Lagrangian Mechanics Review

In this section of pages, we will review the basics of Lagrangian mechanics, which is the foundation of classical field theory.

Overview

In classical mechanics, we often describe the dynamics of a system using the Lagrangian formalism, which is a reformulation of Newton's laws of motion. The Lagrangian formalism is based on the principle of stationary action, which states that the true path of a system is the one that minimizes the action.

We shall first derive the Euler-Lagrange equations, which are the equations of motion derived from the Lagrangian. Then, we will see how it can be applied to simple contexts, such as the length of a curve or a double pendulum.

Afterwards, we will introduce the concept of symmetries and conservation laws, and how they are related to the Lagrangian. These conservation laws are derived from Noether's theorem, which states that for every continuous symmetry of the Lagrangian, there is a corresponding conservation law. Noether's theorem is very important later on in quantum field theory, where symmetries play a central role in the formulation of the theory.

Finally, we will extend our understanding of Lagrangians to more complex systems, such as relativistic systems, general relativity, and eventually, string theory. In general relativity, the Lagrangian is the Einstein-Hilbert action and describes the dynamics of spacetime. Most notably, spacetime is curved by the presence of mass and energy, and this makes it very interesting to study. We will put our knowledge of Lagrangians to the test by deriving the Einstein field equations from the Einstein-Hilbert action.

In string theory, the Lagrangian describes the dynamics of strings, which are the fundamental building blocks of the universe. At a basic level, imagine that instead of an object tracing a worldline in spacetime, it is a string (think of a rubber band) that traces a world-area, or a "worldsheet". We can extend the action from an integral over the worldline to an integral over the worldsheet, and this gives us the dynamics of the string. The action in string theory is called the Nambu-Goto action.