Consider the family of functions on $[a,b]$ given by $y(x; \epsilon) : = u(x) + \epsilon \eta (x)$, where the functions $\eta = \eta (x)$ are twice continuously differentiable and satisfy $\eta(a) = \eta(b) = 0$. Here $\epsilon$ is a small real parameter,..

What does this mean? What is the meaning and use of the "small real parameter". I'm currently doing a bit of self-studying, nothing too serious but I am interested in this stuff, but of course it is tough without guidance, and the biggest stumbling block for me is the notation and the jargon.

I've never really encountered the notation $y(x;\epsilon)$, what is the purpose of the semi-colon?

Source: http://www.macs.hw.ac.uk/~simonm/mechanics.pdf , top of the fourth page. Beginning of a proof of the Euler-Lagrange equation.

  • $\begingroup$ "Small" usually means small enough to prove whatever they are trying to prove. $\endgroup$ – Euler....IS_ALIVE Nov 18 '13 at 20:36
  • $\begingroup$ @Phaptitude What's the source of this? $\endgroup$ – Git Gud Nov 18 '13 at 20:36
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    $\begingroup$ @GitGud "An introduction to Lagrangian and Hamiltonian mechanics" -> macs.hw.ac.uk/~simonm/mechanics.pdf This is a part of a proof of the Euler-lagrange equation $\endgroup$ – Phaptitude Nov 18 '13 at 20:37
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    $\begingroup$ @SimenK. But then, later in the proof, $\varepsilon$ is a variable. Disgusting. $\endgroup$ – Git Gud Nov 18 '13 at 20:45
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    $\begingroup$ @Phaptitude Unfortunately no. Analysts take these sort of liberties all the time. $\endgroup$ – Git Gud Nov 18 '13 at 20:46

The word "small" is not a technical term here and therefore has no specific meaning; it could be omitted. What is important is that $u+\varepsilon \eta$ belongs to the domain of $J$ for all values of $\varepsilon$ in question; also the derivative $$\frac{d\varphi}{d\varepsilon}\bigg| _{\varepsilon=0}$$ must exist, so that $\varphi$ (which is a function of $\varepsilon$) must be defined at least in an open interval around $0$, say $(-\delta,\delta)$ for $\delta>0$ (and this is the way it is done in rigorous texts). The semi-colon notation is nonstandard; (a better notation is $y_\varepsilon(x)$) what is meant by it is: let $u$ and $\eta$ be fixed, we define a family of functions $\left\lbrace y_\varepsilon\right\rbrace$ by $y_\varepsilon:=u+\varepsilon\eta$ (of course we also have to say which values are allowed for $\varepsilon$, say for $\varepsilon\in(-\delta,\delta)$). It is important to notice that $y_\varepsilon(a)=u(a)$ and $y_\varepsilon(b)=u(b)$ for all values of $\varepsilon$ in question.

By the way the "useful lemma" on page 5 is commonly referred to as the "fundamental lemma of the calculus of variations".


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