Predicting orbits and masses of as-yet-unknown bodies

[gulfcoastfella]

I read an article on Phys.org (The Strange Case of the Missing Dwarf), and as I'm in the middle of reading and studying Hamiltonian Dynamics, the article made me wonder how the unexplained orbits of existing bodies are used to determine the orbits and masses of as-yet-undiscovered bodies. It sounds suspiciously like Hamiltonian Dynamics. Am I right in thinking so?

 

[Astronuc]

Has one done Lagrangian mechanics before Hamiltonian mechanics?

Here is an interesting point: there is rarely any benefit from using a Hamiltonian instead of a Lagrangian to solve a standard mechanics problem.
http://www.people.fas.harvard.edu/~djmorin/chap15.pdf

I find examples of both Lagrangian and Hamiltonian dynamics with respect to planetary systems or binary star systems to globular clusters.

 

[gulfcoastfella]

Thanks for the reply, Astronuc. When I said Hamiltonian Dynamics, I really meant Hamilton's Principle and Lagrangian Dynamics/Mechanics. I'm currently reading through chapter 7 (Hamilton's Principle - Lagrangian and Hamiltonian Dynamics) of Marion & Thornton's "Classical Dynamics of Particles and Systems, 5th Ed." I've also put a dent in Robert Weinstock's "Calculus of Variations: with Applications to Physics and Engineering."

In Weinstock's text, I've already read about the use of Variational methods to find curves that begin or end at points, but end or begin on curves, respectively. It was this indication that Variational methods can be used with more constraints than just points that got me wondering if similar methods can be used to predict orbits (varied functions) of as-yet-undiscovered bodies based on unexplained motions in already-known bodies.

 

[mfb]

Neptune and Pluto were predicted that way, with a nice precision for Neptune and a bit more messy search for Pluto.

 

[gulfcoastfella]

mfb, I read that Neptune was discovered through the application of perturbation methods. Does this mean a combination of perturbation and variational methods, or did variational methods not enter into the discovery?

 

[mfb]

The prediction is over 150 years old - all those concepts were quite new back then (Hamilton was 41 at the time Neptune was discovered), I don't know how exactly they did the calculations. It looks like the references here have more details.