The Most Important Idea in Physics: The Principle of Least Action - Ask a Spaceman!

there are laws of physics and then there are laws of physics and what i'm going to tell you about today is like the mother law of physics this is the originator of all the laws of physics this is how physicists create entire laws of physics is crazy and it's called the least action principle to get started if you've had any sort of stem-ish education in college or even took a high school physics class you probably learned about newtonian mechanics you know the three laws of motion uh f equals m a all the good stuff and these are incredibly powerful rules of the universe of how the universe works they allow us to explain so many different un completely unrelated phenomena so newton's laws of motion if i throw a ball at you newton's laws of motion tells you how that that ball will reach you it tells us about the orbits of the planets it tells us about uh ocean currents it tells us about how airplanes work it's all in there baked into those three laws of motion and in particular f equals m a uh this force equals mass times acceleration this classic third law of motion of newton this is what's called an equation of motion this tells us how an object will move all we need to do is add up all the forces that are acting on a certain object and if we know the mass we can calculate its acceleration and from there we can follow its trajectory and we can predict where it's going to go if i throw a ball with a certain amount of force and then we have the gravitational force of the earth pulling on it down like this we know the path it will take to reach you everyone knows newtonian mechanics but not a lot of people know that there are different ways of formulating the exact same physics newton's analysis the analysis that newton used uh focused on forces and masses and accelerations and it's incredibly useful and f equals ma is really really powerful and technically it applies to every situation in classical physics um but sometimes it can get really unwieldy if there are a lot of forces or there are different uh like friction terms and rotational terms and in in real life situations it can get very messy and get very cumbersome and so after newton other physicists and mathematicians came along to develop alternative ways of describing the exact same scenario in the exact same physics that turn out to be mathematically equivalent and most people don't know it and there are two major ones one developed by joseph louis lagrange and we call these lagrangian mechanics and then another set developed by sir rowan hamilton sir william rowan hamilton not not the hamilton the musical different hamilton and from there we have hamiltonian mechanics and the basis for this for lagrangian and hamiltonian physics is that when i look at a physical situation say a ball being thrown there are all sorts of properties of the ball properties of the situation that i might care about and just looking at it completely naively and ignorantly i don't know which properties matter in which don't i don't know what quantities i need to measure and how those connect to other things newton discovered that f equals m a is incredibly powerful and incredibly useful that force and mass and acceleration are all connected to each other but there are different ways of approaching the same problem and lagrangian and hamiltonian mechanics focus instead not on forces and masses and accelerations but on kinetic and potential energies so the kinetic energy of a ball being thrown at you is just one half the mass times the velocity squared and then the potential energy of a ball being thrown at you is just its height above the ground times its mass times the the gravitational constant that's it and from this from instead of looking at forces and masses and accelerations you can find discover relationships between kinetic and potential energies and from there you can develop an equivalent picture of this problem of this situation and the lagrangian picture in particular is very very useful because it turns out to have a superpower so the lagrangian itself the basic core equation of lagrangian mechanics is simply the kinetic minus the potential energy that's it just like the core equation of newtonian physics is f equals ma the core of lagrangian physics is kinetic minus potential energies now where do we go from there in all cases it's a choice of whether you use newtonian mechanics with f equals m a or lagrangian mechanics with kinetic minus potential energies they are mathematically equivalent and proven to be so so it's just a matter of which one is easier which one is less wieldy which one is easier to solve or easier to handle the the the particular scenario and it's no big deal but lagrangian mechanics does something cool in lagrangian mechanics we have a certain quantity we call it the action and the action is the sum total difference between the kinetic and potential energies of an object in motion so if i were to throw a ball at you it would start with some initial kinetic energy because i threw it and some initial potential energy because it's a certain height off the ground then as it travels its potential energy will go up and its kinetic energy will go down it will slow down and then it will come racing towards your face and i hope you can catch it in time and it at the end it will have a certain amount of kinetic and potential energies you can just chart that all along you can measure that difference between the kinetic and potential energies across that entire path and then you can add up all those differences across the entire path to get a single number that we call the action for you stem nerds out there the action is the integral of the lagrangian the integral of the kinetic minus potential energies but you don't have to worry about that if you don't know what an integral is okay so what's the big deal about the action what's the big deal why do we care so much about lagrangian mechanics in the action well here is where this episode is going to start to blow your mind if i were to throw a ball at you as soon as you saw it in motion you'd have some general picture of what it's going to do you you put your hands in roughly the right spot maybe you can't respond fast enough especially if you're like me and like and don't play sports a lot uh but if you play a lot of sports you have a lot of practice so you see okay the ball is coming this way and i think it's gonna land here and i can catch it right here how do you know that how does your brain know that well your brain knows that because it's seen lots and lots and lots of balls thrown at you over the course of your life in every ball tends to have a predictable path once you launch a ball it follows a very specific path it follows a parabola that is the name of the path that we give to an object you know being thrown in typical earth gravity each one is slightly different it depends on the angle and the initial speed and all that but in general it's a parabola you you can write down an equation for the path that the ball takes now i can ask you where does that path come from of all the paths that the ball could have taken why did he choose that one it could have taken a path that shot up straight up into the sky and danced around a little bit and then came shooting down it could have taken a path where it went backwards behind me and looped around and then came to you it could have come and then stand still for a little moment and then and then reach you like why does it follow this parabola instead of literally any other path you know nature had an infinity of paths to choose from and it picked the parabola why we would say because f equals m a this is the law of nature this is the equation of motion force equals mass times acceleration we know the forces on the ball and then this is the only path that f equals m a allows okay why is f equals m a correct why is newtonian physics correct why is f equals ma the correct formula instead of one half m a or m a squared or m minus a or a to the fourth divided by m why this relationship and not any other well this is where lagrangian physics gives us an insight because the path that goes like this that follows f equals m a that has a parabola we can calculate an action for that path and the path of the ball that does something weird like goes off to the side here loops around someone's head and then hits you we can calculate an action a total amount of action for that path and then the action for the path that shoots off to the moon and the action for the path that goes down to the ground and hovers for the while and for a while and then shoots up out right in front of your face we can calculate the actions for all of those in the parabola one the one we expect that one has the least amount of action that one has the least amount of action the path the ball actually takes is the path with the least amount of action the reason that newton's laws are correct and that f equals m a is the correct formula is because that is the formula that results in the motion that has the least amount of action but it gets wilder you know i i used this example of a ball being thrown as one particular example and where you can add up you can calculate all the actions and then the actions for the other pass you know and you realize that the parabola the path given by f equals m a is the one with the least amount of action but we can broaden this we can broaden this because we can write down the k the formula for the kinetic and potential energies regardless of a situation for any object we can write down its kinetic and potential energies it's just one half the mass times the velocity squared the potential energy in a gravitational field is just the mass times its height times the gravitational constant so i can write down very generic very broad formulas for that and i can use a mathematical trick called the calculus of variations to compute equations of motion in any scenario when i write down the lagrangian of a generic situation of just a object in motion in a gravitational field that lagrangian is kinetic minus potential energies i can use something called the calculus of variations to find what the minimum action is in that super generic scenario and you know what pops out what pops out is f equals m a lagrangian mechanics in the principle of least action creates equations of motion i can take the lagrangian and i can derive newton's laws now we know why f equals m a is correct now we know why newton's laws are correct because those are the formulas those are the equations of motion that satisfy the principle of least action and it gets crazier folks folks any time i can write down the kinetic and potential energies of a system anytime i can write down the lagrangian i can derive the equations of motion and equations of motion are also known as physics this is how these are the equations that physicists use to predict motion so kinetic and potential energy i don't know of say a charged particle in an electromagnetic field i can write down that lagrangian i can apply the principle of least action and what comes out are maxwell's equations for electromagnetism i can write down the kinetic and potential energies of quantum particles interacting via the strong and weak nuclear forces i can apply the least action principle i can find the minimum amount of action and what comes out is the standard model of particle physics i can write down the kinetic and potential energies of mass and energy moving in a space-time metric that deforms in response to mass and energy i apply the least action principle and i get general relativity i get the einstein field equations the least action principle is a generator of physics the least action principle is a creator of physics it is a mother principle that allows physicists to generate laws of physics and equations of motion it's right there folks you can write down a lagrangian with around half a dozen terms just half a dozen you can print it on a t-shirt and if you were to apply the least action principle to that lagrangian you would get all of physics every physicist in the modern world operating today no matter their field astrophysics cosmology string theory high energy particle condensed matter thermodynamics everybody every single physicist is working from the same lagrangian in the same principle of least action it's all just different corners just different facets of the exact same principle now another question comes up when we're talking about the principle of least action is how do we connect this to the quantum world you know the quantum world is different than the classical world like the least action principle everything i just said it completely describes all of classical physics but how do we approach this in the quantum perspective where where there's always probabilities and you never know where a particle is going to go or what it's going to do and it depends on the observer and the measurement and all that usual quantum fuzziness well the answer here one answer to it is what's called feynman's path integral approach or the sum over histories approach in a quantum system if instead of throwing a ball at you i shoot an electron at you and an electron is a very quantum mechanical object that electron in fineman's perspective takes every single possible path so there's one electron that shoots right to you there's one electron that goes up to the moon and comes back there's one electron that like dances around the floor and signs out your name first and then comes to you there's electrons that do all of that but in quantum mechanics you have to assign a probability to everything that a particle does and when you properly assign the probabilities to all these different paths all these different trajectories they end up canceling each other out they just wash away and all that's left is that classical path given by the least action principle so this is how feynman made a connection between classical and quantum physics and how he was able to take the least action principle that we know and love from classical physics and import it into the quantum world and then this allows you to reach into the quantum world where sometimes particles do take multiple paths at once this gives you an analysis tool to to handle that uh getting into feynman's path integral approach deserves a whole other episode but at the end of the day no matter what whether it's a classical system or a quantum system whether it's general relativity or electromagnetism or good old-fashioned throwing a ball at your friend the least action principle is in charge thank you so much for watching i hope you enjoyed i'll see you next time in the meantime please go to patreon.comsutter to keep supporting the show

Channel: Dr. Paul M. Sutter

Views: 35,395

Keywords: space, cosmos, universe, astronomy, physics, astrophysics, cosmology, science

Id: UuqpCBZoX3M

Length: 17min 13sec (1033 seconds)

Published: Wed Aug 03 2022