Lagrangian mechanics and planetary formation

[Catria]

I am disappointed by my graduate-level classical mechanics course, and especially the treatment of Lagrangian/Hamiltonian mechanics. Now, I scanned my notes and some crazy idea popped into my head, further fueling my discontent towards this course, because all the problems covered in class were with time-independent masses, even central-force problems.

I picture a planet in its early stages of formation as a central-force system whose mass is time-dependent so that [itex]\dot{m}\neq 0[/itex], orbiting about its star under the effect of a central force. Is the problem actually tractable with Lagrangian/Hamiltonian mechanics (even numerically) or would it require usage of Hamilton-Jacobi or Newtonian mechanics?

 

[gtoguy97]

The problem of a time-dependent mass orbiting a star under the effect of a central force is indeed tractable using Lagrangian/Hamiltonian mechanics. This is due to the fact that any central-force system can be described by a Lagrangian, and that the mass of the system does not affect the functional form of the Lagrangian. The only difference is that the mass of the system will appear as a parameter in the Lagrangian, rather than as a variable. Therefore, it is still possible to use the same Hamiltonian and canonical equations of motion for the central-force system with a time-dependent mass.It is also possible to solve the problem numerically using a method such as the Runge-Kutta algorithm, which can be used to solve the equations of motion for a system with a time-dependent mass. Alternatively, one could use Hamilton-Jacobi or Newtonian mechanics to solve for the equations of motion, although this may be more complicated and less efficient for this particular problem.