The Hamiltonian function H is defined as the total energy of a system, the sum of kinetic and potential energy. Lagrangian function L is defined as the potential energy subtracted from the kinetic energy.
Both functions are essential in solving for predicted movement in a many bodied problem of celestial mechanics. The Hamiltonian is conserved, while the change of Lagrangian with time is minimized in each interval.
That energy is conserved is intuitive. Minimizing the change of Lagrangian takes a bit of explanation that is usually not given. It has to do with the way the action S is calculated from L. One explanation is that in the nature of objects in a group moving under their combined gravities and individual momentums, the path of least resistance is the one in which the least possible conversion of energy between kinetic and potential occurs. This is the simplified explanation for the non specialist.
It's a bit of mystery why a physical system in vacuum space time would be biased against conversion of energy between kinetic and potential. The Lagrangian is suggesting the potential and kinetic energies compete with each other for control of curvature.
In an extreme high gravity the space curvature is tending to enclose the source in an event horizon, like a concave curvature. The opposite must be a convex curvature with respect to the source of kinetic energy. This difference may be a physical cause of bias against conversion between potential and kinetic energy. It is inconvenient in the mechanisms of stress energy and is avoided when another path is available.
In the present question it is recognized and well known in publications that a Legendre transformation is able to compute the Hamiltonian from the Lagrangian subtracted from the change of Lagrangian with logarithm of velocity, for a fixed location. Maybe some additional character can be deduced of space time and the objects it processes, especially how velocity and acceleration relate H to L. Conventional GRT has no such bias, but classical Celestial Mechanics does.
Why Can The Hamiltonian Be Computed From The Lagrangian?
Perhaps you know already, but given the Lagrangian for a system, you can construct the Hamiltonian as described in:
https://physics.stackexchange.com/questions/190471/constructing-lagrangian-from-the-hamiltonian
This discussion about gravity begs another question. How do you calculate the potential and kinetic energy of gravity fields absent any mass? I.e. gravtiy waves? They should result in a speed of light result when combined. I can't recall ever seeing this calculated as a set of field equations in the sense of Maxwell's field equations. I've been thinking about this problem but have no answer to it.
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