Hello,
I've been making my way through Landau-Lifshitz's "Mechanics" book, and I've come across a bit of math I'm not too sure about.
What I'm confused about is here:
https://archive.org/details/Mechanics_541
On Page 93 (of the book, not the PDF) under Motion in a rapid oscillating field. The derivation is simple up until you get to the part "Substituting (30.3) in (30.2) and expanding in powers [itex]\xi[/itex] as far as the first order terms...".
The equation they end up with is:
[itex]m\ddot{X}+m\ddot{\xi}=-\frac{dU}{dx}-\xi\frac{d^2U}{dx^2}+f_{(X,t)}+\xi\frac{\partial f}{\partial X}[/itex]
So I'm wondering how they get this using the substitution:
[itex]x_{(t)}=X_{(t)}+\xi_{(t)}[/itex]
into the equation:
[itex]m\ddot{x}=-\frac{dU}{dx}+f_{(t)}[/itex]
where [itex]f_{(t)}=f_1 \cos{\omega t} + f_2 \sin{\omega t}[/itex]
It seems to me there is some strange derivative such as [itex]\frac{d}{d(X_{(t)}+\xi_{(t)})}[/itex] which so far I have had no luck figuring out. Also it seems as though [itex]f_{(t)} → f_{(X,t)}[/itex] which probably has something to do with the transformation. I also believe some sort of Taylor expansion is happening. If anyone has any ideas, please let me know.
I've been making my way through Landau-Lifshitz's "Mechanics" book, and I've come across a bit of math I'm not too sure about.
What I'm confused about is here:
https://archive.org/details/Mechanics_541
On Page 93 (of the book, not the PDF) under Motion in a rapid oscillating field. The derivation is simple up until you get to the part "Substituting (30.3) in (30.2) and expanding in powers [itex]\xi[/itex] as far as the first order terms...".
The equation they end up with is:
[itex]m\ddot{X}+m\ddot{\xi}=-\frac{dU}{dx}-\xi\frac{d^2U}{dx^2}+f_{(X,t)}+\xi\frac{\partial f}{\partial X}[/itex]
So I'm wondering how they get this using the substitution:
[itex]x_{(t)}=X_{(t)}+\xi_{(t)}[/itex]
into the equation:
[itex]m\ddot{x}=-\frac{dU}{dx}+f_{(t)}[/itex]
where [itex]f_{(t)}=f_1 \cos{\omega t} + f_2 \sin{\omega t}[/itex]
It seems to me there is some strange derivative such as [itex]\frac{d}{d(X_{(t)}+\xi_{(t)})}[/itex] which so far I have had no luck figuring out. Also it seems as though [itex]f_{(t)} → f_{(X,t)}[/itex] which probably has something to do with the transformation. I also believe some sort of Taylor expansion is happening. If anyone has any ideas, please let me know.