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Hilbert's 19th problem asks:

Are the solutions of regular problems in the calculus of variations always necessarily analytic?

This was proven to be true (through the work of Sergei Bernstein, Ennio de Giorgi, John Nash, among others).

My question probably stems mostly from my elementary knowledge of the subject, but I am wondering what exactly we gain from this result -- in as-close-to-layman's terms as possible. And from what I gather, it's a good thing that the answer to Hilbert's question is in the affirmative, correct?

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Variational principles are incredibly important in physics. See the book "Variational Principles in Physics". For example, particle physics is typically done with a "Lagrangian".

Newton began with equations of motion such as F=ma rather than calculations based on energies or action. I've been told that our modern understanding of energy came about 100 years after Newton. Variational methods would be later still. The more modern methods are not taught in introductory physics classes and so I suppose it's possible to get an education as a mathematician without appreciating their use in physics.

I suppose Hilbert included the problem for its use in theoretical physics. When Hilbert proposed it around 1900, variational principles were already extremely important in classical mechanics. As general relativity and quantum mechanics were developed they continued to be used. They're also used in electricity and magnetism, but I'm not sure when that usage started.

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