Classical Dynamical Systems
2026-10-30
a phase space \(x \in \mathbb{R}^{n}\), \(n\) the degrees of freedom \[\frac{d x}{dt} := \dot{x} = f(x)\]
\(\Phi: T \times \mathbb{R}^n \rightarrow \mathbb{R}^n\)
\(\Phi(t,x)\) is the trajectory
\(x=(p,q) \in \mathbb{R}^{2d}\)
\[H(x)=T(p)+U(q)=E\]
degrees of freedom (dof) \(2d \to 2d-1\) ::: {.columns}
\[\dot{p}=-\frac{\partial H}{\partial q}\]
\[\dot{q}=\frac{\partial H}{\partial p}\]
:::
\(x=(p,q) \in \mathbb{R}^{2d}\), and \(p,q \in \mathbb{R}^d\)
\[\begin{pmatrix}\dot{p}\\\dot{q}\end{pmatrix}= \begin{pmatrix}0 & -\mathbb{1} \\ \mathbb{1} & 0\end{pmatrix} \begin{pmatrix} \tfrac{\partial H(x)}{\partial p} \\ \tfrac{\partial H(x)}{\partial q}\end{pmatrix}\]
\(x=(p,q) \in \mathbb{R}^{2d}\)
\[\dot{x}= S \frac{\partial H(x)}{\partial x}\]
\[ \begin{aligned} H(x) &= T(p_L)+T(p_R) \\ &\quad + U_{int}(q_L,q_R)+U_{\mathcal{B}}(q_L,q_R) \end{aligned} \]
\[T(p_i)=\frac{p^2}{2m_i} \geq 0\]
\[U_{int}(q_L,q_R) = \frac{C}{|q_L-q_R|}\]
\[ U_{\mathcal{B}}(q_L,q_R)= \begin{cases} 0 & \text{for } (q_L,q_R) \in \mathcal{B} \\ \infty & \text{else} \end{cases} \]
Reflecting boundary: \[x \in \mathcal{S} = \mathbb{R}^{2d} \cap \mathcal{B}\]
A trajectory is \(\Phi(t,x)\) solves \(\dot{x}=f(x)\):
Use surface of section to find periodic orbits
\(\mathcal{S}\) is a \((2d-1)\) dimensional surface, (constrained by Energy \(E\))
define a \((2d-1)\)-surface in phase space \(\Sigma \subset \mathcal{S}\)
$H(x)=E p_L(E,p_R, q_L,q_R) $
\[\begin{align*}H(x)= T(p_L)+T(p_R) + U(q_L,q_R)&=E \\ T(p_R) + U(q_L,q_R) &\leq E \\ T(p_R) + U(\partial \mathcal{B}_{L2}, q_R) \leq E\end{align*}\]
\(\Sigma = \{(p_R, q_R) | \Phi(t,x(v))=(p_L,p_R, \partial \mathcal{B}_{L2}, x_R) \}\)