From Newtonian physics, to relativity, to quantum physics. All of the most useful and important theories of physics are based on this one principle known as the "Principle of Least Action", also known as the "Principle of Stationary Action". And yet, I don’t think we talk about it enough.
When we learn high school physics, we often start by studying objects that exert forces on each other. But there is another way to find what the equations of motion of objects in a system. We can do this by considering energy rather than forces.
We start by defining the Lagrangian of our system. In simple cases this is just the kinetic energy minus the potential energy in the system. The Lagrangian allows us to define an Action - the integral of the Lagrangian over time, between when we start and stop studying our system.
The action is just a scalar quantity - the complicated integral spits out one single number at the end. And it directly related to the trajectory that our system takes (through its speed dependence due to kinetic energy, and position dependence due to potential energy).
The principle of least action says that our system takes the trajectory for which the action is minimised, i.e. the kinetic and potential energies that result in the lowest possible action are the ones that our system will have in real life. It's worth noting that sometimes the relevant quantity is the stationary action, not just least action.
We look at the logic physicists use to calculate the least action. This is done through understanding the calculus of variations, and how to minimise a functional such as the action. However in this video we keep it very simple and understand it intuitively.
The principle of least action breaks down when our system contains dissipative forces (such as friction, air resistance, etc). However, these forces are simply large-scale simplifications of more complex small-scale interactions. So if we considered those small-scale interactions (e.g. particles bumping into each other), then the principle of least action continues to work perfectly.
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Chapters:
0:00 - The most important physics principle?
0:42 - Forces vs. energy
1:44 - Defining the "Lagrangian", and "Action"
2:53 - The Principle of Least Action
4:35 - Where the principle is used, and where it breaks down
Videos in Cards:
2:05 - youtube.com/watch?v=KpLno70oYHE