From 9efdce646af6acfda8265404ebad31c89dd0672b Mon Sep 17 00:00:00 2001 From: Krishna Ayyalasomayajula Date: Sat, 2 May 2026 12:52:02 -0500 Subject: [PATCH] =?UTF-8?q?fix(HJ):=20A.10=20=E2=80=94=20Add=20cross-refs?= =?UTF-8?q?=20to=20A.06=20and=20U9?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- concepts/advanced/uniform-e-field-hj.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/concepts/advanced/uniform-e-field-hj.tex b/concepts/advanced/uniform-e-field-hj.tex index 1759188..4a17423 100644 --- a/concepts/advanced/uniform-e-field-hj.tex +++ b/concepts/advanced/uniform-e-field-hj.tex @@ -1,6 +1,6 @@ \subsection{Charged Particle in Uniform Electric Field} -This subsection solves the Hamilton--Jacobi equation for a charged particle in a uniform electric field, showing that Jacobi's theorem reproduces the parabolic motion dictated by the constant electric force $\vec{F} = q\vec{E}$. +This subsection solves the Hamilton--Jacobi equation for a charged particle in a uniform electric field, showing that Jacobi's theorem reproduces the parabolic motion dictated by the constant electric force $\vec{F} = q\vec{E}$. The problem is formally identical to the projectile motion treatment in~A.06: the separation ansatz, the characteristic function, and the Jacobi inversion follow exactly the same algebra, with the gravitational acceleration $g$ replaced by $-qE_0/m$. Likewise, the uniform field between parallel plates studied in Unit~9 (e9-3) produces a constant electric force that accelerates the particle uniformly; the HJ solution presented here applies directly to that configuration as well. \dfn{Hamiltonian for a charged particle in a uniform electric field}{ A particle of mass $m$ and charge $q$ in a uniform electric field $\vec{E} = E_0\,\hat{\bm{z}}$ (with $\vec{B} = 0$) is described by the scalar potential $\varphi = -E_0 z$ and zero vector potential. The Hamiltonian is