I was reading about the principle of least action and how to derive Newton's second out of it.(adsbygoogle = window.adsbygoogle || []).push({});

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

so the author defines a variation in the path, [tex]x(t) \longrightarrow x'(t) = x(t) + a(t), a \ll x[/tex]

[tex]a(t_1) = a(t_2) = 0[/tex]

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

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

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

then

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

[tex]\equiv S + \delta{S}[/tex]

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# About calculus of variation and lagrangian formulation

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