# Finding the hamiltonian of a projectile

#### pentazoid

1. Homework Statement

Using cartesian coordinates, find the Hamiltonian for a projectile of mass m moving under uniform gravity. Obtain Hamiltonian's equation and identify any cyclic coordinates.

2. Homework Equations

3. The Attempt at a Solution

I think I will just have trouble determining my coordinates for the position; my coordinates are x and y. I think I can finished the rest of the problem once I know my coordinates for the position

would x=x = v t*cos (theta) , t=x/(v*cos(theta)) and
y=v*t*sin(theta)-.5*g*t^2?

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#### jambaugh

Gold Member
Cyclic coordinates does not mean polar coordinates... why are you working with sines and cosines of theta?

In determining your Hamiltonian you should treat coordinates (and momenta) as independent variables. You then solve Hamilton's equations for the coordinates and momenta as functions of time.

Cyclic coordinates are coordinates which do not appear in the Hamiltonian. For example the Hamiltonian of a free particle moving in the absence of any forces is:
$$H = \frac{1}{2m}\left(p_x^2 + p_y^2 + p_z^2\right)$$
that is to say the Hamiltonian is just the kinetic energy. Since there are no (coordinate dependent) forces none of the coordinates, (x,y,z) appear in the Hamiltonian and so all of them are Cyclic.

#### pentazoid

Cyclic coordinates does not mean polar coordinates... why are you working with sines and cosines of theta?

In determining your Hamiltonian you should treat coordinates (and momenta) as independent variables. You then solve Hamilton's equations for the coordinates and momenta as functions of time.

Cyclic coordinates are coordinates which do not appear in the Hamiltonian. For example the Hamiltonian of a free particle moving in the absence of any forces is:
$$H = \frac{1}{2m}\left(p_x^2 + p_y^2 + p_z^2\right)$$
that is to say the Hamiltonian is just the kinetic energy. Since there are no (coordinate dependent) forces none of the coordinates, (x,y,z) appear in the Hamiltonian and so all of them are Cyclic.
H=T+V, why did you leave out V? I am using cosine and sines of theta because I need to determined x and y?

#### turin

Homework Helper
I am using cosine and sines of theta because I need to determined x and y?
No. x and y are undetermined, and you want to keep them that way. You seem to be approaching this problem as a first year student, but you need to approach this problem in the more advanced context of the action principle. You are not supposed to determine a trajectory.

The first thing to do is to right down the Lagrangian and determine if there are any ignorable coordinates. Since the problem asks for the Hamiltonian in Cartesian coordinates, then the problem is a straightforward application of the definitions of potential energy, kinetic energy, canonical momentum, and symmetry.

#### jambaugh

Gold Member
H=T+V, why did you leave out V? I am using cosine and sines of theta because I need to determined x and y?
I am not going to give you your Hamiltonian. I gave you the example of a Hamiltonian for a different system, namely a particle without any force due to gravity or otherwise and so V=0. I did this to show you what is meant by cyclic coordinates.

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