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.