- #1
Peeter
- 305
- 3
Homework Statement
Attempting a mechanics problem from Landau's mechanics (3rd edition) I get a different answer, as shown below. Error by me, or typo in the textbook? I can't find any errata page for the text, but since it's an older book, perhaps no such page is maintained.
Chapter 1 problem 3a is to calculate the Lagrangian of a pendulum where the point of support is moving in a circle (figure in this google books url).
Homework Equations
See below.
The Attempt at a Solution
The coordinates of the mass are[tex]p = a e^{i \gamma t} + i l e^{i\phi},[/tex]
or in coordinates
[tex]p = (a \cos\gamma t + l \sin\phi, -a \sin\gamma t + l \cos\phi).[/tex]
The velocity is
[tex]\dot{p} = (-a \gamma \sin\gamma t + l \dot{\phi} \cos\phi, -a \gamma \cos\gamma t - l \dot{\phi} \sin\phi),[/tex]
and in the square
[tex]\dot{p}^2 = a^2 \gamma^2 + l^2 \dot{\phi}^2 - 2 a \gamma \dot{\phi} \sin\gamma t \cos\phi + 2 a \gamma l \dot{\phi} \cos \gamma t \sin\phi=a^2 \gamma^2 + l^2 \dot{\phi}^2 + 2 a \gamma l \dot{\phi} \sin (\gamma t - \phi).[/tex]
For the potential our height above the minimum is
[tex]h = 2a + l - a (1 -\cos\gamma t) - l \cos\phi = a ( 1 + \cos\gamma t) + l (1 - \cos\phi).[/tex]
In the potential the total derivative [itex]\cos\gamma t[/itex] can be dropped, as can all the constant terms, leaving
[tex]U = - m g l \cos\phi, [/tex]
so by the above the Lagrangian should be (after also dropping the constant term [itex]m a^2 \gamma^2/2[/itex]
[tex]\mathcal{L} = \frac{1}{{2}} m \left( l^2 \dot{\phi}^2 + 2 a \gamma l \dot{\phi} \sin (\gamma t - \phi) \right) + m g l \cos\phi.[/tex]
This is almost the stated value in the text
[tex]\mathcal{L} = \frac{1}{{2}} m \left( l^2 \dot{\phi}^2 + 2 a \gamma^2 l \sin (\gamma t - \phi) \right) + m g l \cos\phi.[/tex]
It looks like an innocent enough typo (text putting in a [itex]\gamma[/itex] instead of a [itex]\dot{\phi}[/itex]). Also oddly, there's a second reference after that point that also doesn't make sense where they refer to the omission of the total derivative [itex]m l a \gamma \cos( \phi - \gamma t)[/itex] ... a term that I didn't have when multiplying out my velocity?
Is there consensus that there are a pair of typos here, and if not, can somebody spot the error in my calculation?
Last edited: