What are Lagrangian and Hamiltonian mechanics?

[Matt2411]

Only thing I know about them is that they are alternate mechanical systems to bypass the Newtonian concept of a "force". How do they achieve this? Why haven't they replaced Newtonian mechanics, if they somehow "invalidate" it or make it less accurate, by the Occam's razor principle?

Thanks in advance for all the help (and please, if it's not too much to ask, bear in mind I'm not a physics student, so I'd appreciate simplicity).

 

[Twigg]

When solving a problem, we often have the choice to use either Newtonian, Lagrangian, or Hamiltonian mechanics. In principle, they are all complete statements of classical mechanics, but sometimes it just makes more sense to use one over the other, or it's just plain easier to use one of them.

Examples:

1. Find the trajectory of a ball rolling on a parabolic incline given it's initial height and the equation of the parabola.

It would be a pain to solve this with Newtonian mechanics because the normal force's line of action depends on the surface normal, which changes at every point on the incline (because it's a curved surface). But in Lagrangian/Hamiltonian mechanics, it's no big deal, because we only work in scalar quantities, and we can use a single parametric coordinate (say, x) instead of using both the x and y dimensions where they aren't necessary. It's just more efficient.

2. Find the orbital period of a charged particle moving in a circular path of given radius in a uniform magnetic field.

Langrangian mechanics is serious overkill here, as all you need is the centripetal acceleration and the magnetic of the magnetic force to figure it out. The force-based analysis is much faster.

In short, the reason we use three theories instead of one is for greater flexibility.

 

[Matt2411]

When solving a problem, we often have the choice to use either Newtonian, Lagrangian, or Hamiltonian mechanics. In principle, they are all complete statements of classical mechanics, but sometimes it just makes more sense to use one over the other, or it's just plain easier to use one of them.

Examples:

1. Find the trajectory of a ball rolling on a parabolic incline given it's initial height and the equation of the parabola.

It would be a pain to solve this with Newtonian mechanics because the normal force's line of action depends on the surface normal, which changes at every point on the incline (because it's a curved surface). But in Lagrangian/Hamiltonian mechanics, it's no big deal, because we only work in scalar quantities, and we can use a single parametric coordinate (say, x) instead of using both the x and y dimensions where they aren't necessary. It's just more efficient.

2. Find the orbital period of a charged particle moving in a circular path of given radius in a uniform magnetic field.

Langrangian mechanics is serious overkill here, as all you need is the centripetal acceleration and the magnetic of the magnetic force to figure it out. The force-based analysis is much faster.

In short, the reason we use three theories instead of one is for greater flexibility.

So they're only different in their practical use? There's no conceptual difference whatsoever between them?

I'm very ignorant of these issues, but I thought I heard Newton's framework was eventually proved to work only in very special circumstances. Or does that have to do with Einstein's relativity? Again, I apologize if I'm saying something ridiculously wrong. I seriously haven't been able to study this.

 

[Twigg]

So they're only different in their practical use? There's no conceptual difference whatsoever between them?

I'm very ignorant of these issues, but I thought I heard Newton's framework was eventually proved to work only in very special circumstances. Or does that have to do with Einstein's relativity? Again, I apologize if I'm saying something ridiculously wrong. I seriously haven't been able to study this.

No need to apologize. As theories of classical rigid body dynamics, Newtonian, Lagrangian, and Hamiltonian mechanics are equivalent, in that they all predict Newton's Laws. However, Newton's laws break down at high speeds and near extremely massive objects, due to the effects of special and general relativity. There are analogous laws of relativistic physics using each formalism, but their predictions are not the same. Note: classical Lagrangian mechanics and classical Hamiltonian mechanics don't predict relativity. However, there are relativistic Lagrangian mechanics and relativistic Hamiltonian mechanics that strongly resemble the classical versions in the way equations are written out. The mathematical strategy is the same. I'd recommend you make sure you're confident with the classical theories before trying your hand at the relativistic versions.

 

[robphy]

Newtonian, Lagrangian, and Hamiltonian methods provide different methods at studying a dynamical system.
Newtonian methods focus on forces.
Lagrangian and Hamiltonian methods are energy methods with a focus on (respectively) "position"-and-velocity and "position"-and-momentum.

Some systems are better suited to one method over another.

Symmetry and conserved quantities (Noether theorem) and constrained systems are aspects that Lagrangian and Hamiltonian methods are better at than Newtonian methods. Some numerical methods exploit symmetry to manage numerical errors.
Lagrangian and Hamiltonian methods also allow "generalized coordinates" that need not be rectangular or spherical polar, etc... which can make problems easier than what their Newtonian version would suggest.
In fact, Lagrangian and Hamiltonian methods are not restricted to mechanics...
One can exploit dynamical analogies... and solve problems in optics, circuits, electromagnetism, and other classical field theories.
In many cases, Lagrangian and Hamiltonian methods generalize to quantum mechanics more easily than using Newtonian methods.

Of course, these energy methods are more abstract and require more mathematical ability than Newtonian methods.

 

[wrobel]

The Lagrange formalism allows to avoid of calculating of reactions of ideal constraints. If the constraints are ideal and holonomic and the active forces are potential (in generalized sense) then the Newton laws imply the Lagrange equations of the form $\frac{d}{dt}\frac{\partial L}{\partial \dot q}-\frac{\partial L}{\partial q}=0$. These equations are equivalent to the Hamilton equations. The main feature of the Hamilton equations is that their phase flow respects the symplectic structure $dp_i\wedge dq^i$. This fact gives possibility for very deep and subtle analysis of dynamics in Hamilton formalism.

For beginning one should just understand that the Hamiltonian formalism does not work in general for nonholonomic systems and for the systems with dissipative forces like friction etc

 

[Matt2411]

Thank you guys, but there's something I still don't get: if Hamilton's and Lagrange's mechanics dispel the notion of a force, why do we still teach Newtonian forces in fundamental physics courses? Is it because of their simplicity? Because it seems to me as if defining motion in terms of energy is more "complete" than in terms of forces...

 

[robphy]

Thank you guys, but there's something I still don't get: if Hamilton's and Lagrange's mechanics dispel the notion of a force, why do we still teach Newtonian forces in fundamental physics courses? Is it because of their simplicity? Because it seems to me as if defining motion in terms of energy is more "complete" than in terms of forces...

They don't dispel the notion of force.
Lagrangian mechanics uses https://en.wikipedia.org/wiki/Generalized_forces .
As I said earlier, it gives different viewpoints on the dynamics.
Each method has its strengths and its weaknesses.
In simple cases, they may seem equivalent... but they're not in general.
So, you can't really give one up.