# Hamiltons equations for a satellite

• anubis01
In summary, the homework statement asks for the Hamiltonian for a satellite orbiting a star. Using polar coordinates, the student found the kinetic energy and potential energy equations. For cartesian coordinates, the student found the kinetic energy and potential energy equations, but also found that the y component of velocity counted towards the kinetic energy.
anubis01

## Homework Statement

Find Hamiltons canonical equations for a satellite of mass m moving about a star of mass M at the origin. Consider motion in a plane, using polar coordinates r,$$\theta$$.

Repeat the process using cartesian coordinates x,y from the start.

H=T+V

## The Attempt at a Solution

I did both questions but I'm not exactly sure if I defined the kinetic and potential energies correctly for each case. For polar coordinates I found T=m/2 * (r$$\dot{}$$2 +r2 $$\theta$$$$\dot{}$$2) and potential energy V=V(r)

For cartessian coordinates I found the Kinetic T=m/2(x$$\dot{}$$2)
and potential energy V=$$\frac{-GMm}{\sqrt{x^2+y^2}}$$.

So if anyone can confirm if I set up the kinetic and potential energy equations correctly that would be great.

Last edited:
That is the correct kinetic energy in the Lagrangian formulation, what did you have for your kinetic energy and how did you form the Hamiltonian?

Taking care to rearrange the equation I found H=$$\frac{1}{2m}$$ [pr2+$$\frac{p\theta^2}{r^2}$$]. Using this I got the hamilton equations, which there 4 of in this case. I'm sure i did that part correctly i just wasn't 100% sure about setting up the kinetic and potential energy equations correctly.

Oh and when doing the same problem in cartessian coordinates was I correct in defining the Kinetic energy T=m/2(x$$\dot{}$$2)
and potential energy V=$$\frac{-GMm}{\sqrt{x^2+y^2}}$$.

anubis01 said:
Taking care to rearrange the equation I found H=$$\frac{1}{2m}$$ [pr2+$$\frac{p\theta^2}{r^2}$$]. Using this I got the hamilton equations, which there 4 of in this case.

Not quite. H=T+V, and you only included the kinetic energy. The potential energy, V=-GMm/r, must also be included.

Oh and when doing the same problem in cartessian coordinates was I correct in defining the Kinetic energy T=m/2(x$$\dot{}$$2)
and potential energy V=$$\frac{-GMm}{\sqrt{x^2+y^2}}$$.

The potential energy is correct, but don't forget the y component of velocity counts towards the kinetic energy!

Ah, that makes more sense. Thanks for the help, its much appreciated.

## What are Hamilton's equations for a satellite?

Hamilton's equations for a satellite are a set of differential equations that describe the motion of a satellite orbiting around a central body, such as the Earth.

## How are Hamilton's equations derived?

Hamilton's equations are derived from Hamilton's principle, which states that the true path of a system is the one that minimizes the action integral. The action integral is the difference between the kinetic and potential energies of the system.

## What are the variables in Hamilton's equations for a satellite?

The variables in Hamilton's equations for a satellite are the position and velocity of the satellite, as well as the mass and position of the central body.

## What is the significance of Hamilton's equations for a satellite?

Hamilton's equations for a satellite are significant because they provide a mathematical framework for predicting the behavior of a satellite in orbit. They can also be used to design and control satellite trajectories.

## What are some limitations of Hamilton's equations for a satellite?

Some limitations of Hamilton's equations for a satellite include the assumption of a two-body system, neglecting the influence of external forces, and not accounting for relativistic effects. Additionally, they may not accurately describe the behavior of satellites in highly elliptical or chaotic orbits.

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