Do you already have your Hamilton function and your generalised coordinates?
No no,,I Don't have.......
What I understand is,,,,,,,1st I will have to find the Lagrangian by finding the Potential and kinetic energy of the particle of mass m, and then I will use the lagrangian to find the hamilton function and hamilton equations.........I have solved problems before this one but lagrangian was already given......Here I will have to little talk about the motion of a particle in plane under the conservative force,,,,From where I will get the lagrangian......But how to apply all this.........How I will get the Lagrangian??????????
And if I am wrong......Help me,,,,,,,
first of all: why so complicated, just use H = T + V (sum from kinetic and potential energy which is the Hamiltonian of your system). What you have to know is the form of your potential, of course.
Then you just have to use the Hamiltionian relations between H and your generalized coordinates.
The best way is to use the Cartesian coordinates, since they give the Hamiltonian directly. The momentum is the usual formula.
H = (p_x^2 + p_y^2) / 2m + V(x, y)
p_x = m x'
p_y = m y'
The equations of motion are given by the canonical equations:
q' = @H/@p, p' = -@H/@q.
The general method starting from the Lagrangian is to first write the Lagrangian. It is a function of q and q', where q is any system of coordinates, and is equal to the kinetic energy minus the potential one. From there one gets the generalized momenta, given by: p = @L/@q'. The Hamiltonian is given by the Legendre transform of the Lagrangian, or more simply, it is the total energy of the system expressed as a function of q and p.
@Claude Massé: actually you usually use generalised coordinates (mostly in dependence of the angle and the distance, because the metric of your coordinate system should be independent from your choice of coordinates, as long as you have an orthogonal base).
so, it is most convinient to express your cartesian coordinates through the generalised coordinates and write your Hamiltonian directly (but of course you may also start with the Lagrangian, but why bother with that?).
I am assuming that you are familiar with Lagrangian
function.
Your problem is two dimensional because the particle
is moving on a plane. But you have not said whether the
potential has a symmetry. In some cases it is possible to
write the potential more easily in a particular coordinate
system. For example if the potential has spherical symmetry
it is more convenient to write potential in (r,theta) coordinates.
In this way one chooses a coordinate system to express
the Hamiltonian in a convenient form. These are called
generalized coordinates as opposed to Cartesian coordinates
(x,y). Once the generalized coordinates are chosen, there is
a prescription for writing down the Hamiltonian function
of the system. Firstly write down the Lagrangian L in terms
of generalized coordinates and velocities q_i and q'_i. Here
the (total) time derivative is expressed by the symbol '
Calculate conjugate momenta for each generalized
velocity by using,
p_i= del L/del q'_i
Now Hamiltonian function is,
H= ( sum_i q_i' p_i ) - L
Now use the equations given above by Claude. I am writing
them again,
del H / del q_i = - p'_i
del H / del p_i = q'_i
There is an additional equation involving partial
derivatives w.r.t time,
del H / del t = - del L / del t
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