Understanding Derivation of Euler-Lagrange

I am trying to understand the derivation of the Euler-Lagrange equation. I drew a graph below. So, according to the graph,

$$\int_{t_1}^{t_2} L(x+\delta{x},\dot{x}+\delta\dot{x}\,t) dt - \int_{t_1}^{t_2} L(x,\dot{x},t) dt = \delta{S}$$

That is, we take the difference between the integral of the true path and the integral of the variated path, where S is the action defined by $$\int_{t_1}^{t_2} L(x,\dot{x},t) dt$$ and $$\delta{S}$$ is the variation in action (in the graph it is the distance between the blue line and the red line at any t value). Where do I proceed from there? I don't understand this line in the derivation:

$$\int_{t_1}^{t_2} L(x,\dot{x},t) dt + \frac{d{L}}{d{x}}\delta{x} + \frac{d{L}}{d{\dot{x}}}\delta{\dot{x}}+O((\delta{x})^2)- L(x,\dot{x},t) dt = \delta{S}$$

Namely, what is the function O? Where are the partial derivatives coming from? The only thing I know to do is to combine the two integrals because the times and the integration variables are the same. How do I get all the mess to the left of $$L(x,\dot{x},t)$$ in the equation above?

The partial derivatives appear as a result of the chain rule for partial derivatives; this takes place implicitly when the variation is applied to the Lagrangian functional. The big O notation says that there are higher order terms which can be ignored in the variational limit.

The web page shows a couple of ways to obtain the result. Euler used purely geometric methods, and Lagrange invented the variational technique with the delta process.

• The partial derivatives come from the Taylor series – Oswald Apr 5 '16 at 4:02
• Although $x, \dot x$ are dependent, they *can be treated as independent variables *to find $\delta S$ – Narasimham Jan 16 at 21:12
• @Narasimham: x,x˙ are linearly independent. – Peter Diehr Jan 16 at 23:27
• I meant to say that though they are differentially related, they are linearly independent and on that base calculus of variation is built. Any takers? – Narasimham Jan 18 at 15:07

It might help to think about the definition of a derivative,

$$\frac{df}{dx} = \lim_{\delta x\rightarrow 0} \frac{f(x+\delta x)-f(x)}{\delta x}.$$

If you rearrange this equation you'll see that

$$f(x+\delta x) \approx f(x) + \frac{df}{dx} \delta x,$$ when $$\delta x$$ is small. As $$\delta x$$ gets smaller the approximation becomes more exact. The term $$\mathcal{O}((\delta x)^2)$$ represents some quantity that scales like $$(\delta x)^2$$ as $$\delta x$$ goes towards 0. So another way to write the above the equation might be $$f(x+\delta x) =f(x) + \frac{df}{dx} \delta x + \mathcal{O}((\delta x)^2).$$

• This is good for intuition, but the EL equations uses functional derivatives. – InertialObserver Mar 19 at 18:03
• The way I've always seen it, the way to rigorously define these functional derivatives is in terms of ordinary derivatives. I.e. you define a new path as $x(t)$ by $x(t) = x_0(t) + \epsilon \delta x(t)$, where $x_0(t)$ is the path that minimizes the action and $\delta x(t)$ is an arbitrary but fixed function (that vanishes at the end points) and $\epsilon$ is a number. Then you plug in $x(t)$ into Lagrangian, consider the action as a function of $\epsilon$, and take an ordinary derivative with respect to $\epsilon$. – Alex Mar 19 at 22:39
• No, you're right, but you might want to mention that in your answer. – InertialObserver Mar 19 at 22:39
• ... or equivalently to doing the derivative wrt $\epsilon$: at each value of $t$ in the integrand you expand the integrand to first order in $\epsilon$ and demand that the variation in the action is 0 (at first order in $epsilon$) for every possible perturbation $\delta x(t)$. – Alex Mar 19 at 22:42

Namely, what is the function O? Where are the partial derivatives coming from? The only thing I know to do is to combine the two integrals because the times and the integration variables are the same. How do I get all the mess to the left of L(x,x˙,t) in the equation above?

These are good questions. The usual derivation is a heuristic derivation and typically a rigorous derivation of the EL equations in the calculus of variations, at the level of rigour, that a first year undergraduate mathematics degree will approach the calculus of one real variable appears to be fairly rare in the literature.

Abraham & Marsden, in their book, Foundations of mechanics, take a geometric approach, which allows them to deduce the EL equations in a rigorous manner, admitting that an analytic derivation is difficult.