Understanding Lagrangian/Hamiltonian Mechanics I am trying to understand Lagrangian/Hamiltonian mechanics, and I have a few small questions based on guesses I have had. I admit that I do not have all of the background in topology to understand a lot of what I have seen, but I am pretty comfortable with diff eq, etc.
These are more like guesses based on what I have been reading so far (mostly Mechanics by Landau and Lifshitz, as well as various papers on PDEs and both calculus of variations and analytic mechanics. If you'd like, I can try to dig up my reading material, but it might be difficult to find everything). Please let me know if these are correct interpretations, as if they are not, I am confident that I am sorely missing something.
Also, these are specifically for mechanics problems.

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*The Lagrangian functional is at most quadratic in $q(t)$, as any higher degree would eliminate the possibility of only one $q(t)$ minimizing the action (I'm certain that this would be true if $q(t)$ was a scalar, but I don't know anything about function space).


*The action functional must have definite bounds, otherwise a similar problem occurs as did in my first assumption. In a sense, the necessity of defined bounds can be thought of as a constraint.


*This one is more of a question, I just haven't found a good explanation of it using jargon I understand:
Is the Lagrangian function created when one uses the Lagrange multiplier method for optimization sort of the same as the Lagrangian function $L=T-U$, where $T$ is a function to be optimized and $U$ is a constraint function? I'm not really sure about this, but it could be helpful to understand a lot of necessary requirements in Lagrangian/Hamiltonian mechanics as direct results of adapting Lagrange's method of optimization for functionals rather than scalar functions.
Please let me know if I am completely missing the point, criticism is very helpful.
 A: I am only really half-knowledgeable about this, but I have been thinking about it recently and do think I can address your questions.

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*No, to the best of my knowledge a Lagrangian can be non-quadratic (and indeed can even contain non-polynomial expressions in terms of the inputs.)  You're right that this can complicate the search for extrema.  However, it's usually assumed you are in a region that is locally convex, so that an extrema exists. I've read that "most" Lagrangians you encounter in nature are like that.


*I don't know what you mean by "definite bounds" on the action functional. The action functional is a line integral which, by definition has a starting and stopping point, if that's what you mean.


*My experience has been that the most typical case is the Lagrangian is $K-V$ where $K$ is the kinetic energy of a system and $V$ is a potential.  When further physical constraints are at play, then the method of Lagrange multipliers can be used to layer these constraints onto the Lagrangian, resulting in a different function with different solutions (ones which conform to the constraints.)
