Is the lagrangian always K.E-PE?

[quantumfireball]

Is the lagrangian always K.E-PE?
I don't think so?
so WHAT REALLY is the physical significance of the lagrangian?
Superficially it appears to be KE-PE,but the lagrangian for a charged particle in a vector potential is 1/2MV.V +qV.A-qW.
where V=velocity
A=vector potential
W=Scalar potential.
If the hamiltonion for the above lagrangian is calculated it turns out to be 1/2MV.V+qW.
So what does the term eV.A represent?
It appears to be the interaction energy.
But why does it dissapaer in the hamiltonion.

 

[Almsco]

The term qV.A represents the interaction energy between the particle and the vector potential, which is also known as the Lorentz force. It disappears in the Hamiltonian because it is a conservative force and its work done is zero. Therefore, it does not contribute to the energy of the system.

 

[charng]

The Lagrangian, in general, is not always equal to the sum of kinetic and potential energy. It is important to note that the Lagrangian is a mathematical quantity used in the field of classical mechanics to describe the dynamics of a system. Its physical significance lies in its ability to describe the equations of motion of a system in terms of generalized coordinates and their derivatives, rather than in terms of forces and accelerations.

In the case of a charged particle in a vector potential, the Lagrangian does not represent just the kinetic and potential energy, but also includes the interaction energy between the particle and the electromagnetic field. This interaction energy is represented by the term eV.A in the Lagrangian, where e is the charge of the particle, V is its velocity, and A is the vector potential. This term is not included in the Hamiltonian because it is not a conserved quantity and therefore does not contribute to the equations of motion.

The physical significance of the Lagrangian lies in its ability to describe the dynamics of a system in a more elegant and compact way, using generalized coordinates and their derivatives. It allows for a deeper understanding of the underlying physical principles governing the behavior of a system. Therefore, the Lagrangian is a powerful tool for scientists and engineers in a variety of fields, from classical mechanics to quantum field theory.