PX285 - G1 - inertia tensor

note: from typed notes
the kinetic energy in the most general quadratic form:

T = \frac{1}{2} \sum_{i j} M_{i j} {\dot{q}}_{i} {\dot{q}}_{j}

where, $M_{i j}$ are a number of arbitrary coefficients, a set of which can be thought of as a square matrix

as ${\dot{q}}_{i} {\dot{q}}_{j} = {\dot{q}}_{j} {\dot{q}}_{i}$ , there is not distinction between $M_{i j}$ and $M_{j i}$ , and they can be thought of as a symmetric matrices with commuting indices:

M_{i j} = M_{j i}

the canonical momenta:

\begin{aligned} p_{k} & = \frac{\partial L}{\partial {\dot{q}}_{k}} \\ = \frac{\partial}{\partial {\dot{q}}_{k}} \frac{1}{2} \sum_{i j} M_{i j} {\dot{q}}_{i} {\dot{q}}_{j} \\ = \frac{1}{2} \sum_{i j} (M_{i j} {\dot{q}}_{i} \frac{\partial {\dot{q}}_{j}}{\partial {\dot{q}}_{k}} + M_{i j} {\dot{q}}_{j} \frac{\partial {\dot{q}}_{i}}{\partial {\dot{q}}_{k}}) \\ = \frac{1}{2} \sum_{i j} M_{i j} ({\dot{q}}_{i} δ_{j k} + {\dot{q}}_{j} δ_{i k}) \\ ∴ p_{k} & = \sum_{j} M_{j k} {\dot{q}}_{j} \end{aligned}

$M_{i j}$ is the inertia tensor (or matrix), which relates the canonical momenta to the generalised velocities
this can be used to calculate the hamiltonian

\sum_{i} p_{i} {\dot{q}}_{i} = \sum_{i} \sum_{j} M_{i j} {\dot{q}}_{j} {\dot{q}}_{i} = 2 T

the hamiltonian:

H = \sum_{i} p_{i} {\dot{q}}_{i} - L = 2 T - (T - V) = T + V

which is as required