Lagrangian for a relativistic free particle can be written as
It gives correct expression of Hamiltonian which is
During a lecture today, my professor told me that Lagrangian for a free particle does not have a specific form and can be written in many ways. Then, he wrote another expression of Lagrangian that can also represent a relativistic free particle
My Questions are: What are the other forms that We can write our Lagrangian in? Does It have something to do with Lorentz invariance of the Lagrangian? How does above expression for Lagrangian can give us the correct equation of motion and Hamiltonian?