What is the Lagrangian in Mechanics and How Does it Relate to the Hamiltonian?

[r.a.c.]

Hi. There is just this one thing in mechanics which is lagrangian that I just simply can't grasp physically. I'm taking a mechanics course I simply do not understand what the lagrangian is. There is calculus of variations (at least a tiny but of it) a bit of geodesics and the least action principle which is proving also a bit elusive. Is it possible that someone may just give me a quick few sentences about the lagrangian, physically and a bit mathematically? I have tried looking up online pages but they all pretty much say the exact same thing.

 

[Prologue]

I am not sure what you want. But, the lagrange-euler equation is merely the equation that must be satisfied for the 'action', or whatever you are measuring, to be minimal. So, you have an equation of the form

$$I=\int_{t_{0}}^{t_{1}}L(y,\dot{y},t)dt$$

And by postulate you believe that a minimization/maximization of that integral will give you the right answer (least action, whatever, things of that nature). In order for the integral to be a max or min, this equation must be satisfied

$$\frac{\partial L}{\partial y}-\frac{d}{dt}(\frac{\partial L}{\partial\dot{y}})=0$$

This result comes from calculus of variations. No matter what, y has to satisfy that equation.

 

[vertices]

If a body wants to get from A to B, the path the body will take is such that the action (as defined by prologue) is extremised. This is called the "principle of least action" and the EL equations come from this idea..

 

[r.a.c.]

First off, thanks for replying.
Bringing up the subject of calculus of variations. I'm not sure which one(L or I) but I think I[L] is a functional...right!? May I ask what exactly is a functional. Because they said that a functional is a function from the space of functions to R...by space of functions do they mean L|R|...the L-norm of R? Just trying to get a mathematical grasp on it.
And what exactly is the physical meaning of the action!? Or L for that matter!?

 

[waht]

The Lagrangian is the difference between kinetic energy and a potential energy: L = T - V

If there is kinetic energy T then a particle of mass is in motion. If there is potential energy V then it has the ability to store and release energy like a buffer so to speak. The Lagrangian in a sense captures these forms of energies mathematically. If a particle is moving and you want to slow it down, then its kinetic energy must go somewhere. And likewise, if a particle is at rest, and you want to accelerate it, the energy must come from somewhere. Hence the Lagrangian is accounting for motion and potential energy interacting with motion. The units are also in Joules.

The action

$$S = \int L dt$$

integrates Lagrangian with respect to time. Units : J-s

If you do it from t0 to t1, the action adds up all Lagrangians of a system evolution from t0 to t1.

If you drop a ball from a height, then due to the gravitational potential energy it will start to accelerate and fall down to the ground. But suppose that the ball drops half way to the ground, then magically accelerates to the top again, then falls down 3/4 quarters way, goes up half way, and then floats down to the ground. Is this a possible path for the ball?

Not in this universe. But this effect can be captured by the Lagrangian if you plot it with respect to time. At every point in time the Lagrangian will vary in some way. It turns out if you take a path of least action, that is minimize it, then will it lead to the only possible motion of a particle, and that is it will accelerate down to Earth in this example.

 

[Prologue]

The idea idea of a functional in this case goes something like this: Input a function, then the functional chugs out a number (because it does a definite integral). This means, input a function into the functional, get out a number. Whenever that number is min, you have the right function.

 

[Pengwuino]

The Lagrangian isn't something physical, by the way. While the Hamiltonian is the total energy, H = T + U, the Lagrangian, L = T - U, is not a physical quantity. As far as I know, nothing physical exists that IS the difference between the kinetic and potential energy. However, as you must know, it proves vital to solving classical systems.

 

[vela]

The Lagrangian isn't something physical, by the way.

That's true in the classical framework. In quantum mechanics, however, Feynman showed that the classical action for a given path gives the quantum phase of its contribution to the amplitude of reaching the endpoint. This explains why a classical object follows a path that minimizes the action. Only trajectories close to that path don't cancel with others.

 

[FredericGos]

Hi. There is just this one thing in mechanics which is lagrangian that I just simply can't grasp physically. I'm taking a mechanics course I simply do not understand what the lagrangian is. There is calculus of variations (at least a tiny but of it) a bit of geodesics and the least action principle which is proving also a bit elusive. Is it possible that someone may just give me a quick few sentences about the lagrangian, physically and a bit mathematically? I have tried looking up online pages but they all pretty much say the exact same thing.

Have a look at the susskind lectures on youtube if you have some time. They are great imho. When I watched these the first time, I was a bit confused too about all this stuff, but I think they cleared up a lot of things. Great stuff.

http://www.youtube.com/view_play_list?p=189C0DCE90CB6D81&search_query=susskind

cheers
/Frederic

 

[Born2bwire]

That's true in the classical framework. In quantum mechanics, however, Feynman showed that the classical action for a given path gives the quantum phase of its contribution to the amplitude of reaching the endpoint. This explains why a classical object follows a path that minimizes the action. Only trajectories close to that path don't cancel with others.

It is still a non-physical calculation though. The "paths" that you sum up over in the path integral are not to be taken as a physical interpretation of any actual trajectory, especially since you can have trajectories that cannot be physically realized but still contribute to the path integral. In a similar vein, the Lagrangian is not a causal calculation.

 

[fluidistic]

The Lagrangian isn't something physical, by the way. While the Hamiltonian is the total energy, H = T + U

Just curious. I'm taking CM next semester. Isn't the Hamiltonian then constant if no dissipative force is present?

 

[r.a.c.]

The hamiltonian is a constant if no dissipative forces are present. That's how you can build a phase space for idealistic examples, by looking at the hamiltonian.

The Susskind lectures are really good. Is this the same for all stanford (or any other university for that matter) courses?

 

[r.a.c.]

And the lagrangian (as stated by my physics professor) is physical but it has something to do with high energy physics.

 

[fluidistic]

The hamiltonian is a constant if no dissipative forces are present. That's how you can build a phase space for idealistic examples, by looking at the hamiltonian.

Thanks :)