- #1
physlad
- 19
- 0
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}
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}