Interesting calculus of variations problems?

[Montrealist]

Hi, I would like to know if anyone has good ideas for problems involving calculus of variations, other than the classic textbook questions (brachistochrone, Fermat, catenary, etc..) that I could create as a classical mechanics class project? Thank you

 

[bbbeard]

Hi, I would like to know if anyone has good ideas for problems involving calculus of variations, other than the classic textbook questions (brachistochrone, Fermat, catenary, etc..) that I could create as a classical mechanics class project? Thank you

How about verifying that the path of a projectile in a uniform gravitational field really does minimize the action?

Since we know the path from Newtonian mechanics is a parabola

y = h - ax² if x is measured from the peak y=h,

it would be interesting to pick a one-parameter family of curves, say

y = h - b_nxⁿ

that has the same endpoints. Then calculate the action (the time integral of kinetic minus potential energy) as a function of n and show that it is minimized for n=2.

BBB

 

[stargazer_iq]

check the "structure and interpretation of classical mechanics" book (available online)

 

[ZapperZ]

All discussion on such topic must be done in the HW/Coursework forum.

Zz.

 

[pmacias]

.

As a fellow scientist, I find the topic of calculus of variations to be quite fascinating as it allows us to find the optimal solution to a problem by minimizing or maximizing a certain functional. In terms of interesting problems, I would suggest looking into the application of calculus of variations in economics, where it is used to find the optimal production or consumption levels for a given system. Another interesting area to explore could be in fluid mechanics, where the calculus of variations is used to derive the equations of motion for a fluid flow. Additionally, you could also look into problems involving the principle of least action in classical mechanics, which is closely related to the calculus of variations. I hope these suggestions help in creating a unique and challenging project for your class. Good luck!