Minimisation Problem (Euler-Lagrange equation)

In summary: It becomes a trivial equation, as all terms involving y and its derivatives cancel out, leaving everything equal to 0. This is not a useful solution, and so you need to find a different approach to solve the problem.
  • #1
Plaetean
36
0

Homework Statement


http://i.imgur.com/BV5gR8q.png

Homework Equations


d/dx ∂F/∂y'=∂F/∂y

The Attempt at a Solution


I have no problem with the first bit, but the second bit is where I get stuck. Since the question says the speed is proportional to distance, I have taken v(x)=cx where c is some constant of proportionality, then tried to proceed with solving the EL equation, however I end up with everything just being equal to 0 so I'm not sure how to proceed.

http://i.imgur.com/hDNPyG7.jpg
 
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  • #2
Hi Plaetean:

I think you forgot that y' is a function of x.

Your (d/dx) ∂F/dy' does not include any dy'/dx = y''.

Hope this helps.
Regards,
Buzz
 
  • #3
Plaetean said:

Homework Statement


http://i.imgur.com/BV5gR8q.png

Homework Equations


d/dx ∂F/∂y'=∂F/∂y

The Attempt at a Solution


I have no problem with the first bit, but the second bit is where I get stuck. Since the question says the speed is proportional to distance, I have taken v(x)=cx where c is some constant of proportionality, then tried to proceed with solving the EL equation, however I end up with everything just being equal to 0 so I'm not sure how to proceed.

http://i.imgur.com/hDNPyG7.jpg
What does your Euler-Lagrange equation boil down to if F is not a function of y?
 

1. What is the minimisation problem in the context of the Euler-Lagrange equation?

The minimisation problem in the context of the Euler-Lagrange equation refers to finding the function or curve that minimises a given functional. This functional represents a quantity that depends on the function or curve and its derivatives. The solution to this problem is obtained by solving the Euler-Lagrange equation, which is a differential equation that characterises the extrema of the functional.

2. What is the significance of the Euler-Lagrange equation in minimisation problems?

The Euler-Lagrange equation is significant in minimisation problems because it provides a systematic method for finding the function or curve that minimises a given functional. This equation is derived from the calculus of variations and is a necessary condition for the existence of an extremum of the functional. It also serves as a powerful tool for optimisation and has applications in various fields such as physics, engineering, and economics.

3. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived by applying the calculus of variations to a given functional. This involves finding the functional's stationary points, which are the points where the functional does not change its value under small variations of the function or curve. By considering these stationary points, the Euler-Lagrange equation is obtained, which is a second-order, non-linear differential equation that characterises the extrema of the functional.

4. What are the assumptions made in the Euler-Lagrange equation?

The Euler-Lagrange equation makes several assumptions, including the existence of a functional that is continuous and has well-defined derivatives, the existence of a unique extremum, and the existence of boundary conditions. These assumptions are necessary for the derivation of the equation and for obtaining a unique solution to the minimisation problem.

5. What are some applications of the Euler-Lagrange equation?

The Euler-Lagrange equation has numerous applications in physics, engineering, economics, and other fields. It is used to find the path of a particle that minimises the time taken to travel between two points, to determine the shape of a cable hanging under its own weight, and to optimise the trajectory of a rocket. It is also used in the calculus of variations, optimal control theory, and field theory.

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