Talk:Hamilton–Jacobi equation

Attention needed in the "Hamilton's principal function" def section

This part doesn't make sense. Particularly, that we're going nxn Hessian matrix to inverted matrix which I have never heard of to a vector equality. This seriously needs to be elaboration.

Let the Hessian matrix ${\textstyle H_{\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)=\left\{\partial ^{2}{\cal {L}}/\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}\right\}_{ij}}$ be invertible. The relation

${\frac {d}{dt}}{\frac {\partial {\cal {L}}}{\partial {\dot {q}}^{i}}}=\sum _{j=1}^{n}\left({\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}}}{\ddot {q}}^{j}+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {q}^{j}}}{\dot {q}}^{j}\right)+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial t}},\qquad i=1,\ldots ,n,$

shows that the Euler–Lagrange equations form a $n\times n$ system of second-order ordinary differential equations. Inverting the matrix $H_{\cal {L}}$ transforms this system into

${\ddot {q}}^{i}=F_{i}(\mathbf {q} ,\mathbf {\dot {q}} ,t),\ i=1,\ldots ,n.$

Ehaarer (talk) 03:21, 27 January 2023 (UTC)

Another particular issue here is that the notation ${\textstyle {\cal {L}}}$ is pulled out of nowhere. A line could easily be added to the "Notation" section about this. Mosher (talk) 17:24, 31 March 2023 (UTC)