The system is described by the Euler-Lagrange equations. So basically the Lagrangian is the description of the system. However the Euler-Lagrange equations can be the same for different Lagrangians.

Now you can go from a Lagrangian description to a Hamiltonian via Laplace transform which is an one to one process.

The system is described by the Euler-Lagrange equations. So basically the Lagrangian is the description of the system. However the Euler-Lagrange equations can be the same for different Lagrangians. The field will be the solution of the Euler-Lagrange equations. They are differential equations so the boundary conditions can change the solution.

Now you can go from a Lagrangian description to a Hamiltonian via Laplace transform which is an one to one process.

The system is described by the Euler-Lagrange equations. So basically the Lagrangian is the description of the field. Different Lagrangian means different field. However the Euler-Lagrange equations can be the same for different fields.

Now you can go from a Lagrangian description to a Hamiltonian via Laplace transform which is an one to one process.

The system is described by the Euler-Lagrange equations. So basically the Lagrangian is the description of the system. However the Euler-Lagrange equations can be the same for different Lagrangians.

Now you can go from a Lagrangian description to a Hamiltonian via Laplace transform which is an one to one process.