The Lagrangian is a quantity we define for a system of particles and is denoted by $\mathcal L$. It’s most often defined as

$$ \mathcal L=T-U $$

Where $T$ is the kinetic energy for a system of particles and $U$ is the total potential energy for that system

Why do we use the difference instead of the total energy like we’re used to?

→ It’s difficult to provide an intuitive motivation for this, but it turns out we need the difference in order for Hamilton’s Principle to give us the right path that matches with Newton’s Laws

We use the Lagrangian to find the Action, $\mathcal A$, (or sometimes denoted by $S$) ****which describes how the Lagrangian (and thus our system) changes over the course of its trajectory.

$$ \mathcal A=\int_{t_1}^{t_2}\mathcal L ~dt $$

Where $t_1$ is some starting time and $t_2$ is some end time. The Lagrangian is based upon the path that’s taken in between $t_1$ and $t_2$. We say that the action is a functional because the value of $\mathcal A$ depends on the functional form of the path taken. The action will in general be different if the path taken is a straight line vs if the path is some swirly helical path for instance.

→ Our goal with Lagrangian Mechanics is to find what that path is

→ All possible paths will share the same boundary conditions, that is they all will have the same value at $t_1$ and $t_2$ respectively. Essentially we’re considering all possible paths between two fixed points

Hamilton’s Principle

In general Hamilton’s Principle states that the actual path taken by a system of particles is the one that minimizes the action for deviations that are consistent with the constraints of the system.

→ Now there’s a lot to unpack here. The core ideas we need to make this principle usable are:

• how do we minimize a functional like $\mathcal A$? → For this we will need the calculus of variations which involves looking at variations for a functional
• What do we mean by deviations consistent with the constraints of the system? → This will lead us to generalized coordinates, one of the most powerful benefits of the Lagrangian method
• Why is this true? → It turns out that based on the calculus of variations and the definition of the Lagrangian, the action will be minimized when the path taken satisifies Newton’s Second Law
  • This principle is principally based on Newton’s Second Law, which we will recall is only valid for inertial reference frames. Therefore we must define our Lagrangian in an inertial reference frame
  • However, while we must define our Lagrangian in an inertial reference frame by expressing the position of the particles of the system in the inertial reference, we will find that we still have the freedom to use noninertial coordinates

Let’s break down each of these ideas:

Minimizing Functionals and the Calculus of Variations

Generalized Coordinates

Proof of Hamilton’s Principle