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I am asked to compute Hamilton's equations and check that the solutions to said equations are circular paths, centered at the origin, with angular velocity $\xi \in R^3$. The Hamiltonian is given by $H(q,p) = (q \times p)\cdot \xi$ in this example. So, after letting $ q=(q_1,q_2,q_3)$ and $p=(p_1,p_2,p_3)$, we end up with $H(q,p) = \xi_1(q_2p_3-p_2q_3)+\xi_2(p_1q_3-q_1p_3)+\xi_3(q_1p_2-p_1q_2)$. So Hamilton's equations end up being

$q_1'(t)= (\xi_2q_3-\xi_3q_2)(t)$

$q_2'(t)= (\xi_3q_1-\xi_1q_3)(t)$

$q_3'(t)= (\xi_1q_2-\xi_2q_1)(t)$

$p_1'(t)= (\xi_2p_3-\xi_3p_2)(t)$

$p_2'(t)= (\xi_3p_1-\xi_1p_3)(t)$

$p_3'(t)= (\xi_1p_2-\xi_2p_1)(t)$

Now, I am tasked in showing that the solutions to the above equations are circular paths about the origin. Not surprisingly, you get two linear 1st order ODEs:

$p'(t) = Ap(t)$

$q'(t) =Aq(t)$

where $A = \begin{pmatrix} 0 &-\xi_3 &\xi_2 \\\xi_3 &0 &-\xi_1 \\ -\xi_2&\xi_1&0 \end{pmatrix}$

My next step would be to calculate the solution using the matrix exponential, but that seems long and cumbersome. Is there an easier way to show that the solutions are circular paths?

Qmechanic
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Vegeta the Prince of Saiyans
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    You'll likely find the following useful: https://physics.stackexchange.com/a/74014/19976 and also this: https://physics.stackexchange.com/a/85989/19976 – joshphysics Jan 29 '19 at 04:58

2 Answers2

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Hints:

  1. Show that if $\vec p$ satisfies the differential equation you wrote, then \begin{align} \frac{d}{dt}|\vec p| = 0. \end{align}
  2. Show that if $\vec p$ satisfies the differential equation you wrote, then \begin{align} \frac{d}{dt}\left|\frac{d\vec p}{dt}\right| = 0. \end{align}
  3. Think about what these two equations mean.
joshphysics
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$$\vec{\dot{p}}=A\,\vec{p}\tag 1$$

$$\vec{\dot{q}}=A\,\vec{q}\tag 2$$

The solutions of equation (1) and (2) is:

$\vec{p}=e^{A\,t}\,\vec{p}_0\quad,\vec{q}=e^{A\,t}\,\vec{q}_0$ ,where $\vec{p}_0\,,\vec{q}_0$ the initial condition

the requirement for a circular motion :

$$\vec{p}^2+\vec{q}^2=\text{const}\tag 3$$

so is equation(3) fulfill ?

Edit

$\vec{p}^2=\vec{p}^T\,\vec{p}=\vec{p}_0^T\,e^{A^T\,t}\,e^{A\,t}\,\vec{p}_0= \vec{p}_0^T\,e^{\left(A^T+A\right)t}\,\vec{p}_0=\vec{p}_0^T\,\vec{p}_0=\text{const}$

with:

$A^T+A=\mathbb{0}\quad $ ,$A\quad$ is skew matrix!

Eli
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