# Calculus of Variation: Extremum & Further Variances

• LagrangeEuler
In summary, the conversation discusses the conditions for a functional to have an extremum, specifically a minimum or maximum. It is stated that if δI=0, the functional is at an extremum. However, if δ^2I>0, it is a minimum and if δ^2I<0, it is a maximum. If δ^3I>0, it is a minimum. It is also mentioned that finding further variations is necessary in some cases, and the conversation suggests referring to external sources for a better understanding of these conditions.
LagrangeEuler
If for some functional ##I##, ##δI=0## where ##δ## is symbol for variation functional has extremum. For ##δ^2I>0## it is minimum, and for ##\delta^2I>0## it is maximum. What if
##δI=δ^2I=0##. Then I must go with finding further variations. And if ##δ^3I>0## is then that minimum? Or what?

Finding further variations is useless from this point of view.
The stationarity of the functional, i.e. δI=0 , occurs for maxima, minima and saddles.

So then how I could know? Is it minimum or maximum?

Here you will find a better explanation than I could give on sufficient and necessary conditions for minima http://www.math.utah.edu/~cherk/teach/12calcvar/sec-var.pdf
If you have the book "introduction to Calculus of Variations" by Fox you will find there a thorough discussion of the second variation: yes further variations are to be computed.
I really do apologise for my previous reply which was wildly inaccurate due to a misunderstanding of mine.

## 1. What is the calculus of variations?

The calculus of variations is a mathematical theory that deals with finding the optimal solution to a functional, which is a function that takes in other functions as input. It involves finding the extrema of the functional, which are the maximum and minimum values, and the corresponding function that produces those values.

## 2. What is an extremum?

An extremum is a point where a function reaches either a maximum or minimum value. In the calculus of variations, we are interested in finding the extrema of a functional, which is a function that takes in other functions as input.

## 3. How is the calculus of variations used in real-world applications?

The calculus of variations has numerous applications in various fields such as physics, engineering, and economics. It is used to optimize various systems and processes, such as finding the shortest path between two points, determining the shape of a bridge that can withstand the most weight, and minimizing energy consumption in mechanical systems.

## 4. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in the calculus of variations that is used to find the extrema of a functional. It is derived from the principle of least action and provides a necessary condition for a function to be an extremum of a functional.

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

One limitation of the calculus of variations is that it cannot always find the global optimum solution, especially for non-convex functions. It may also be computationally expensive to solve for the extremum, especially for complex functions with multiple variables. Additionally, the calculus of variations may not have a unique solution and may yield multiple extrema for a given functional.

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