Hamiltonian Dynamics in Bijected Vector Space

Starting with a regular Lagrangian of the form

L (q, \dot{q}) = \frac{1}{2} \sum_{i} m_{i} {\dot{q}}^{2} + U (q)

bijecting it into another vector space so $Q = f (q)$ and $q = f^{- 1} (q)$

Using the chain rule we can see that

{\dot{q}}_{i} = \frac{d q_{i}}{d t} = \sum_{j} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} \frac{d Q_{j}}{d t} = \sum_{j} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} {\dot{Q}}_{j}

Then substituting back into the Lagrangian

L (Q, \dot{Q}) = \frac{1}{2} \sum_{i} m_{i} (\sum_{j} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} {\dot{Q}}_{j})^{2} + U (f^{- 1} (Q))

= \frac{1}{2} \sum_{i} m_{i} \sum_{j} \sum_{k} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{k}} {\dot{Q}}_{j} {\dot{Q}}_{k} + U (f^{- 1} (Q))

= \frac{1}{2} \sum_{j} \sum_{k} (\sum_{i} m_{i} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{k}}) {\dot{Q}}_{j} {\dot{Q}}_{k} + U (f^{- 1} (Q))

Then define a matrix

G_{j k} (Q) = \sum_{i} m_{i} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{j}} \frac{\partial f_{i}^{- 1} (Q)}{\partial Q_{k}}

Making

L (Q, \dot{Q}) = {\dot{Q}}^{⊤} 𝐆 (Q) \dot{Q} + U (f^{- 1} (Q))

Then do the Legendre transform to make it a Hamiltonian

P_{i} = \frac{\partial L}{\partial {\dot{Q}}_{i}} = \frac{1}{2} \sum_{j} 𝐆_{i j} {\dot{Q}}_{j} + \frac{1}{2} \sum_{j} 𝐆_{j i} {\dot{Q}}_{j}

And $𝐆$ is symmetric so

= \sum_{j} 𝐆_{i j} {\dot{Q}}_{j}

P = 𝐆 (Q) \dot{Q}

The Legendre transform requires that it's invertable so if the matrix is invertable then

\dot{Q} = 𝐆^{- 1} (Q) P

Then with the Legrendre transform

H (Q, P) = P^{⊤} \dot{Q} - L (Q, \dot{Q})

= P^{⊤} 𝐆^{- 1} (Q) P - L (Q, 𝐆^{- 1} (Q) P)

= P^{⊤} 𝐆^{- 1} (Q) P - \frac{1}{2} ((𝐆^{- 1} (Q) P)^{⊤} 𝐆 (Q) (𝐆^{- 1} (Q) P)) + U (Q)

= P^{⊤} 𝐆^{- 1} (Q) P - \frac{1}{2} (P^{⊤} 𝐆^{- 1} (Q) 𝐆 (Q) 𝐆^{- 1} (Q) P) + U (Q)

= P^{⊤} 𝐆^{- 1} (Q) P - \frac{1}{2} (P^{⊤} 𝐆^{- 1} (Q) P) + U (Q)

= \frac{1}{2} P^{⊤} 𝐆^{- 1} (Q) P + U (Q)