Justification of the euler lagrange equation.?

Question

I understand why the euler lagrange equation has to be equal to zero on the open interval between both points. Now I have an idea as to why the minimizing functional has to satisfy it but seeing as how it is only necessarily true on the open interval how do we know wether the solutions are absolute minima(or maxima) or just stationary points. also if there is no solution that doesn't mean there are no stationary points is the euler lagrange equation only has to work on the interval. Can someone please clear this all up for me. please don't be mean I'm only 15.

Oh I didn't mean it that way exactly. I know that they are stationary integral values. but the euler lagrange equation only equals zero on the open interval. so how do we know that the soltuions describe ALL stationary poinbts as they have to satisfy it entirely. Say the abnsolute minima only satisfies it on an interval it won't be included in the solutions of the euler lagrange equation. I may be way off though. Maybe i just suck at expressing myself.

I followed the derivation of the euler lagrange differential equation and it made sense til i hit a bit of a snag. Now it obviously applies on the open interval but the resource i read from says that it's a necessary condition and therefore staitfies it. but if it satisfies it that means that the equation applies over the entire function. Of course we know that it's only a necessary condition on the open interval. What if the absolute minimum does not have that property all over it? Can someone please clear this up for me? gosh i'm dumb...

Please give me more help!

Answer 1

I think you may be confused between functions and functionals. Take a quick look at the wiki article on functionals, where it describes it as a function of functions, as its argument. The principle behind variational calculus is a generalization of calculus of functions. When one differentiates a function, one is examining the values of the function about a given point (x, f(x)), which is just a line of points with a single value f(x) for each x. In variational calculus, on the other hand, one is examining entire functions about the original function, frequently expressed as f(x) + g(x), where g(x) is a range of pertubative functions. The Euler-Lagrange equation is called a functional equation for this reason, and it looks for functions whereby the integral value of the Euler-Lagrange equation is "stationary", or approximately constant, for a given range of perturbative functions g(x) added to f(x). We don't speak of "stationary points" along the interval, but that the min/max function itself produces stationary integral values, relatively unaffected by perturbative functions.

It's in fact an infinite-dimensional generalization of calculus of functions. Check out Hilbert Space.

Hey, when I was 15, I used to brag about reading this sort of stuff too. You're pretty good if you're even this far, I hope you keep it up.

Addendum: Maybe you do suck at expressing yourself clearly, but I will try again. I think you're wondering if a function meets the functional minimum from, say, point 0 to point 1, what about the interval from point 0.25 to point 0.75. The answer is, "yes, it's a functional mininum for that interval as well". Indeed, the Principle of Least Action (or the more restricted version, the Principle of Least Time) would be in trouble if this weren't true. We're actually addressing geodestics upon some landscape, including those in configuration spaces. If I have a shortest path from A to B, then it's also the shortest path between any two points in the interval from A to B. One of the more interesting things about mathematical physics is that sometimes a global property is actually a reflection of local properties. A geodesic, for example, may be the shortest distance between A and B, but at every point between A and B it's entirely determined locally.

Answer 2

The Lagrangian is greater counter intuitive when you consider that we are calculating natures shortest direction applying a distinction between energies, yet this simplifies looking a gadget's equation(s) of action relative to diverse coordinate structures in one blow. that's less complicated to appreciate than Newtonian formulations in lots of instances. some straight forward Intuitive recommendations I even have: alongside the least action direction the place L=T(qi')-U(qi) a million. d(?L / ?qi')dt - ?L / ?qi = 0 ie: Newton's rules are obeyed F_ma - F_u = 0 The Non-conservative case can be added. 2. Conservation of capability and momentum on which the Lagrangian (physics) is predicated are on the middle of least action. 3. Superimpose some random direction over the path of least action and ask the question: What direction greater advantageous conforms to nature's rules. this is kinda elementary to work out that many paths does not comply with what nature intends to do. interior the tip, this is purely a much less complicated way of awareness nature whilst nonetheless compiling with Newton's rules.