Physical interpretation of the Hamiltonian

[zush]

When dealing with the Euler-Lagrange equation in a physical setting, one usually uses the Hamiltonian L=T-V as the value to be extremized. What is the physical interpretation of the extremizing of this value?

 

[Cleonis]

When dealing with the Euler-Lagrange equation in a physical setting, one usually uses the Hamiltonian L=T-V as the value to be extremized. What is the physical interpretation of the extremizing of this value?

Actually, (T - V) is the Lagrangian.
It doesn't matter all that much, the Lagrangian and the Hamiltonian are very closely related

As to the Lagrangian (T - V), Here is what I replied in a thread from january 2010:

Try reading the article "www.eftaylor.com/pub/ActionFromConsEnergy.pdf"[/URL][/u] written by Jozef Hanc and Edwin F. Taylor.
They're putting in a serious effort to build up to an intuitive understanding of the principle.
There's more where that came from, the authors have collaborated on several other articles. Links to those articles are available on the [u][PLAIN]http://eftaylor.com/leastaction.html" page on Taylor's website.



In november 2010 I posted a discussion in a thread where someone asked https://www.physicsforums.com/showthread.php?t=443711"
The physical interpretation stuff is in post #10 of that thread.

I believe that discussion is good, but of course you must make your own judgement.
An expanded version of that discussion is in the http://www.cleonis.nl/physics/phys256/least_action.php" article on my own website.

When it comes to physical interpretation of the Lagrangian there are very few sources. I get the impression that textbook authors copy what earlier authors have written. That's not necessarily bad, but there's no innovation, I think.

 

[vanhees71]

Although I learned yesterday, that it is not allowed to answer with quantum theory in the classical-physics section of this forum, I cannot resist to mention that one possible very intuitive "explanation" for the validity of Hamilton's principle comes from Feynman's path-integral formulation of quantum mechanics. Here, the classical limit is given by the stationary-phase approximation of the path integral. I.e., if the action becomes large compared to $$\hbar$$, the path integral is well approximated by consdering only the trajectories in phase space, which are very close to the classical trajectory.

The Lagrangian formulation follows then for a often applicable special case, where the Hamiltonian is quadratic in the canonical momenta with space independent coefficients.

 

[zush]

thank you I found what I was looking for

 

[PJC01]

The Hamiltonian, which is the difference between the kinetic energy (T) and potential energy (V), has a physical interpretation as the total energy of a system. When the Hamiltonian is extremized, it represents the state of the system where the total energy is at its minimum or maximum value. This is important in physics because it allows us to determine the equilibrium state of a system and predict its behavior over time.

In the context of the Euler-Lagrange equation, extremizing the Hamiltonian means finding the path or trajectory in which the system's total energy is conserved. This is known as the principle of least action and is a fundamental principle in classical mechanics. It states that a system will follow the path that minimizes the action, which is the integral of the Lagrangian (L=T-V) over time.

In summary, the physical interpretation of extremizing the Hamiltonian in the context of the Euler-Lagrange equation is that it represents finding the path or trajectory in which the system's total energy is conserved, and the principle of least action governs the system's behavior. This has important implications in understanding the dynamics of physical systems and predicting their behavior.