7.1. Special Methods for ODE IVPs of Second Order and Conservative Systems#
Created on December 8, 2025.
7.1.1. Introduction#
This is for now just a stub; the section will introduce special “direct” methods for second order equations of the form
\[
u'' = f(t, u, u')
\]
and the important special case of a conservative equation
\[
u'' = f(u)
\]
which has the conserved “energy”
\[
H(\mathbf u) = \frac{u'^2}{2} + P(u)
\]
where \(P' = -f\); \(P\) is the “potential energy” for the equation
7.1.1.1. Systems of second order equtions#
This all extends straightforwardly to systems
\[
\mathbf u'' = \mathbf f(t, \mathbf u, \mathbf u')
\]
with the one extra condition that in the conservative case
\[
\mathbf u'' = \mathbf f(\mathbf u)
\]
the force \(f\) must be conservative: it must be of the form
\[
\mathbf f = -\nabla P
\]
which is true if and only if it satisfies the cross partials condition
\[
\frac{\partial f_i}{\partial u_j} = \frac{\partial f_j}{\partial u_i}
\]
for all pairs \(i \neq j\).
Then the conserved energy is
\[
H(\mathbf u) = \frac{\|u'\|^2}{2} + P(\mathbf u)
\]