Physics+ Hamiltonian Intro: UNIZOR.COM - Physics+ 4 All -Hamiltonian

Duration: 18:41 1885 transcript words EN summary

Video summary

The video introduces Hamiltonian mechanics as a refined mathematical framework designed to simplify and improve upon the traditional Lagrangian approach used in classical physics. While previous chapters focused on deriving equations of motion using the Euler-Lagrange method based on the difference between kinetic and potential energy, this lecture highlights several drawbacks of that older model. The primary issues include the reliance on second-order differential equations, which are more difficult to solve than first-order ones, and a lack of clear physical intuition regarding why the Lagrangian is defined as the subtraction of energies rather than their sum. To address these challenges, the presenter proposes shifting perspective by treating generalized coordinates and generalized momentum as independent variables, thereby transforming complex second-order problems into a set of simpler, coupled first-order equations. The core transformation involves rewriting kinetic energy in terms of momentum instead of velocity, allowing for a new function called the Hamiltonian to be defined as the sum of potential and kinetic energies rather than their difference. This redefinition is particularly effective because, in simple mechanical systems where forces are conservative, this total energy remains constant over time. By substituting the Lagrangian with the Hamiltonian, the equations of motion split into two distinct but symmetric sets: one describing how momentum changes based on potential energy gradients and another describing how coordinates evolve based on kinetic energy relationships. This symmetry not only makes the mathematical structure more elegant but also ensures that each equation is a first-order differential equation, which significantly reduces computational complexity compared to solving single second-order equations for position alone. The significance of this approach extends far beyond classical mechanics into modern theoretical physics fields such as quantum mechanics and statistical mechanics, where these simplified formulations are essential tools. In contemporary applications like the Schrödinger equation or path integral formulation in quantum theory, variables often need to be treated independently without assuming a direct relationship between position and momentum at every instant, making Hamiltonian formalism indispensable. The lecture concludes by emphasizing that while this method introduces twice as many equations—once for coordinates and once for momenta—the resulting system is far easier to manage because each component deals with first-order dynamics rather than the more cumbersome second-order derivatives found in Lagrangian mechanics. Ultimately, adopting the Hamiltonian perspective provides a clearer physical interpretation of energy conservation and offers a powerful foundation for advancing into advanced areas of physics research.

Read the full video transcript

Hi, I'm Zohar. Welcome to New Zohar Education. Uh today we will start a new chapter called Hamiltonian. So, today is Hamiltonian introduction. Kind of short. Um just to basically introduce you to main concept of Hamiltonian mechanics. Um and this is more, I would say, mathematical rather than physical. Um however, it's very important to actually understand that um it leads to a some more physical um approach to um equations of motion. What do we know as of now from the previous uh chapters, Lagrangian and uh Noether's theorem? We have related um the equations of motion with Lagrangian. Now, equations of motion is basically Euler-Lagrange equations, and let me just write it down. Um it's uh partial derivative of Lagrangian by coordinate is equal to time derivative of partial derivative of Lagrangian by generalized velocity. So, this is the main Euler-Lagrange equation, which we basically derived many times, explained, etc. Now, what is Lagrangian? Well, in simple, relatively, you know, in classical, simple cases, Lagrangian is difference between um kinetic and potential energy. Now, this difference, which is called, well, it's it's Lagrangian, but I mean, what what's the physical meaning of this? Well, to tell you the truth, I feel very uncomfortable with in >> [clears throat] >> interpreting t minus t t minus u from the physical standpoint. I I don't know basically what what physical sense is in this difference. So, it it feels uncomfortable. What else is uncomfortable? Well, this is the second order differential equation. C1 derivative, another derivative. Uh so, it's the second order. It's always difficult to deal with, much easier to deal with one with the first order with the the first order differential equations. So, there are certain draw points in this approach. Now, Hamiltonian mechanics kind of eases out a little bit. Um so, we will convert basically with this model into another model, which is Hamiltonian mechanics, which looks simpler. And going forward, I can tell you that Hamiltonians uh mechanics and all these whatever uh formulas are related are much more suitable for contemporary um theoretical physics like um quantum mechanics, statistical mechanics, and and others. So, this is basically the motivation for us. Now, this chapter uh of this particular course is dedicated to basically research what is the Hamiltonian mechanics and how it really looks like from mathematical standpoint with equations, etc. And this is all part of the course called Physics Plus. It's presented on unizor.com totally free website, no subscription, no advertisement, no signing in, just pure knowledge for your consumption. So, let me basically start with this thing and try to to do it maybe a little bit better. Okay, so first of all, let's just concentrate on one very simple thing. In the previous lectures, primarily in description of the Noether's theorem, we were talking about generalized momentum, which we defined as partial derivative of Lagrangian, which is basically function of time, generalized coordinates, and generalized velocities. By general by generalized velocities. So, we we have defined it as um momentum, generalized momentum. It's a definition, basically. Now, obviously when I'm writing just Q, I mean Q1, Q2, Q3, etc. It's generalized uh coordinates. Okay? Now, why is this um making sense? Well, look at the Euler-Lagrange equation. What can we do right now is to rewrite it as dL by dt Q is equal to uh time derivative of this, of the momentum. So, time derivative is in Newtonian uh notation is just a dot. So, dot is time derivative here and here. So, this looks This looks much better than this, right? Well, what's the problem? The problem is that we have only n uh equations of the second order. Here we have uh also n equations of the second order, but here we have n different unknown variables, which are functions q of time t. Here we have two n functions, q as a function and p as a function. If we want to consider this as this simple thing, we should consider them to be independent. So, this is basically a very important moment. We are considering momentum, generalized momentum, and generalized coordinates independently. So, this is something we should really have to uh kind of understand because in classical mechanics, we know that momentum is mass times velocity, and velocity is time derivative of coordinate. So, they are related. Here we are considering them independently because it's not the old-fashioned classical mechanics. So, if we are considering them independently, because otherwise we cannot really bring it to a first-order equation. Then we have a problem of not having sufficient number of equations. We have two n uh independent variables, but only n equations for each i, basically, right? So, it's for each i. For each coordinate. So, that's the problem. And now I'm introducing a very easy solution to this problem, which leads to Hamiltonian equations. And here it is. Let's just recall that in most of the um classical mechanical simple um models simple mechanical systems, we have this thing where T is a function of what? Of velocities. And U is a function of coordinates. Kinetic energy does not depend on position. It's basically remember MV squared. And V is actually Q with a dot, time derivative of coordinate. And the potential energy depends only on position. Which means on coordinates. So. In classical mechanics, we know that Q and P are related. So, I would like actually to to have it rewritten as function of P somehow. So, what is this? Well, the kinetic energy is MV I squared. For all um different It's kinetic energy of the whole system. So, VI is the uh the velocity of each particular component of that system. Now, if instead of this, I will put this. That's the same thing, right? P is uh mv. Square would be m squared v squared divided by m, it would be this. Uh I really have to put 1/2 here. I'm sorry. So, that's the definition of T, kinetic energy. Okay, fine. So, now if I'm using this particular definition for this, I get come up with the following thing. So, if T is equal to 1/2 sigma um 1 over m i p i squared, then dT by dP uh i would equal to So, for one particular uh i, so it's a square and 1/2, so it would be 1 over m i p i. Right? Okay. And what is this? Back to P is equal to mv. So, if I divide it by m, it would be v, right? So, it's q with a dot. Now, this is an interesting equation. Now, this is also an interesting equation. So, let's rewrite this as T minus U. Now, we know that kinetic energy is independent of coordinate. So, this is actually minus dU by dQ i. T is independent of q1. Q depends on qi. It depends on the velocity. So, we have an interesting two equations. One is this one. pi is equal to minus du by dqi. And another is this one. Q Uh sorry. We have Is it dot? qi with a dot is equal to dt by dpi. Look at the symmetry of this. It's definitely much better. And now we have two n equations. What's the problem? Well, the problem is that the first one is a function of of potential energy. And the second one is a function of kinetic energy. And these are two different functions and this is really doesn't look that doesn't look good. However, we can definitely very easily repair it. Again, T is independent of Q. It's function only of uh of momentum. Now, U is independent of momentum and is function of coordinates. So, what I can do is I can put here instead of U I can put T plus U. And it doesn't really change anything. Because T is independent of coordinate. Instead of this, I can put also T + U. Because U is independent of uh momentum. It depends only on uh position. Now, I have the same function. Right? I call this function Hamiltonian. And now, my 2N equations now look like this. Now, we have two independent 2N independent equations where QI and PI are independent variables and variables each and N variables this. We have 2N different differential equations of the first order and quite frankly, it looks much beautiful much more beautiful than this one. So, this is the main kind of road which introduced us to Hamiltonian mechanics. And what's one of the very important issues for myself, what is this? Well, this is the sum of potential and kinetic energy which is total energy of the system. So, in these relatively simple mechanical systems, whatever whenever this thing actually is an energy, we have some meaningful meaningful um uh system of coordinates a system of equations where the function from which we do this partial differentiation is energy. I mean, it has certain meaning much more than this one. The difference between kinetic and potential. So, this is the main uh road where we introduced a concept of Hamiltonian. In classical simple cases, it's full energy of the system, mechanical system. Um again, everything basically is related to conservative mechanical systems. Um we have certain forces like gravitation and some some something like this. The the the the forces which which have the field and you it's uh conservative forces. Um so, in all these cases, like planets are moving around the sun uh along their orbits. I mean, this is typical example of conservative system. So, and in these cases, we have this particular thing. So, obviously, it's very natural to generalize it to generalized coordinates. And uh in as much as we need the Lagrangian as a function uh which characterize the behavior of the system, we need for this theory, for Hamiltonian mechanics, we need Hamiltonian to basically describe the system. And then, the system of equations would be much much more um uh I don't know. Beautiful, if you wish. I mean, I do like how it looks much better than than this one. Although, there are two n equations and only n here, but each equation seems to be much simpler. So, the more complicated problem we have divided into more less complicated problems. And then each particular problem is easier to resolve because it's a first-order rather than a second-order equation. So, as a result, I can say that the Hamiltonian mechanics is um used in many different contemporary theoretical physics theoretical physics fields. For example, the statistical mechanics, quantum mechanics, and and some others which I don't really know myself about. Okay, anyway. So, that's it for today. I do suggest you to read notes for this lecture. Uh every lecture on unizor.com has video part and textual part. Textual part is similar to a textbook. So, you I suggest you to read. That would be you go to unizor.com, you choose physics plus course. Uh the chapter is called Hamiltonian, and then that's the first lecture called introduction um to Hamiltonian. That's where you will read whatever I was just talking about. So, thank you very much and good luck.