About calculus of variation and lagrangian formulation

[physlad]

I was reading about the principle of least action and how to derive Newton's second out of it.

at a certain point I didn't follow the calculations,

so the author defines a variation in the path, $$x(t) \longrightarrow x'(t) = x(t) + a(t), a \ll x$$

$$a(t_1) = a(t_2) = 0$$

Now, $$S \longrightarrow S' = \int_{t_1}^{t_2} (m/2 (\dot{x} +\dot{a})^2 - V(x +a)) dt$$

$$= \int_{t_1}^{t_2} {1/2 m\dot{x}^2 + m\dot{x}\dot{a} - [V(x) + aV'(x)]} dt + O(a^2)$$

(what happened exactly here? could anybody tell me??)

then

$$= S + \int_{t_1}^{t_2} [m\dot{x}\dot{a} - aV'(x)] dt$$

$$\equiv S + \delta{S}$$

 

[Hurkyl]

I assume you meant

$$S \longrightarrow S' = \int_{t_1}^{t_2} (m/2 (\dot{x}(t) +\dot{a}(t))^2 - V(x(t) + a(t))) dt$$

? It looks like they simply used a differential approximation. (i.e. a first-order Taylor series)

 

[physlad]

Yes, I got it. Thanks