Variational principle

In summary, to find the critical points, we need to take the variation of the functional with respect to \eta and \phi, which leads to the two coupled equations \partial_t \eta - g\eta = 0 and \partial_t \eta - h \partial_x^2 \phi = 0.
  • #1
dirk_mec1
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Homework Statement



Consider the variation principle for the space-time functional of the variables [tex] \eta, \phi [/tex]

[tex] A( \eta, \phi) = \int \int \phi \partial _t \eta -\frac{1}{2}g \eta ^2 -\frac{1}{2} h ( \partial_x \phi ) ^2\ \mbox{d}x \mbox{d}t [/tex]Derive the two coupled equations for the critical points.

Homework Equations


[tex] \eta, \phi [/tex] are functions x and t and are the surface elevation and the potential at the surface respectively, g is gravitation constant.

Fluid depth is h and h=h(x).

The Attempt at a Solution


Do I need to find the dervatives of this double integral? To be honest I don't know how to start.
 
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  • #2


it is important to approach problems in a systematic and logical manner. In this case, the first step would be to carefully read and understand the given functional and its variables.

The variation principle states that for a functional to have a critical point, its variation must be zero. In this case, we are looking for the critical points of the space-time functional A( \eta, \phi).

To find these critical points, we need to take the variation of the functional with respect to \eta and \phi separately. This means finding the derivatives of the functional with respect to \eta and \phi.

Taking the derivative of the functional with respect to \eta, we get:

\frac{\partial A}{\partial \eta} = \int \int \partial_t \eta - g\eta \ \mbox{d}x \mbox{d}t

Similarly, taking the derivative of the functional with respect to \phi, we get:

\frac{\partial A}{\partial \phi} = \int \int \partial_t \eta - h \partial_x^2 \phi \ \mbox{d}x \mbox{d}t

Now, for the functional to have a critical point, both of these derivatives must be equal to zero. This leads to the following two coupled equations:

\partial_t \eta - g\eta = 0

\partial_t \eta - h \partial_x^2 \phi = 0

These are the two coupled equations for the critical points of the given space-time functional.
 

1. What is the variational principle?

The variational principle is a mathematical concept used in physics and engineering to find the best approximation to the solution of a problem. It states that the actual solution to a problem is the one that minimizes a certain functional, or mathematical expression, that represents the problem.

2. How is the variational principle used in science?

The variational principle is used in various fields of science, such as mechanics, electromagnetism, quantum mechanics, and fluid dynamics. It is used to simplify complex problems and find accurate solutions to physical systems.

3. What is the difference between the variational principle and the principle of least action?

The variational principle and the principle of least action are closely related but not interchangeable. The variational principle is a mathematical concept that applies to a wide range of problems, while the principle of least action specifically applies to the motion of particles in a physical system. The principle of least action is a special case of the variational principle.

4. Who first proposed the variational principle?

The variational principle was first proposed by the mathematician and physicist Pierre Louis Maupertuis in the 18th century. However, it was later refined and popularized by other scientists, such as Lagrange and Hamilton, in the 19th century.

5. What are some real-life applications of the variational principle?

The variational principle has many practical applications in science and engineering. Some examples include finding the most efficient path for a spacecraft to travel in space, determining the shape of a bridge that minimizes stress and maximizes stability, and predicting the behavior of atoms and molecules in chemical reactions.

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