1 Hamiltonian Mechanics
A configuration space \(M\) is the space defined by the coordinates that define the configuration of a physical system (e.g. Cartesian space \(\mathbb{R}^3\), spherical coordinates \(S^2 \subseteq \mathbb{R}^3\) etc).
Let \((M,\mathcal{L})\) be a mechanical system consisting of a configuration space and a Lagrangian which is a function defining the dynamics of the system. We select a standard coordinate system on \(M\) denoted \((\boldsymbol{q}, \dot{\boldsymbol{q}}):=((q_{1}, \dots, q_{n}), (\dot{q}_{1}, \dots, \dot{q}_{n}))\) from which we define the momenta \(\boldsymbol{p}\) as \[ p_{i}(\boldsymbol{ q}, \dot{\boldsymbol{q}},t) := \frac{{\partial \mathcal{L}}}{\partial \dot{\boldsymbol{q}}^i}. \] For a time instant \(t\) the [[legendre-transform]] of \(\mathcal{L}\)