Solution of Euler-Lagrange equation

In summary, the given conversation discusses the Lagrangian, Euler-Lagrange equations, and the problem of finding constant and static solutions for a real scalar field in spacetime. The solution of constant solutions is relatively easy, while the solution for static solutions can be found by solving a stationary nonlinear Schrodinger equation. There may be other approaches to solving this differential equation, such as finding a first integral or using physical explanations. Additionally, there may be a soliton-like solution for this problem. The conversation also mentions the use of a Hamiltonian and the existence of homoclinic and heteroclinic orbits in the phase plane of the second order equation.
  • #1
parton
83
1
I have the following Lagrangian:

[tex] \mathcal{L} = 1/2 \partial_{\mu} \varphi \partial^{\mu} \varphi - 1/2 b ( \varphi^{2} - a^{2} )^{2} [/tex], where [tex] a,b \in \mathbb{R}_{>0} [/tex] and [tex] \varphi [/tex] is a real (scalar) field and x are spacetime-coordinates.

I calculated the Euler-Lagrange eq. and get: [tex] \square \varphi + 2 b ( \varphi^{2} - a^{2} ) \varphi = 0 [/tex]

My problem is now to find constant solutions and static ones like [tex] \varphi(x) = f(x-x_{0}) [/tex] where [tex] x_{0} [/tex] is constant. But I don't know how to solve the differential equation above. Does anyone have an idea?
 
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  • #2
Constant solutions are easy to calculate (there are 3 of them), as [itex]\square \varphi[/itex] is zero if [itex]\varphi[/itex] is constant.

Now, for the other question, I'm assuming that for static solution you mean independent of time. If so, [itex]\square \varphi = \nabla^2 f[/itex], which leads to the stationary NLS ecuation with constant potential, and there is plenty of literature on how to solve it.
 
  • #3
AiRAVATA said:
Constant solutions are easy to calculate (there are 3 of them), as [itex]\square \varphi[/itex] is zero if [itex]\varphi[/itex] is constant.

I recognise the solution if [itex]\varphi[/itex] is constant.

[itex]\varphi = a[/itex] , [itex]\varphi = -a[/itex] and [itex]\varphi = 0[/itex]. :biggrin:
 
  • #4
Thank you for your replies!

Remark: I think it should be: [tex] \square \varphi = - \nabla^{2} f [/tex]

But isn't there any other (easier) approach to solve this differential equation? I thought of something like finding a first integral or something like that, but I can't. Or maybe there is a (simple) physical explanation which could lead to a solution?
 
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  • #5
There should be a soliton-like solution arising from from energy conservation. For example, let [itex]x \in \mathbb{R}[/itex], then

[tex]-f'' + 2b(f^2-a^2)f=0.[/tex]

Multiplying by f' and integrating,

[tex]-\frac{f'^2}{2} + b(\frac{f^2}{2}-a^2)f^2 = C,[/tex]

where C is a constant of integration determined by the boundary conditions (i.e. [itex]f,f'\rightarrow 0[/itex] as [itex]x \rightarrow \infty[/itex]). The remainding equation is separable and solvable (in terms of elliptic functions maybe?), and if you study the face plane of the second order equation, you can prove that there is a soliton-like solution (depending of the existence of homoclinic or heteroclinic orbits).

In more variables, you should be able to build a Hamiltonian and see if that approach gives you some additional info.
 
  • #6
AiRAVATA said:
[tex]-\frac{f'^2}{2} + b(\frac{f^2}{2}-a^2)f^2 = C,[/tex]

Yeap, it is really difficult to expressed f in term of x. Soliton?
wave that maintains its shape while it travels at constant speed. http://en.wikipedia.org/wiki/Soliton" [Broken] without equation!

But what are face plane? homoclinic or heteroclinic orbits ? :cry:
 
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  • #7
If C=0 is not that hard to solve, the equation is separable and i think that the change of variable f = sech² x will do the trick. Even if the constant is different than zero, you can solve the integral in terms of elliptic functions (I believe).
Regarding the phase plane or portrait (my native language betrayed me there), you can read all about it in a basic DE book like the one of Braun or Boyce, homoclinic and heteroclinic orbits can be seen in more advanced books, like the second volume of Hubbard's dynamical systems book.
 
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Question 1: What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical formula used to find the extrema (maximum or minimum) of a functional, which is a mathematical expression that depends on a function. It is commonly used in the field of calculus of variations to solve optimization problems.

Question 2: What is the significance of the Euler-Lagrange equation in science?

The Euler-Lagrange equation is significant in science because it helps us find the optimal solution for a given problem. It has applications in various fields such as physics, engineering, economics, and more. It allows us to optimize processes and systems in a mathematical and rigorous way.

Question 3: How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived from the calculus of variations, which involves finding the extrema of a functional. It is a result of the fundamental lemma of the calculus of variations, which states that the functional derivative of the functional must equal zero at the extremum.

Question 4: What are the assumptions made in the derivation of the Euler-Lagrange equation?

The derivation of the Euler-Lagrange equation assumes that the function being optimized is continuous and has continuous first-order derivatives. It also assumes that the boundaries of the function are fixed and that it is differentiable within these boundaries.

Question 5: Can the Euler-Lagrange equation be solved analytically?

In most cases, the Euler-Lagrange equation cannot be solved analytically and requires numerical methods to find a solution. However, there are some special cases where an analytical solution can be found, such as for simple systems with linear or quadratic functionals.

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