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Opticmist
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Homework Statement
Greetings! This is an example problem at the end of Chapter 1 in Mechanics (Landau):
A simple pendulum of mass m whose point of support oscillates horizontally in the plane of motion of the pendulum according to the law [tex] x=acos(\gamma t) [/tex].
Find the Lagrangian.
Homework Equations
The Attempt at a Solution
[tex] x = a cos(\gamma t) + l sin(\phi) ; y = l cos(\phi) [/tex]
Where, [tex] \phi [/tex] is the angle the pendulum makes with the vertical. Then:
[tex]
\dot{x}= -a\gamma sin(\gamma t)+l cos(\phi) \dot{\phi} ; \dot{y} = -l sin(\phi) \dot{\phi}[/tex]
[tex]
L = T - U = \frac{1}{2} m ( \dot{x}^2 + \dot{y}^2) - U
[/tex]
Which gives (ignoring full time derivative terms):
[tex]
L = \frac{1}{2} ml^2 \dot{\phi}^2 - m l a \gamma sin (\gamma t) cos(\phi)\dot{\phi} + m g l cos(\phi)
[/tex]
Answer given in the book:
[tex]
L = \frac{1}{2} ml^2 \dot{\phi}^2 + m l a \gamma^2 cos (\gamma t) sin(\phi) + m g l cos(\phi)
[/tex]
My solution is different from the one given in the book. Could someone please tell me where I'm going wrong?