In this video, I solve the catenary problem. A catenary is a curve that describes the shape of a string hanging under gravity, fixed on both of its ends. Here, I determine the equation of the catenary for a uniform string with both ends fixed at the same height using the techniques of Variational Calculus (with constraints).

I begin by deriving the two integrals necessary to solve the constrained variation problem. The first integral is the constraint integral, according to which the length of the string is fixed. The second integral is formed by evaluating the total gravitational potential energy of the entire string (the functional to minimize).

After setting up the integrals, I create a composite functional K which includes my potential energy + (Lagrange multiplier)*(constraint integral). Applying the Euler-Lagrange equation/Beltrami Identity to K and solving the resulting differential equation gives me the equation for my catenary. Finally, I compute the 3 unknown constants in the catenary equation using the constraint and the two boundary conditions, and I show that the equation of a catenary takes the form of a hyperbolic cosine.

Questions/requests? Let me know in the comments!

Pre-reqs: This playlist (especially the video on constraints + multiple dependent variables): [ Ссылка ]_

Lecture Notes: [ Ссылка ]
Patreon: [ Ссылка ]
Twitter: [ Ссылка ]

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- Jose Lockhart
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