(w(r), r, r, r))^2:

a1 := EL(F):

a2 := VectorCalculus:-Laplacian(EL(f), 'polar[r, t]'):

simplify(a1-a2)

8*r^6*(diff(diff(diff(diff(diff(diff(diff(diff(w(r), r), r), r), r), r), r), r), r))+248*r^5*(diff(diff(diff(diff(diff(diff(diff(w(r), r), r), r), r), r), r), r))+2582*r^4*(diff(diff(diff(diff(diff(diff(w(r), r), r), r), r), r), r))+10910*r^3*(diff(diff(diff(diff(diff(w(r), r), r), r), r), r))+17786*r^2*(diff(diff(diff(diff(w(r), r), r), r), r))+8192*r*(diff(diff(diff(w(r), r), r), r))-92*(diff(diff(w(r), r), r))

 (1)

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