On Landau vol.1 Pg.5 (Question about conclusion drawn by Landau)

Thread starter zhuang382
Start date Aug 18, 2020
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Calculus of variation Landau Mechanics

Aug 18, 2020

zhuang382

Homework Statement: How do we deduce the conclusion of (3,2)? This is a topic inertial frame.

Relevant Equations: Law of inertia

I understand that d/dv(L) = constant, and L is only dependent on v, but how do we get to the fact that v = constant?

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Aug 19, 2020

anuttarasammyak

Gold Member

Say
\frac{\partial L}{\partial v}=f(v),

\frac{d}{dt}f(v)= \frac{df(v)}{dv}\frac{dv}{dt}=0

Usually ##\frac{df(v)}{dv} \neq 0## so

\frac{dv}{dt}=0

Aug 19, 2020

zhuang382

Thanks!

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Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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