This has been used in molecular dynamics computer simulation since the mid seventies, and the subject has become a major part of chemistry and physics.
To: EMyrone@aol.com
Sent: 25/02/2017 16:33:44 GMT Standard Time
Subj: Re: The Basics of the Euler Lagrange TheoryAgreed, the method of Lagrangian multipliers is more basic and can for example be applied when functions are not continuously differentiable. I will try a solution of eqs. (17-19) of note 371(2).
Horst
Am 25.02.2017 um 12:47 schrieb EMyrone:
I reviewed these basics as described by Marion and Thornton. The number of proper Lagrange variables is equal to the number of degrees of freedom, but it is also possible to use the method of undetermined multipliers when the number of Lagrange variables is greater than the number of degrees of freedom. However, a simple solution to the problem is to solve Eqs. (17) to (19) of Note 371(2) simultaneously, to give r1, r2 and r3 in terms of theta, phi and chi. These are the required orbits. There are three dimensions (degree of freedom) and three proper Lagrange variables, r1, r2, and r3. So there are three differential equations in three unknowns, an exactly determined problem. The orbits are r1(theta, phi, chi), r2(theta, phi, chi) and r3(theta, phi, chi). Finally use
r squared = r1 squared + r2 squared + r3 squared
to find r(theta, phi, chi) and its precessions. In the planar limit it should reduce to a conic section without precession.