Information contained in Lagrangians and actions

Question

I've been looking into analytical mechanics with the intention of finding out more about Lagrangians and actions. As far as I currently understand it, the Lagrangian is formed with positions and velocities and then operated on using the Euler-Lagrange equation to determine the laws of motion. However, talking to more experienced physicists than myself, they always say that there's more than just equations of motion in a Lagrangian. Also, when I look at Lagrangians defined for fields, I cannot interpret them (probably because I'm coming at it from a mechanical point of view). I guess that's two questions in one, but the overriding question I have is what information can be obtained from a Lagrangian, and taking it beyond mechanical systems what does it generally represent? Links to websites and book recommendations would be greatly appreciated.

Answer 1

The Lagrangian, L, of a dynamical system is a mathematical function that summarizes the dynamics of the system. For a simple mechanical system, it is the value given by the kinetic energy of the particle minus the potential energy of the particle but it may be generalized to more complex systems. It is used primarily as a key component in the Euler-Lagrange equations to find the path of a particle according to the principle of least action.

What, firstly the lagrangian gives you - in classical mechanics- is the ability to look to a problem where you wish to predict the movement of a system without getting involved with resistance forces(like the forces from a rope or the floor) but only with physical forces(gravity and other,if you work on abstract problems potential originated forces). Also, in this interpretation you don't see the forces as vectors, because to derive the lagrangian, let's say in classical mechanics, you have L=T-V where T the kinetic energy and V the potential. You only have to understand correctly the bonds of the system.

From the understanding of the bonds of movement we have an interpretation of geometralization of the problem at stake. With not dealing with the meaning of force but by seeing the problem like a geometry problem with solution via an energy function of the system(lagrangian) not only we make easy for our selves not to search what the force is but we find in the end the principle of least action: The principle remains central in modern physics and mathematics, being applied in the theory of relativity, quantum mechanics and quantum field theory, and a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

What, in a first reading says is this: When a dynamical system moves from a position 1 at time t1 to a position 2 at t2 with generalized coordinates q the path the system will follow is that where the $$\int {L(q,q',t)dt}$$ from 1 to 2 takes the least value, that is the statical value $$δ \int{Ldt}=0$$

From here on, the use is huge. For example, in quantum mechanical interpretation of light the Feynman's Quantum Electrodynamics comes by applying accordingly the above mentioned principle.