Finding the hamiltonian of a projectile

In summary, the projectile has a Hamiltonian of:H=\frac{1}{2m}\left(p_x^2 + p_y^2 + p_z^2\right)with cyclic coordinates of:x=x = v t*cos (theta) , t=x/(v*cos(theta)) and y=v*t*sin(theta)-.5*g*t^2?
  • #1
pentazoid
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0

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.

Homework Equations





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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  • #2
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:
[tex] H = \frac{1}{2m}\left(p_x^2 + p_y^2 + p_z^2\right)[/tex]
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.
 
  • #3
jambaugh said:
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:
[tex] H = \frac{1}{2m}\left(p_x^2 + p_y^2 + p_z^2\right)[/tex]
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?
 
  • #4
pentazoid said:
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.
 
  • #5
pentazoid said:
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.

Reread my post carefully!
 

What is the Hamiltonian of a projectile?

The Hamiltonian of a projectile is a mathematical function that describes the total energy of a projectile in a given system. It takes into account both the kinetic energy and potential energy of the projectile.

How do you calculate the Hamiltonian of a projectile?

The Hamiltonian of a projectile can be calculated using the equation H = T + V, where H is the Hamiltonian, T is the kinetic energy, and V is the potential energy.

What is the significance of the Hamiltonian in projectile motion?

The Hamiltonian is important in projectile motion because it helps us understand how the energy of the projectile changes over time and how it is affected by external forces. It also allows us to make predictions about the future motion of the projectile.

How does the Hamiltonian change for different types of projectiles?

The Hamiltonian may change for different types of projectiles depending on the system in which they are moving. For example, the Hamiltonian for a projectile in a vacuum will be different from that of a projectile moving through a medium with air resistance.

Can the Hamiltonian of a projectile be used to predict its trajectory?

Yes, the Hamiltonian can be used to predict the trajectory of a projectile by solving the equations of motion derived from it. By calculating the Hamiltonian at different points in time, we can determine the position and velocity of the projectile at any given time.

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