Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

Consistency of calculations in notes 413(5) and 414(1) with Lagrangian

OK thanks, in UFT414 the coordinate transform to ( r , phi’) is used throughout, so we simply replace phi in the usual plane polar coordinates by phi’, defined throughout by phi’ = phi + omega sub 1 t. The exact cross check in UFT414(4) shows that the kinematic and lagrangian methods give the same results exactly.

Calculations in notes 413(5) and 414(1) with Lagrangian
To: Myron Evans <myronevans123>

In the notes you wrote the terms for L and Omega_r (eqs. 4 and 7 in 414(1)) with overall positive signs. However the kinetic energy for the Lagrangian variable phi’ is defined by

(where phi corresponds to phi’). Therefore there are negative terms in the constant of motion:

Here is dphi’/dt = omega (or more correctly omega’).
From the Lagrange equation for r (eq.1, to be written with phi’) follows


There is a sign change in front of the factor 2 omega.
Concering radius shrinking form L=const, I guess that omega counteracts the term t*d omega_1/dt as long as possible, keeping the radius constant. I will check this by examples.
It seems that in the notes the variables phi and phi’ have accidentally be interchanged. The relevant transformation is:

phi = phi’ – omega_1*t.

Another point: I had erroneausly added the spin connection term to the full equation for r dot dot. According to the above derivation, using the spin connection is only a formal re-writing. The equations remain the same. I will do further checks and write up a preliminary version of UFT 413,3 hopefully before my holiday next week.


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