- #71
etotheipi
Apologies for a late reply; the pubs were open again today
Given any frame one can define the time derivative with respect to of an arbitrary vector It's easy to show, given another frame , that where and is the orthogonal, time-dependent matrix such that .
Now let us show how to obtain the equations of motion for a particle in an arbitrary reference system. Begin with an inertial reference system , in which the Lagrangian is simply where . Let's now consider a second reference system and define , such that . Operating with yields, upon defining and It follows that The new Lagrangian is then written where we have re-written . Let us now compute the derivatives of the Lagrangian The Euler-Lagrange equation thus implies the following equation of motion in the reference system, after converting from suffix notation back into vector notation, where is the translational acceleration of the -system with respect to the -system and is the rotational acceleration of the -system with respect to the -system. And of course is the acceleration of the particle with respect to the -system.
This is the equation of motion of a particle in an arbitrary reference system ; the additional terms arising from the co-ordinate transformation are not forces, although they are sometimes affectionally referred to as 'inertial forces'.
Given any frame
Now let us show how to obtain the equations of motion for a particle in an arbitrary reference system. Begin with an inertial reference system
This is the equation of motion of a particle in an arbitrary reference system
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