Calculus of Variation

• HAMJOOP
In summary, the conversation discusses the use of Euler's equation in calculus of variation to minimize integrals. It is mentioned that there is no algebraic relation between a function and its derivative, which is why boundary conditions are necessary to solve differential equations. However, it is argued that the real reason for treating y and y' as independent is to obtain an expression for the difference in terms of partial derivatives. Additionally, it is noted that while y and y' may appear to be independent variables, they are actually dependent on the function f(x) being sought.

HAMJOOP

In calculus of variation, we use Euler's equation to minimize the integral.

e.g. ∫f{y,y' ;x}dx

why we treat y and y' independent ?

Because there is no algebraic relation between a function and its derivative.

This is why you need boundary conditions to solve differential equations.

UltrafastPED said:
Because there is no algebraic relation between a function and its derivative.

This is why you need boundary conditions to solve differential equations.

Sorry, but this is a bogus answer. A function may depend on another function non-algebraically, and that is perfectly fine as far as functional dependency goes. Not to mention that the dependency may perfectly well be algebraic.

The real reason is that we use the partial derivatives to obtain an expression for the difference ## F(z + \Delta z, y + \Delta y, x) - F(z, y, x) ##, which is approximately ## F_z \Delta z + F_y \Delta y ## when ##\Delta z## and ##\Delta y## are sufficiently small. This expression is true generally, and is true when ## z ## represents the derivative of ## y ## - all it takes is that the variations of both must be small enough. If ## y = f(x) ##, its variation is ## \delta y = \epsilon g(x) ##, and ## \delta y' = \epsilon g'(x)##. If ## \epsilon ## is small enough, then using the result above, ## F((y + \delta y)', (y + \delta y), x) - F(y', y, x)) \approx \epsilon F_{y'}g'(x) + \epsilon F_y g(x) ##, where ##F_{y'}## is just a fancy symbol equivalent to ##F_z##, meaning partial differentiation with respect to the first argument. Then we use integration by parts and convert that to ## \epsilon (-(F_{y'})' + F_y) g(x)##. Observe that we do use the relationship between ## y ## and ## y' ## in the final step.

Would the following also be correct reasoning?

We want to find the least action for:

##S = \int_{x_1}^{x_2} f(y,y',x) \, dx##

While this may look as though y, y' and x are simple independent variables, since we are actually looking for the function f(x) that provides this least action, what this notation really means is this:

##S = \int_{x_1}^{x_2} f[y(x), \frac d {dx} y(x), x] \, dx##

So y and y' are not truly independent.

1. What is the calculus of variations?

The calculus of variations is a branch of mathematics that deals with finding the optimal value of a function, also known as the extremum, by considering variations in the function. It is used to solve problems involving optimization, such as finding the shortest path or minimizing energy usage.

2. What is the difference between the calculus of variations and traditional calculus?

The main difference between the two is that traditional calculus deals with finding the maxima and minima of a function with a fixed set of variables, while the calculus of variations considers variations in the function itself. Traditional calculus is also concerned with finding exact solutions, whereas the calculus of variations focuses on finding the optimal solution within a certain range of values.

3. What are some common applications of the calculus of variations?

The calculus of variations has various applications in physics, engineering, economics, and other fields. Some examples include finding the path of least resistance in fluid dynamics, determining the shape of a hanging chain, and optimizing the shape of an airplane wing for maximum lift.

4. How does the calculus of variations relate to the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in the calculus of variations that is used to find the extremum of a functional. It relates the derivative of the function to the function itself and its derivatives. Solving this equation yields the optimal solution for the given problem.

5. Are there any limitations to the calculus of variations?

While the calculus of variations is a powerful tool for solving optimization problems, it does have some limitations. It can only be applied to functions that have a smooth and continuous behavior, and it does not always guarantee finding the global extremum. Additionally, some problems may have no solution or an infinite number of solutions, making it difficult to determine the optimal value.