Lagmental Vicfred

Symplectic Euler Preserves Phase-Space Geometry Better than Energy by Vicfred

Last updated: Mon 18 May 2026

tags: mathematics, dynamical systems, chaos, Lyapunov exponents

A first-order semi-implicit step is symplectic even though its energy usually oscillates. I will separate the object being defined from the consequence being claimed.

Definitions first

A discrete dynamical system iterates \(x_{n+1}=F(x_n)\), while a flow solves \(\dot x=V(x)\). Fixed points, periodic orbits, and invariant sets organize long-term behavior.

$$ H(q,p)=T(p)+V(q) $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ p_{n+1}=p_n-h\nabla V(q_n),\qquad q_{n+1}=q_n+h\nabla T(p_{n+1}) $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ D\Phi_h^{\mathsf T}\begin{pmatrix}0&I\\-I&0\end{pmatrix}D\Phi_h=\begin{pmatrix}0&I\\-I&0\end{pmatrix} $$

The reusable statement

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad H(q,p)=T(p)+V(q),\\[5pt] \mathsf{C}\;&:\quad p_{n+1}=p_n-h\nabla V(q_n),\qquad q_{n+1}=q_n+h\nabla T(p_{n+1}). \end{aligned} $$

A nearby false statement

Sensitive dependence is not the same as randomness. A deterministic system may be chaotic while remaining exactly specified by its initial condition.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] p_{n+1}=p_n-h\nabla V(q_n),\qquad q_{n+1}=q_n+h\nabla T(p_{n+1}) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sat 28 December 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.