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
T=

(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.

Horst

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