I am using the Runge Kutta implicit midpoint method $$m_{n+1}=m_n + \frac{h}{2}(\frac{m_n + m_{n+1}}{2} \times (T^{-1}\frac{m_n + m_{n+1}}{2}).$$ To solve a free rigid body problem, where $T$ is the diagonal inertia tensor, $m(t)$ is the angular momentum and $h$ is the step size.
This method supposedly preserves the energy and momentum, i.e. it holds that $$m_{n+1}^T m_{n+1}=m_n^Tm_n$$ and $$\frac{1}{2}m_{n+1}^T(T^{-1}m_{n+1})=\frac{1}{2}m_n^T(T^{-1}m_n).$$ How can one go about proving these properties? I am thinking i should take an induction approach, but I am struggling to get started.
Just use that $$ a^Ta-b^Tb=(a-b)^T(a+b). $$ And of course that $u^T(u\times v)=\det([u,u,v])=0$.