What Are Hamilton's Equations of Motion for a Quadratic Potential?

[physics-?]

Having a bit of trouble with this question, if anyone could help?

For the following questions we assume the Hamiltonian to
be of the generic form
H(r, p) = T(p) + V (r) = p²/2m+ V(r)
where T(p) and V (r) denote kinetic and potential energies, respectively.
Find Hamilton's equations of motion for a general quadratic potential V (r) = [(r,Ar)]/2
A = A^T , a symmetric 3-by-3 matrix.
Which force do you obtain?
Is this force conservative?

Thanks!

 

[tiny-tim]

welcome to pf!

hi physics-?! welcome to pf!

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[physics-?]

Well if I'm honest I don't really know where to begin!
I am having trouble understanding it.
Sorry that's not much use to you!

 

[tiny-tim]

if A was diagonal, (r,Ar) would be ax² + by² + cz²

the off-diagonal elements add some xy and yz and zx to that

 

[paranormal]

Hamilton's equations of motion are a set of equations that describe the evolution of a physical system over time. They are derived from Hamilton's principle, which states that the action of a system is minimized along the path that the system takes.

In this case, we are given a Hamiltonian of the form H(r,p) = T(p) + V(r), where T(p) is the kinetic energy and V(r) is the potential energy. We are also given a specific quadratic potential V(r) = [(r,Ar)]/2, where A is a symmetric 3-by-3 matrix.

To find Hamilton's equations of motion, we first need to calculate the conjugate momenta, defined as p = ∂L/∂r, where L is the Lagrangian of the system. In this case, the Lagrangian is L = T(p) - V(r). Therefore, the conjugate momenta are given by p = ∂T/∂p = p/m and p = -∂V/∂r = Ar.

Next, we can use the Hamiltonian equations of motion, which state:

dr/dt = ∂H/∂p = p/m
dp/dt = -∂H/∂r = -∂T/∂r + ∂V/∂r = -∂V/∂r = -Ar

These equations describe the motion of the system over time, with the first equation giving the velocity and the second equation giving the acceleration. We can see that the force obtained from these equations is -Ar, which is a conservative force since it can be derived from a potential energy function.

In conclusion, Hamilton's equations of motion for this system are given by dr/dt = p/m and dp/dt = -Ar, with a conservative force of -Ar. These equations can be used to study the behavior of the system and predict its motion over time.