Why Lagrangian Mechanics is BETTER than Newtonian Mechanics F=ma | Euler-Lagrange Equation | Parth G

hey everyone parth here now many of you have asked me to discuss lagrangian mechanics in one of my videos and so i thought we'd take a look at the basics if you enjoyed this video then please do hit the thumbs up button and subscribe for more fun physics content hit that bell button if you'd like to be notified when i upload and please do check out my patreon page if you'd like to support me on there alright let's get into it now lagrangian mechanics revolves around one central quantity known as the lagrangian named after joseph louis lagrange it is denoted with the capital letter l although sometimes when handwritten we'll use a squiggly curly l rather than a normal uppercase l and this quantity the lagrangian is defined as the kinetic energy of the system that we happen to be studying minus the potential energy of that system in this case kinetic energy is represented by capital t and potential energy by capital v now there are a couple of things worth mentioning here the type of lagrangian mechanics that we will be discussing is based on classical physics so we're using the classical kinetic energy and potential energy we're not looking at any relativistic or quantum mechanical additions to lagrangian mechanics so basically everything we'll be looking at today is based on stuff like you know newton's laws of motion or the fairly intuitive physics that came well before relativity or quantum mechanics and secondly the lagrangian is indeed defined as the kinetic energy minus the potential energy the lagrangian is not necessarily a physically useful quantity it is a mathematically useful quantity what do i mean by this well basically the kinetic energy minus the potential energy of a system doesn't really represent anything physical per se unlike another quantity known as the hamiltonian which is defined as the kinetic energy plus the potential energy which can be thought of as the total energy of the system but the lagrangian t minus v kinetic energy minus potential energy is a mathematically useful quantity as we will see later by the way there's also a whole nother branch of physics that deals with the hamiltonian and meltonian mechanics and you'll also see this crop up in a different form in quantum mechanics but that discussion is for a separate video today we'll stick with the lagrangian so the lagrangian deals with the kinetic and potential energies of a system let's take a look at an example of how we go about doing that let's consider one of my favorite basic systems the classic mass attached to a spring let's also keep things simple and assume that we've got an ideal system which means that all of the mass is in the mass we don't worry about the mass of the spring and let's also assume that the mass block is on a frictionless surface which means there's no friction between the mass and the floor the system can therefore just ping back and forth once we set it oscillating now these idealizations are obviously just there to make the scenario easier to understand and the maths easier to follow so let's look into the maths let's find the lagrangian for this system what is the kinetic energy of the system well let's recall that kinetic energy is defined as half mv squared where m is the mass of the object we happen to be considering and v is the speed with which it moves at this point we can also define the distance moved by our mass away from its natural position and we'll call this distance x the natural position of course refers to when the spring is neither extended nor compressed and then we can recall that the speed of an object is equal to the distance moved per unit time some of you may be familiar with the expression v is equal to delta x divided by delta t or delta x is the change in the distance moved by an object and delta t is the time taken for that change to occur or using calculus we can say that v the speed is equal to dx by dt the rate of change of the distance moved by the object or simply how quickly the object moves now just as a little notation thing another way to write dx by dt is simply as x with a dot on top this is like i say just a notation thing it just makes our mass look a bit cleaner but the dot represents the d by dt bit so we can say that the kinetic energy of this system now is equal to half m x dot squared and the reason we write it in terms of x dot rather than just the velocity v will become clear very shortly now what about the potential energy of our system well this is where we really need to think about the spring let's recall that the potential energy stored in a spring is equal to half kx squared where k is known as the spring constant or stiffness of the spring and x is either the compression or extension of the spring away from its natural position and at this point we can combine our two expressions for the kinetic and potential energies and stick them into our definition of the lagrangia l is equal to t minus v the lagrangian is equal to kinetic energy minus potential energy which is equal to half m x dot squared minus half k x squared and that's how we find the lagrangian for this particular system now what well this is where things take off a little bit in terms of intensity and complexity for lagrangian mechanics as it turns out the lagrangian is a quantity that heavily features in an equation known as the euler lagrange equation it looks something like this this equation looks super complicated but we'll be looking at some of its most important aspects first of all we can notice that it contains the lagrangian now the euler lagrange equation is actually consistent with newtonian classical mechanics we won't talk about why that's true we'll save that for another video but let's just assume that it is for now if you're keen to find out more about this equation though i'll leave some resources in the description box below and i highly recommend you look up calculus of variations anyway so we've got this generic equation the euler lagrange equation and we can take our lagrangian for a specific system and plug it in to our equation this will tell us something about that system in fact when we plug in the lagrangian what we'll find is known as the equation of motion for our system go through the details of this derivation in a second but essentially what we find is that the mass of the block m multiplied by the acceleration x double dot that's the second derivative or d2 x by dt squared is equal to minus kx of course k is the spring constant and x is the displacement of the block now this equation might look a little bit familiar we've already mentioned that x double dot is the acceleration of the block which means that we're just stating something about the forces acting on our system essentially the net force on our system m a is equal to the force exerted by the spring minus kx and the spring force is just negative because it acts in the opposite direction to the displacement of the block now this equation is known as the equation of motion and we could have easily just found it by considering the forces in our system and saying that the net force is equal to all of the forces acting on our system in this case just the spring force so why did we go the long way around finding the lagrangian plugging it into the euler lagrange equation dealing with lots of complicated calculus and then arriving at something which we could have done very quickly if we'd considered forces well there's a few different reasons for this firstly the lagrangian method actually gives us the equation of motion without having to think about forces at all we only need to consider energies in some cases this is actually much more convenient and certainly it's just another way of getting to the same result but actually there's a lot more to it than that for more complicated systems it can be very finicky and annoying to have to deal with lots of different forces and it's much easier in many cases to deal with energies obviously not in this case but certainly is true for other cases additionally when we were looking at our system we were only considering one coordinate the x-coordinate but in other systems where there's motion going on in lots of different directions x y and z or r theta phi depending on which coordinate system you're using it becomes much easier to use the euler lagrange equations because they're very good at dealing with multiple coordinates and they actually give us an equation for each coordinate in this case we've got an equation in x and then we could also have an equation in y and z if we had motion in those directions now as we've mentioned already this is just one rather simple example of how we can apply the euler lagrange equations to any system we happen to be considering the more complicated the systems get the more useful the euler lagrange equations become and the more handy it becomes to work with lagrangian mechanics but lagrangian mechanics goes even deeper than anything we've discussed today for example some of you may have heard of notre's theorem named after eminota which says that there is a fundamental link between any symmetries in our universe and conservation laws like conservation of energy and conservation of momentum those are directly related to specific types of symmetry in our universe now that is a very hand-wavy statement from me i haven't even described what i mean by symmetries but we'll talk about that in a separate video about notice theorem the point though is that lagrangian mechanics is extremely handy extremely clever and extremely deep and so i think the big takeaways are the fact that we have a quantity known as the lagrangian defined as the kinetic energy minus the potential energy the fact that this is just a mathematically useful quantity rather than a necessarily physical one and the fact that there is this equation the oil lagrange equation which at the moment looks like it came out of nowhere but it is consistent with newtonian mechanics and classical physics basically and with all of that being said thank you so much for watching if you enjoyed this video please do hit the thumbs up subscribe for more fun physics content and hit that bell button if you'd like to be notified when i upload please do check out my patreon page if you'd like to support me on there thanks very much for all your support as always and i will see you very soon [Music] you

Channel: Parth G

Views: 278,603

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Keywords: Lagrangian mechanics, physics, classical physics, euler lagrange equation, euler-lagrange equation, lagrangian mechanics explained, lagrangian basics, parth g, newtonian mechanics

Id: KpLno70oYHE

Length: 9min 44sec (584 seconds)

Published: Tue Feb 23 2021