Missing step: Euler-Lagrange equations for the action integral

[alfredblase]

Hi its me again, stuck once more. Sorry guys and gals :P
Ok a problem I found on http://en.wikipedia.org/wiki/Action_(physics)
In a 1-D case how do we get from:
$$\delta S = \int_{t_1}^{t_2} [L(x + \varepsilon, \dot{x} + \dot{\varepsilon})-L(x,\dot{x})]dt$$
to:
$$\delta S = \int_{t_1}^{t_2} \left(\varepsilon \frac{\pd L}{\pd x} + \dot{\varepsilon} \frac {\pd L} {\pd \dot{x}}\right)dt$$
where $$\varepsilon = x_1(t) - x(t)$$
and where the first order expansion of L in ε and ε′ is used? I don't even know what that last phrase means, so if someone could explain that to me too, that would be great.
Thankyou very much.

 

[George Jones]

Consider a real-valued function of 2 variables, say $$f = f(x,y)$$. Taylor's theorem for multivariable calculus gives

$$f(x + \Delta x , y + \Delta y) - f(x,y) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + ...$$

In your variation principle, $$L$$ is a function of the "independent variables" $$x$$ and $$\dot{x}$$, so

$$L(x + \varepsilon , \dot{x} + \dot{\varepsilon}) - L(x,\dot{x}) = \frac{\partial L}{\partial x} \varepsilon + \frac{\partial L}{\partial \dot{x}} \dot{\varepsilon} + ...$$

Neglecting the ... (i.e., the terms higher than first-order) gives the desired result.

Note: the "independent variables" actually are functions themselves.

Regards,
George

 

[alfredblase]

Thanks George! =)

Edit: just posting a version of the last formula that shows correctly;

$$L (x + \varepsilon , \dot{x} + \dot{\varepsilon}) - L (x, \dot{x}) = \frac{\partial L}{\partial x} \varepsilon + \frac{\partial L}{\partial \dot{x}} \dot{\varepsilon} + ...$$

 

[alfredblase]

need more help on this one sorry...

I still don't understand how to get from one to other. I thought boning up on the Taylor theorem for multiple variables wouldn't be too hard, but I was wrong.
Can some one post a step by step, dummies guide to getting from:
$$f(x + \Delta x , y + \Delta y) - f(x,y)$$
to:
$$\frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + ...$$
please?
thanx
38

 

[alfredblase]

Actually I just found a source in MathWorld that cleared it all up for me. =)

It was: http://mathworld.wolfram.com/TaylorSeries.html