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

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Homework Help Overview

The discussion centers around Hamilton's equations of motion in the context of a quadratic potential. The original poster is seeking assistance with deriving these equations given a specific Hamiltonian and potential form.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to start the problem and seeks guidance on the derivation process. Some participants suggest examining the structure of the potential when the matrix A is diagonal, indicating a focus on understanding the implications of the quadratic form.

Discussion Status

The conversation is ongoing, with participants exploring different aspects of the problem. There is an invitation for the original poster to share their attempts, which may help guide further discussion.

Contextual Notes

The original poster has not provided specific attempts or insights into their understanding, which may limit the direction of the discussion. The nature of the potential and its implications for force and conservativeness are also under consideration.

physics-?
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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 = AT , a symmetric 3-by-3 matrix.
Which force do you obtain?
Is this force conservative?

Thanks!
 
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welcome to pf!

hi physics-?! welcome to pf! :wink:

show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 
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!
 
if A was diagonal, (r,Ar) would be ax2 + by2 + cz2

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

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