Hamiltonian/Lagrangian Mechanics

[Aidan130791]

Can anybody please explain these theories, and what they do so differently than Newton's mechanics? I can't really get my head round them, and I'm only doing my A-Levels at college, so i haven't been told anything about them properly. I only know what i can learn by reading things, and they are really confusing, I can't find anything that is understandable. Help please? Also, what is their significance in QM and QFT?

Thanks

 

[ZapperZ]

Can anybody please explain these theories, and what they do so differently than Newton's mechanics? I can't really get my head round them, and I'm only doing my A-Levels at college, so i haven't been told anything about them properly. I only know what i can learn by reading things, and they are really confusing, I can't find anything that is understandable. Help please? Also, what is their significance in QM and QFT?

Thanks

The quick and dirty reply: you deal with energies (scalar) and not forces (vectors). Any time you can juggle one less ball, it tends to be easier.

The long and detailed answer can be complicated, especially if you only have A-level knowledge so far. The Hamiltonian/Lagrangian approach is based on the principle of least action. To some people, that is more "fundamental" than dealing with "forces".

The best way for you to catch up on this is to read about this at http://www.eftaylor.com/leastaction.html" , who is one of the figures that have been trying to push this approach even at the elementary level.

Zz.

 

[Cyrus]

Lagrange makes things so much easier. You should always avoid using a Newtonian method if you can. Newtons method is pulling teeth compared to Lagrange. Find the energy, take the derivatives, and plop, out comes the equations of motion.

You don't have to worry about internal forces between this body and that body, which way does this force point, what is the cosine or sine of the force direction, etc etc etc. Newtons method is archaic in comparison.

 

[Aidan130791]

Ok, that's nice and helpful. But why the significance for quantum physics? Also, is there any books that you know of that I would understand, at least partially, with my A-Level knowledge?

 

[Cyrus]

Whats an A-level?

 

[ZapperZ]

The link I gave you should have an article on why this is important to QM. It is not a mere coincidence that the Schrödinger Equation is often called (in some variation) the Hamiltonian.

Zz.

 

[nrqed]

Can anybody please explain these theories, and what they do so differently than Newton's mechanics? I can't really get my head round them, and I'm only doing my A-Levels at college, so i haven't been told anything about them properly. I only know what i can learn by reading things, and they are really confusing, I can't find anything that is understandable. Help please? Also, what is their significance in QM and QFT?

Thanks

That's a very deep question. It's the kind of question that as the years pass you will keep coming back to again and again and always feel that there is a new layer to uncover. Others have made already good points. Let me add a quick comment.

As already said, the Lagrangian approach (I won't talk about the Hamiltonian approach for now) involved working with scalare quantities (energy) rather than the vector quantities (forces, position, acceleration) of the Newtonian approach. Also, the Lagrangian approach is inherently nonlocal. You work with possible *paths* of a particle and pick up the one that minimizes (or in general extremizes) the action. In Newtonian mechanics, you start at a point, look at the initial velocity and use the forces to find the new position at an infinitesimal time later, and so on. So you always work locally.

The Lagrangian approach is better suited to study wave phenomena and interference (one can look at the different paths the wave can go through and see which path is constructively reinforced) and even for a classical wave it's better to not have to work with forces acting on a wave but rather with its energy. In quantum mechanics, one never works with forces acting on particle or with an actual well-defined trajectory, which is why one cannot use at all the Newtonian approach but one has to rely on the Lagrangian approach (which leads to the path integral formulation fo quantum mechanics).

Just my two cents.

 

[smallphi]

Ok, that's nice and helpful. But why the significance for quantum physics?

When quantizing a system we have to guess its equations of motion. We cut down on the possibilities by assuming those equations obey certain symmetries like Lorentz invariance etc. It's hard to write from scratch an equation with certain symmetries built in. On the other hand it's much easier to write a Langrangian with certain symmetries. Once a Lagrangian is guessed that obeys the assumed symmetries, it automatically generates equations of motion that obey the same symmetries.

 

[Aidan130791]

Thanks very much. I think I am going to go an buy a book on this, I think the Schaum's outline one looks managable. Any opinions? And A-Levels are what you do at college in the uk between 16-18 years old.