# Is Newtonian Mechanics more general than Hamiltonian Mechanics?

1. Sep 4, 2010

### nonequilibrium

Apparently things like the Lorentz' force can't be handled as a hamiltonian system. I heard other people describe the hamiltonian mechanics as an equivalent characterization of classical mechanics, but this is wrong, then?

2. Sep 5, 2010

### Dr Lots-o'watts

3. Sep 5, 2010

Staff Emeritus
Where did you hear that?

If I didn't drop a minus sign,

$$L = \frac{1}{2}mv^2 - q\phi + \frac{1}{c}q({\mathbf v} \cdot {\mathbf A})$$

4. Sep 5, 2010

### nonequilibrium

Oh, my apologies, I interpreted "In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant." (wikipedia) as meaning that the force can't be dependent on the speed of a particle... What does the quoted sentence say?

5. Sep 5, 2010

Staff Emeritus
Last edited by a moderator: Apr 25, 2017
6. Sep 5, 2010

### Phyisab****

That statement is strange, but let me guess what they are trying to say. The Hamiltonian is not equal to the total energy of the system when the force depends on the spatial derivative. The system can however still be described with a Hamiltonian, you just have to use the real definition and not H = T + V.

7. Sep 5, 2010

### nonequilibrium

Much appreciated. The weird thing is: I've read a (serious) article where they were working with what looked like classical systems (no relativity, quantum mechanics) but stated that they weren't hamiltonian systems, because the Theorem of Liouville (a certain theorem proven for hamiltonian systems) didn't apply to their systems, thus proving it wasn't a hamiltonian system by reductio ad absurdum. Is this plausible? Are there such classical systems? Or do you need to go to quantum mechanics for it to "stop working"?