I'm reading Lanczos: 'The variational Principles of Mechanics'. I need help resolving a paradox - which is probably trivial
Lanczos (page 100 Dover edition) introduces a system, S', rotating at angular velocity [itex]\vec \Omega[/itex] about a fixed point with respect to inertial system S. The radius vectors [itex]\vec R[/itex] and [itex]\vec R'[/itex] in the two systems are, he says, the same: [itex]\vec R = \vec R'[/itex]. [I don't have trouble with this: it's fundamental to the idea of a vector that the same vector can be expressed in terms of different basis vectors, in this case, [itex]\vec i, \vec j, \vec k[/itex] and [itex]\vec i', \vec j', \vec k'[/itex].]
Nevertheless, he says, the velocities and accelerations measured in both systems differ from each other because rates of change observed in the two systems are different. If a certain vector [itex]\vec B[/itex] is constant in S' it rotates with the system and thus, if observed in S, undergoes in the time dt an infinitesimal change [itex]d \vec B = (\vec \Omega \times \vec B) dt[/itex]. [Again, easily seen - especially for [itex]\vec R[/itex] itself.]
Hence [itex]\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)[/itex] while at the same time, Lanczos says, [itex]\frac{d' \vec B}{dt} = 0 [/itex].
Here, Lanczos has introduced the notation [itex]\frac {d'}{dt}[/itex] which refers to the operation of observing the rate of change of a quantity in the moving system S'.
If you've read as far as this, well done and thank you. Now here's my problem. Regarding [itex]\vec R[/itex] and [itex]\vec R'[/itex] as functions of time, we can surely write:
[tex]\vec R (t) = \vec R' (t).[/tex]
But because [itex]\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)[/itex]
[tex]\vec R (t + dt) = \vec R (t) + (\vec \Omega \times \vec R (t)) dt[/tex]
while because [itex]\frac{d' \vec B}{dt} = 0 [/itex]
[tex]\vec R' (t + dt) = \vec R' (t) + 0.[/tex]
It would therefore seem that at time (t + dt), it is no longer the case that [itex]\vec R = \vec R'[/itex]. Whereas [itex]\vec R'[/itex] has changed over dt, [itex]\vec R'[/itex] has stayed constant.
Where is my reasoning wrong?
Lanczos (page 100 Dover edition) introduces a system, S', rotating at angular velocity [itex]\vec \Omega[/itex] about a fixed point with respect to inertial system S. The radius vectors [itex]\vec R[/itex] and [itex]\vec R'[/itex] in the two systems are, he says, the same: [itex]\vec R = \vec R'[/itex]. [I don't have trouble with this: it's fundamental to the idea of a vector that the same vector can be expressed in terms of different basis vectors, in this case, [itex]\vec i, \vec j, \vec k[/itex] and [itex]\vec i', \vec j', \vec k'[/itex].]
Nevertheless, he says, the velocities and accelerations measured in both systems differ from each other because rates of change observed in the two systems are different. If a certain vector [itex]\vec B[/itex] is constant in S' it rotates with the system and thus, if observed in S, undergoes in the time dt an infinitesimal change [itex]d \vec B = (\vec \Omega \times \vec B) dt[/itex]. [Again, easily seen - especially for [itex]\vec R[/itex] itself.]
Hence [itex]\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)[/itex] while at the same time, Lanczos says, [itex]\frac{d' \vec B}{dt} = 0 [/itex].
Here, Lanczos has introduced the notation [itex]\frac {d'}{dt}[/itex] which refers to the operation of observing the rate of change of a quantity in the moving system S'.
If you've read as far as this, well done and thank you. Now here's my problem. Regarding [itex]\vec R[/itex] and [itex]\vec R'[/itex] as functions of time, we can surely write:
[tex]\vec R (t) = \vec R' (t).[/tex]
But because [itex]\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)[/itex]
[tex]\vec R (t + dt) = \vec R (t) + (\vec \Omega \times \vec R (t)) dt[/tex]
while because [itex]\frac{d' \vec B}{dt} = 0 [/itex]
[tex]\vec R' (t + dt) = \vec R' (t) + 0.[/tex]
It would therefore seem that at time (t + dt), it is no longer the case that [itex]\vec R = \vec R'[/itex]. Whereas [itex]\vec R'[/itex] has changed over dt, [itex]\vec R'[/itex] has stayed constant.
Where is my reasoning wrong?