What do they mean? Basically, the Hamiltonian is the sum of the kinetic and potential energy of a system (and has a constant value in any closed system if generalized to treat rest mass as a form of energy), while the Lagrangian is the kinetic energy of a system minus the potential energy of a system.

Note that while conservation of mass-energy is a fundamental and never broken law of nature, that the absolute value of both the Hamiltonian and the Lagrangian are not not fundamental, because kinetic energy which is 1/2mv^2 and velocity is a function of one's frame of reference, and potential energy can be set with a zero at any arbitrary point, for example, for gravitational potential energy either at the bottom or the top or the middle of a sloped ramp in Earth's gravitational field.

[E]nergy – or the Hamiltonian – is sufficient for formulating all the differential equations that govern the evolution of any realistic physical system in time. In the quantum context, the evolution of other observable quantities is governed by the commutators with the Hamiltonian.

The Lagrangian is deeply related to the "action" in a system:

the Lagrangian is the integrand in an integral called the action,(S), over the time variable. So the action is typically an integral of a "quantity local in time over time" and this quantity is nothing else than the Lagrangian (L). So I have converted the question "what is the defining property of the Lagrangian" to a related question, "what is the defining property of the action".

The answer is the "principle of least action" or, somewhat more generally, the "principle of stationary action". It says:

For a fixed pair of initial and final states (in which we specify just the coordinates, not the velocities or the momenta), every classical system in physics must choose a history (x_i(t)) for which the action (S[x_i(t)]) is stationary. It means that if (x_i(t)) is changed by (epsilon cdot delta x_i(t)) for a small (epsilon) and a function of time (delta x_i(t)) that vanishes at the initial as well as final moment, the action is only allowed to change by terms scaling like (epsilon^2), not terms scaling like (epsilon).

In other words, the action must be equal to a local maximum, a local minimum, or a local saddle point for the allowed history. This allows one to select the whole history as soon as we are told the initial and final configurations. When we do so, we may derive the Euler-Lagrange equations, the differential equations that govern the evolution of a physical system in time.

These equations turn out to be equivalent to Newton's equations, Maxwell's equations, Einstein's equations, the Klein-Gordon equation, the Dirac equation, or whatever other equation describes the evolution of your physical system. All these things may be derived from an action much like they could be extracted from the Hamiltonian.

The linked post does quite a good job at explaining at a conceptual level what is going on with Lagrangians and actions, both in classical systems and in quantum mechanics (with a conceptual understanding of the Feynman path integral being pre-requisite to understanding the quantum mechanical part), but don't try to read it with Internet Explorer which does not convert the coding for the mathematical notation used in the post properly.

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