A calculus of variations question

In summary, the Euler-Lagrange equation derives the extremum of a function with respect to a given parameter.
  • #1
Saketh
261
2
I am trying to learn the calculus of variations, and I understand the mathematical derivation of the Euler-Lagrange equation.

As I understand it, the calculus of variations seeks to find extrema for functions of the form:
[tex]S[q,\dot{q}, x] = \int_{a}^{b} L(q(x),\dot{q}(x), x) \,dx[/tex].

Here is my question: Why is it necessary to have [itex]\dot{q}[/itex] as an argument of [itex]L[/itex]? Isn't the derivative of the function [itex]q[/itex] "included" with the function itself?

Thanks in advance.
 
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  • #2
L is a function of three variables L(z,y,x) where we consider the cases where z = q(x), y = q'(x). Now when we write L(z,y,x), then it makes sense to consider (dL/dy)(z,y,x). If we consider L a function of two variables, writing L(z,x), then how can we make sense of the derivative of L with respect to (dz/dx)(x)? So although the second argument of L ends up being the derivative of the first argument w.r.t. the third (and so, seems to already be "contained" in the other arguments), if we want to do things like take the partial of L with respect to q'(x), we need to have q'(x) occupy its own argument.
 
  • #3
AKG said:
L is a function of three variables L(z,y,x) where we consider the cases where z = q(x), y = q'(x). Now when we write L(z,y,x), then it makes sense to consider (dL/dy)(z,y,x). If we consider L a function of two variables, writing L(z,x), then how can we make sense of the derivative of L with respect to (dz/dx)(x)? So although the second argument of L ends up being the derivative of the first argument w.r.t. the third (and so, seems to already be "contained" in the other arguments), if we want to do things like take the partial of L with respect to q'(x), we need to have q'(x) occupy its own argument.
Thanks for the explanation - I understand now.

I suppose this is something peculiar to functionals, but it's simple enough.
 

1. What is a calculus of variations question?

A calculus of variations question is a mathematical problem that involves finding the optimal solution for a functional. This functional is typically expressed as an integral, and the goal is to find the function that minimizes or maximizes this integral. It is an important tool in optimization and is used in various fields such as physics, economics, and engineering.

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

The main difference between a calculus of variations question and a traditional calculus problem is that the former deals with functions of functions. In other words, instead of finding the optimal value for a given function, we are finding the optimal function itself. This requires a different approach and uses techniques such as the Euler-Lagrange equation.

3. What are some real-world applications of calculus of variations?

Calculus of variations has many real-world applications, including finding the optimal path for a moving object, minimizing energy consumption in a system, and maximizing profit in economics. It is also used in physics to find the path of a particle that minimizes the action, which is a measure of the particle's motion.

4. Is calculus of variations a difficult concept to understand?

Calculus of variations can be challenging for beginners as it requires a solid understanding of traditional calculus concepts. However, with proper instruction and practice, it can be understood by anyone with a strong mathematical background. It is important to have a solid understanding of calculus, integration, and differential equations before attempting to learn calculus of variations.

5. How is calculus of variations related to optimization?

Calculus of variations is closely related to optimization as it is used to find the optimal solution for a given problem. By finding the function that minimizes or maximizes a certain integral, we are essentially finding the best possible outcome for a given situation. This makes it a powerful tool in optimization, and it is commonly used in engineering and economics to solve real-world problems.

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