How to find the minimum of an integral with calculus of variations

In summary, the conversation discusses the need to find the minimum of a given integral, which is subject to a constraint. The integral is calculated over a fixed interval and involves constants and a function y of x. The problem is not a Calculus of Variations problem, but rather a Lagrangian type problem. The integrand is a function of x and y, and can be minimized by setting its partial derivative with respect to y equal to 0 for each x. The resulting function y contains a parameter r, which can be determined by setting the integral of y over the interval equal to a constant N. This may require a numerical approach and for certain values of the constants, there may not be a minimum at all.
  • #1
fedefrance
2
0
I need to find the minimum of this integral

F=∫ (αy^-1+βy^3+δxy)dx

where α, β and δ are constant; y is a function of x

the integral is calculated over the interval [0,L], where L is constant

I need to find the function y that minimizes the above-mentioned integral

The integral is subject to the following constraint

N=∫ydx

where N is a constant and the integral interval is again [0,L]Anyone can help?
Is it possible to find an analytical solution?
Thanks

Ps:Sorry for the bad format, it's my first post
 
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  • #2
fedefrance said:
I need to find the minimum of this integral

F=∫ (αy^-1+βy^3+δxy)dx

where α, β and δ are constant; y is a function of x

the integral is calculated over the interval [0,L], where L is constant

I need to find the function y that minimizes the above-mentioned integral

The integral is subject to the following constraint

N=∫ydx

where N is a constant and the integral interval is again [0,L]


Anyone can help?
Is it possible to find an analytical solution?
Thanks

Ps:Sorry for the bad format, it's my first post

This does not look anything like a Calculus of Variations problem, because dy/dx is not involved in the integrand. Instead, you can just minimize the integrand for each x (to get a function y(x)). More precisely, you can look at the "Lagrangian" type problem, where yu want to minimize int f(x,y) dx + r* int y dx with no constraints; here, r is a "lagrange multiplier" and note that it is a constant, not a function of x. So, your integrand is of the form
[tex]f(x,y) = \frac{a}{y} + b y^3 + c x y + r y,[/tex]
where I have used 'a' instead of , 'b' instead of β and 'c' instead of δ. If a > 0 and b > 0 we can minimize f by setting [itex] \partial f/\partial y = 0 [/itex] for each x and solve for y. There are 4 roots, but for b > 0 it seems there are only two relevant roots, both of which contain the parameter r. Determine r by asking that int y dx = N. (This will be a nasty problem that almost certainly needs a numerical approach for given a, b, c and L.)

If a and/or b < 0, there may not be a minimum at all; we may be able to find a sequence y_n(x) giving int f(x,y_n(x)) dx --> -infinity, while keeping int y_n(x) dx = N for each n. (I am not absolutely sure about this, but I think it is true.)

RGV
 
  • #3
thanks a lot!
 

1. How is the minimum of an integral with calculus of variations found?

The minimum of an integral with calculus of variations is typically found by using the Euler-Lagrange equation, which is derived from the principle of least action. This equation involves taking the derivative of the integrand with respect to the function, setting it equal to zero, and solving for the function that minimizes the integral.

2. What is the significance of finding the minimum of an integral with calculus of variations?

Finding the minimum of an integral with calculus of variations is important in many fields of science and engineering. It allows us to optimize various systems and processes by finding the most efficient or optimal solution. It is also a fundamental tool in the study of mechanics, electromagnetism, and other physical phenomena.

3. Can the minimum of an integral be found analytically?

In some cases, the minimum of an integral with calculus of variations can be found analytically, meaning that an exact solution can be obtained. However, in more complex systems, numerical methods may be required to approximate the minimum.

4. Are there any limitations to using calculus of variations to find the minimum of an integral?

One limitation of using calculus of variations is that it only applies to integrals with a single independent variable. Additionally, the function being minimized must satisfy certain boundary conditions. In some cases, other techniques such as optimization algorithms may be more suitable for finding the minimum of an integral.

5. What are some real-world applications of finding the minimum of an integral with calculus of variations?

The use of calculus of variations to find the minimum of an integral has many practical applications. For example, it can be used to optimize the shape of a bridge or airplane wing to reduce drag and improve efficiency. It is also used in economics to find the most cost-effective solution to a problem, and in image and signal processing to enhance and compress data.

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