21-24 September, 2026
To do!
Units of my system
My hamiltoninan \(H(u)=H(x_1,x_2,p_1,p_2)\) is given by the electromagnetic interaction potential \(V(x_1,x_2)\), and the reflecting boundaries potnetial \(V_b\), and of course the kinetic terms \(T(p_i)\)
\[H(u)=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{\mathcal{C}}{|x_1-x_2|}+V_b(x_1)+V_b(x_2)\]
Where as \(V_b(x_i)\) is to be understood as a refelecting boundary \(\partial \mathcal{B}\) that traps particle \(i\), so \(x_i\in\mathcal{B}\). So when \(x_i\) crosses the boundary the conjugate momentum is inverted \(p_i \to -p_i\). Motivated by thermodynamical settings, in which two isolated systems exchange heat, we considere a partition on \(\mathcal{B}=\mathcal{B}(1) \cup \mathcal{B}(2)\), with no overlap \(\mathcal{B}(1)\cap \mathcal{B}(2)=\emptyset\). We define the boundary in coordinates by an open interval \(\mathcal{B}(i)=(b_i-L,b_i+L)\), with boundaries at \(\partial\mathcal{B}=b_i\pm L\). The partition is created by choosing \(b_1 < b_2\) sattisfying \(b_1+L < b_2-L\). A result of the partitioning is a serperation distance \(\delta = b_1+L- (b_2-L)=b_1-b_2+2>0\).
The boundary potential can then be defined as \[V_b(x_i) =\Bigg\{\begin{matrix} 0 & \text{ if } x_i \in \mathcal{B}(i),\\ \infty & \text{ if } x_i \not\in \mathcal{B}(i). \end{matrix} \]
The hamiltonian for inside the boundary \(x_1,x_2 \in \mathcal{B}\), then becomes \[H(u)=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{C}{|x_1-x_2|}\]
I want to investigate the characteristic units of my system. As my boundary defines a characteristic length scale \(L\), to which the particle positions are confind to. I will write the units of the dimesion in brackets, i.e. meters \([\text{m}]\)
\[ \begin{align} x [\text{m}] \to \hat{x}[\text{L}]&=\tfrac{x}{L} [\text{L}] \\ \Leftrightarrow x [\text{m}] &= L\hat{x} \end{align} \]
How did this effect my Hamiltonina \(H\)? The interaction term gets transformed: \[V(x_1-x_2)= \frac{\mathcal{C}}{|x_1-x_2|}= \frac{\mathcal{C}/L}{|\hat{x}_1-\hat{x}_2|}\]
This results in a natural units of energy to be given by the relation of the characteristic interaction strength \(\mathcal{C}\) per characteristic length scale \(L\). So we considere in relation to the \(\mathcal{C}/L\): \[\frac{H(u)}{\mathcal{C}/L} = \frac{p_1^2}{2m_1\,\mathcal{C}/L}+\frac{p_2^2}{2m_2\,\mathcal{C}/L}+\frac{1}{|\hat{x}_1-\hat{x}_2|}=\frac{E}{\mathcal{C}/L}.\]
With the dimensions of energy given by \(\mathcal{C}/L\).
How about the units of momentum and time? Energy is in units \(E_{\text{sys}}[\mathcal{C}/L]\) and in units of length \(L\), mass \(\text{}\) and time \(\text{t}\), \(E_{\text{SI}}[(L/\text{t})^2\text{m}]\) we can equate these \(E_{\text{sys}}=E_{\text{SI}}\) to obtain the units of time:
\[\begin{align} 1[(L/\text{t})^2\text{m}]&=1[\mathcal{C}/L]\\ 1[\text{t}^2] &=1[L^3\text{m}/\mathcal{C}] \end{align}\]
Here a suggestion, without much thought: As a simple model, I will set \(m_1=m_2=1\text{m}\) so that both point mass are the same, and would simplify my units for the system.
surface of section
Yet again, considering the boundary of the surface of section: The surface of section will be a plane which position depends on the sign of the characteristic interaction strenght \(\mathcal{C}\). If \(\mathcal{C}<0\) then the ineraction is repulsive and the boundaries further apart, are sure to make contact, where as the opposite is the case for \(\mathcal{C}>0\). Thus, without loss of generality, we choose particle \(1\) as the conditional particle. Depending on the sign of the characteristic interaction strenght \(\mathcal{C}\) either \(\partial \mathcal{B}(1,1)\) (\(\mathcal{C}<0\), repulsive) or \(\partial \mathcal{B}(1,2)\) (\(\mathcal{C}>0\), attractive) will mark the point that span the surface of section \(\Sigma=\text{span}(x_2,p_2)\). We shall refere to the boundary of particle \(1\) as \(\partial \mathcal{B}(1,\_)=\partial \mathcal{B}=\{-(1+\delta),-\delta\}\)
Finding the boundary Remember that the kinetic energy term is positive \(T(p_i)\geq 0\) for all \(p_i\). \[\begin{align} H(u)=T(p_1)+T(p_2)+V(x_1,x_2)&=E\\ \Rightarrow T(p_2)+ V(\partial \mathcal{B}(j),x_2)&\leq E \end{align}\]
We are seeking all points \(\{(x_2,p_2)|T(p_2)+ V(\partial \mathcal{B}(j),x_2)-E=0\}\).
The code, scans through the range of \(x_2\in\mathcal{B}(2)\) and determine all \(p_2\) that sattisfy this condition.
function boundary(E; p=(; C=-1.0, m1=1.0, m2=1.0, L=1.0, del=DEL_BOX), n=400)
dB = get_boxes(p)
x1 = p.C > 0 ? dB[1][1] : dB[1][2] # section: particle 1 fixed here
x2 = range(dB[2][1], dB[2][2]; length=n) # sweep all of box 2
p2 = map(x2) do x
K = E - V_int([x1, x, NaN, NaN], p) # kinetic energy available (p1 = 0)
K >= 0 ? sqrt(2p.m2 * K) : NaN # NaN = forbidden, breaks the line
end
upper = Point2f.(x2, p2)
lower = Point2f.(reverse(x2), -reverse(p2))
return vcat(upper, lower, upper[1:1]) # closed curve
endObservations of the two charged particles serperated by boxes
The figure shows the boundary of the Poincaré section for the repulsive two-particle system, with particle 1 fixed at the selected boundary point and particle 2 sampled across its allowed box. The closed curve is obtained by solving the energy constraint \[ T(p_2) + V(\partial \mathcal{B}, x_2) = E \] for each position \(x_2\) in the second box. The accessible states lie inside the enclosed region, while the curve itself marks the phase-space boundary where the kinetic energy available to particle 2 vanishes. This construction makes explicit how the fixed boundary condition on particle 1 defines a reduced two-dimensional section of the full four-dimensional dynamics.
When varying the parameter \(\mathcal{C}\) from negative we observe no converging behaviour, as to be expected.
Looking at phase space
The figure shows the trajectories of both particles, their intersections with the Poincaré section, and the corresponding energy error. The solid and dashed curves represent (x_1) and (x_2), respectively. The section plot shows the accessible ((x_2,p_2)) states for the chosen initial conditions, while the energy plot confirms the numerical conservation of (E).
On the nature of my interaction
I can determine the boundary of my surface of section as a function \(\hat{x}_2(p_2)\).
One obtains a Laplacian expression
\[\hat{x}_2 = \frac{2m_2\mathcal{C}/L}{2Em_2-p_2^2}-\partial \mathcal{B}(\mathcal{C})\]
Here it is interesting to note, lets take \(\text{sign}(\mathcal{C})=-1\), this means our Hamiltonian can have negative eigenvalues \(H(u)=E\). What this means, is that if the energy is negative, \(E<0\) the two charged particles are bound. If the system is on the energy-shell whith positive eigenvalue, \(E>0\), then the electrons are not bound by the interaction potential, though through the boundary conditions \(x_1,x_2\in\mathcal{B}\). So it would be interesting if one observes some kind of transition, of a bound state to a free state.
A sigularity at resonance \(0>E\overset{!}{=} T(p_2)\) (implying \(Em_2-p_2^2=0\)) is avoided, as \(T(p_2)\geq 0\). For \(0<E\overset{!}{=} T(p_2)\), the sigularity is also avoided, as \(V(x_2)\not=0\).