OK many thanks, solving these equations will give a lot of new results for spherical orbits in astronomy. The two other problems being worked on at present are:
1) Gyro in the presence of the general potential energy due to an external torque of any kind. I will send over the expression for the potential energy shortly. This is the rigorous theory of the replicated Laithwaite experiment. I will send over the lagrangian shortly.
2) The nutations and precessions of the earth in the presence of the sun’s gravitational field – the rigorous theory of the Milankovitch cycles. I will send over the lagrangian shortly.
Sent: 02/02/2017 19:50:49 GMT Standard Time
Subj: Re: Checking Final Version of Note 369(7)
You are right, I had forgotten the factor phi dot in the definition of L_phi. All is ok. The canonical form of the Lagrange eequation with L_phi is eq. %o20 of the protocol.
Am 02.02.2017 um 18:31 schrieb EMyrone:
Many thanks again, all is OK because the units of L sub phi r dot / (m r squared) are kgm m per second m squared per second / (kgm m squared) = m per second squared. These look like very interesting equations.
Sent: 02/02/2017 14:47:53 GMT Standard Time
Subj: Re: Final Version of Note 369(7)
I checked the note. There is only one point: In eq.(29) it seems that a factor of phi dot is missing in the first term. I myself prefer the “canonical” form with second derivatives at the lhs, see sect. 4 of the attachment. The phi equation can be rewritten with the constant of motion L_phi, then it takes the form %o20.
Checking eq.(29) of the note required a number of re-arrangements, it can be seen that the first term of (29) contains only one time derivative while the others all contain two time derivatives which gives an inconsistency in physical units.
Am 02.02.2017 um 13:21 schrieb EMyrone:
This corrects typo’s in Eqs. (19), (21) and (28).