- #1
mtak0114
- 47
- 0
Hi
I'm trying to define a Newtonian lagrangian in an
rotating reference frame (with no potential)
Something to note is that the time derivative of in a rotating reference frame must be corrected for by:
[tex] \frac{d {\bf B}}{dt} \rightarrow \frac{d {\bf B}}{dt} + {\bf \omega} \times {\bf B}[/tex]
where B is some vector, this can be found in wikipedia
Therefore I get something like
[tex]L = \frac{1}{2} m(\dot{{\bf x}}+{\bf \omega} \times {\bf x})^2[/tex]
where the dot is the time derivative
and I am expecting to get three ficticious forces: Centrifugal, centripetal and euler forces.
but this does not appear what am I doing wrong?
I believe the answer should be
[tex] m(\ddot{{\bf x}}+2{\bf \omega} \times \dot{{\bf x}}+\dot{{\bf \omega}} \times {\bf x}+{\bf\omega \times (\omega} \times {\bf x}))=0[/tex]
I get this by taking the Newtonian lagrangian in a non rotating frame and calculating the euler lagrange equations of motion, and then transforming into the rotating frame. but the two results do not agree?
any help would be greatly appreciated
I'm trying to define a Newtonian lagrangian in an
rotating reference frame (with no potential)
Something to note is that the time derivative of in a rotating reference frame must be corrected for by:
[tex] \frac{d {\bf B}}{dt} \rightarrow \frac{d {\bf B}}{dt} + {\bf \omega} \times {\bf B}[/tex]
where B is some vector, this can be found in wikipedia
Therefore I get something like
[tex]L = \frac{1}{2} m(\dot{{\bf x}}+{\bf \omega} \times {\bf x})^2[/tex]
where the dot is the time derivative
and I am expecting to get three ficticious forces: Centrifugal, centripetal and euler forces.
but this does not appear what am I doing wrong?
I believe the answer should be
[tex] m(\ddot{{\bf x}}+2{\bf \omega} \times \dot{{\bf x}}+\dot{{\bf \omega}} \times {\bf x}+{\bf\omega \times (\omega} \times {\bf x}))=0[/tex]
I get this by taking the Newtonian lagrangian in a non rotating frame and calculating the euler lagrange equations of motion, and then transforming into the rotating frame. but the two results do not agree?
any help would be greatly appreciated