Differentiating equations for given model

[Railgun]

Homework Statement

I need to derive Euler-Lagrange equations and natural boundary conditions for a given model. I've worked out and broken down the model into the following 5 parts:

J1 = ∫ {ϕ>0} |f(x) − u+(x)|^2dx

J2 = ∫ {ϕ<0} |f(x) − u-(x)|^2dx

J3 = ∫ Ω |∇H(ϕ(x))|dx

J4 = ∫ {ϕ>0} |∇u+(x)|^2dx

J5 = ∫ {ϕ<0} |∇u-(x)|^2dx.

where f : Ω → R and u+- ∈ H^1(Ω) (functions such that ∫ Ω(|u|^2 + |∇u|^2)dx < ∞).

I need to differentiate each of these 5 equations in terms of ϕ,u+ and u-, any assistance would be very appreciated as I'm weak in calculus.

Homework Equations

The Attempt at a Solution

I tried getting the first variation, for example for J1(ϕ),

let v be a perturbation defined in a space V such that J(v) exists.

deltaJ1(ϕ) = lim{epilson->0} [J1(ϕ+epilson.v)-J1(ϕ)]/epilson

= lim{epilson->0} ∫{ϕ+epilson.v>0}lf(x)-u+(x)l^2dx-∫{ϕ>0}lf(x)-u+(x)l^2dx From here onwards I'm not sure how to proceed.

 

[mighty2000]

Thank you for your post. I am happy to assist you in deriving the Euler-Lagrange equations and natural boundary conditions for your given model. Before we begin, it would be helpful to have a clear understanding of the variables and functions involved in your model. Could you please provide some background information or context for your model? This will help me to better understand the problem and provide a more accurate solution.

In general, the Euler-Lagrange equations can be derived by setting up a functional and then finding the critical points of that functional. In your case, you have a functional that is composed of five parts, each of which involves the functions ϕ, u+, and u-. To differentiate each of these five equations, we will need to apply the chain rule and product rule of calculus.

Let's start with J1(ϕ). As you correctly stated, we need to find the first variation of this functional. This can be done by setting up a perturbation v and then taking the limit as epsilon approaches 0. This will give us a general expression for the first variation of J1(ϕ), which we can then simplify.

deltaJ1(ϕ) = lim{epsilon->0} [J1(ϕ+epsilon.v)-J1(ϕ)]/epsilon

= lim{epsilon->0} ∫{ϕ+epsilon.v>0} |f(x)-u+(x)|^2 dx - ∫{ϕ>0} |f(x)-u+(x)|^2 dx

= lim{epsilon->0} ∫{ϕ+epsilon.v>0} (f(x)-u+(x))^2 dx - ∫{ϕ>0} (f(x)-u+(x))^2 dx

= lim{epsilon->0} ∫{ϕ+epsilon.v>0} (f(x)^2 - 2f(x)u+(x) + u+(x)^2) dx - ∫{ϕ>0} (f(x)^2 - 2f(x)u+(x) + u+(x)^2) dx

= lim{epsilon->0} ∫{ϕ+epsilon.v>0} f(x)^2 dx - 2∫{ϕ+epsilon.v>0} f(x)u+(x) dx + ∫{ϕ+epsilon.v>