Variational Calculus
Functional Differentiation
The Euler-Lagrange equations work Lagrangians of the form: DFunc( Id ,y)(phi,x) = delta(x-y)Functional Derivative
The main result is \( \delta f(x)/ \delta f(y) = \delta(x-y) \)Are there axioms like the product rule?
Investigating if there are axioms similar to the product rule for the functional derivative \( \frac{\delta}{\delta f(x)} \)Let \( Delta[L](x) = \frac{\delta}{\delta \phi(x)} \int L(\phi, \phi', \phi'', ...)dx \)
The formula is \( Delta[ \int f(Id, D, DoD, DoDoD,...)] = f^{[0]} - D o f^{[1]} + D o D o f^{[2]} - .... \)
Is there an iterative formula?
delta Id = 1
delta D = 0
delta D o F = 0
delta A x (D o B) = - delta (D o A) x B
delta L(D o A) = - D o L'(D o A) // delta L o (G o A) = G o (delta G) o A ???
For the functional derivative delta/delta phi(x) works with partial derivative. This doesn't work with delt/delta int [L]
Partial derivative satisfy delta D = diracdelta