Hamiltonian Function - Definition & Explanation

[jgrossm1]

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

 

[pradeep reddy]

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

 

[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:
$$\(\frac{\partial x}{\partial t}=\frac{\partial H}{\partial p_{x}}\)$$
$$\(\frac{\partial p_{x}}{\partial t}=-\frac{\partial H}{\partial x}\)$$

 

[Diminique]

I see that a quasi-particlee 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.

 

[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.

 

[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.

 

[TobyC]

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.

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

 

[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 f_xx_t+f_yy_t=0, in the form x_t=f_y and y_t=-f_x or x_t=-f_y and y_t=f_x. So that, there is no physics at all, just simple mathematical analysis. But the physics is reach in phantasies.

 

[A. Neumaier]

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

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

 

[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?

 

[Dickfore]

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

 

[Diminique]

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

 

[Dickfore]

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

Always. See path integral formulation of Quantum Mechanics.

 

[Diminique]

I'll try

 

[Diminique]

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

 

[A. Neumaier]

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?

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

 

[Diminique]

True