Hamiltonian Function

1. Sep 28, 2007

jgrossm1

Hi, I'm just wondering if someone could explain to me exactly what the hamiltonian function is

2. Oct 5, 2007

hamiltonian function is H(p,q,t)
where p is momentum
q is position and t is time
pdot=-dH/dq
qdot=dH/dp
where pdot=dp/dt
this gives explantion to solve eqns or problems more easily than lagrangian and newtonian ways.study canonical transformations and gendre transformations
u can understand well
bye

3. Oct 5, 2007

easttiger2007

in the simple case of a particle of mass m in Cartesian coordinates, the Hamiltonian function of motion is
$$$$\emph{H}(\textbf{r,p})=\frac{1}{2m}(p^{2}_{x}+p^{2}_{y}+p^{2}_{z})+\emph{V}(x,y,z)$$$$

The relations mentioned previously can be derived:
$$$$\frac{\partial x}{\partial t}=\frac{\partial H}{\partial p_{x}}$$$$
$$$$\frac{\partial p_{x}}{\partial t}=-\frac{\partial H}{\partial x}$$$$

Last edited: Oct 5, 2007
4. May 1, 2011

Diminique

I see that a quasi-particale be more interesting to interpret. If one suppose, sir Hamilton knew nothing about these ones, but equations appears and live rights all the time as Got Trues.

5. May 1, 2011

TobyC

The hamiltonian is a quantity which is conserved by a system if that system is invariant under a translation in time.

If the system is composed of point particles described by cartesian coordinates (which is how you can view all systems in classical mechanics fundamentally) then the hamiltonian is just the total energy of the system. If however different coordinates are used then the hamiltonian might not correspond to the total energy of the system.

6. May 1, 2011

Diminique

Not always. An example: cartesian frame, H(p,q)=p*p-q*q=const < M - arbitrary, but the energy E(p,q)=p*p+q*q tends to infinity.

7. May 1, 2011

TobyC

Oh ok I don't really understand that but you could well be right, I don't know too much about Hamiltonian mechanics.

8. May 2, 2011

Diminique

It's easy. Get smooth f(x,y), then try a parameterization along t. Require f=const. If that, then one obtains the Hamiltonian set from the condition fxxt+fyyt=0, in the form xt=fy and yt=-fx or xt=-fy and yt=fx. So that, there is no physics at all, just simple mathematical analysis. But the physics is reach in phantasies.

9. May 2, 2011

A. Neumaier

In Hamiltonian mechanics, it is the function that tells you how the energy depends on position, momentum, and time.

10. May 2, 2011

Diminique

Not always. General case is dynamical system transforming a state to the following state through the Hamiltionian operator. Suppose the state is a wave function of a qusiparticle. What do You say, where are the coordinade and momentum, though we've got an ordinary pattern of the Hamiltonian mechanics?

11. May 2, 2011

Dickfore

The Hamiltonian function is a Legendre transfrom with respect to the generalized velocities of the Lagrange function.

12. May 2, 2011

Diminique

Not always, please try to find any Lagrangiane in quantum mechanics. If absent, how to be with the Legendre transfrom?

13. May 2, 2011

Dickfore

Always. See path integral formulation of Quantum Mechanics.

14. May 2, 2011

Diminique

I'll try

15. May 2, 2011

Diminique

Thanks, it's the first quantum Lagrangiane to my own collection, just analyzing why and where.

16. May 2, 2011

A. Neumaier

In quantum mechanics one doesn't have a Hamiltonian function but a hamiltonian operator.

17. May 2, 2011

True