We found Hamilton's equations by starting with the Lagrangian (Lleft(q_{i}, dot{q}_{i}, t ight)) and using a Legendre

We found Hamilton's equations by starting with the Lagrangian \(L\left(q_{i}, \dot{q}_{i}, t\right)\) and using a Legendre transformation to define the Hamiltonian \(H\left(q_{i}, p_{i}, t\right)\). Now starting with the Hamiltonian and Hamilton's equations, use a reverse Legendre transformation to define \(L\), and show that one obtains the Lagrange equations.