Functional variation of Lagrangian densities

[Cinquero]

Why do we treat a scalar field phi and its derivatives as being independent when trying to find a stationary solution for the action?

Doesn't that give too general solutions?

Where does the restriction that (d_mu phi) is dependent on phi come back in?

 

[dextercioby]

That's the idea. We take things as general as possible. The Lagrangian needs to be a function of positions and velocities treated as independent variables as to allow the passing to the Hamiltonian formalism on one hand and account for the correct description of dynamics on the other. I mean we usually see the lagrangian as the KE-PE function and the KE part involves time derivatives of position (generalized velocities), while the PE part is allowed to depend only on "x" and "t", and not on $\frac{dx}{dt}$.

Setting things so general we have to use the fundamental variational principle of Hamilton which selects the physically realizable mechanical states,which are of course the solutions of Lagrange equations.

Daniel.

 

[sir-pinski]

Functional variation of Lagrangian densities refers to the process of varying the fields in the Lagrangian density to find the equations of motion that yield a stationary action. This is a necessary step in solving physical systems using the principle of least action.

When treating a scalar field phi and its derivatives as independent variables, we are essentially considering all possible variations of the field and its derivatives. This allows us to find a general solution that satisfies the equations of motion for a given system. This is important because it allows us to find the most general and complete solution for a physical system.

The restriction that (d_mu phi) is dependent on phi comes back in when we plug the general solution into the equations of motion. This is because (d_mu phi) represents the momentum of the field and is dependent on the field itself. Therefore, when we solve for the equations of motion using the general solution, we are taking this dependence into account.

It is important to note that while treating phi and its derivatives as independent variables may seem to give too general solutions, it actually allows us to find the most complete and accurate solution for a physical system. This is because it takes into account all possible variations of the field and its derivatives, leading to a more comprehensive understanding of the system.