LegendreTransform

From Lagrangian to Hamiltonian Formulation {#from-lagrangian-to-hamiltonian-formulation .unnumbered}

1. Start with the Lagrangian: $$L = L(q, \dot{q}, t)$$

2. Define the Conjugate Momenta: $$p_i = \frac{\partial L}{\partial \dot{q}_i}$$

3. Perform the Legendre Transformation: $$H = \sum_i p_i \dot{q}_i - L$$ Here, $\dot{q}_i$ should be expressed in terms of $p_i$ and $q_i$, making $H$ a function of $p$, $q$, and possibly $t$: $$H = H(q, p, t)$$

4. Hamiltonian Formulation: In the Hamiltonian formulation, the equations of motion are derived from Hamilton's equations: $$\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}$$

From Hamiltonian to Lagrangian Formulation

1. Start with the Hamiltonian: $$H = H(q, p, t)$$

2. Express the velocities: Use Hamilton's equations to express the velocities $\dot{q}_i$ in terms of $p$ and $q$: $$\dot{q}_i = \frac{\partial H}{\partial p_i}$$

3. Invert to find $\dot{q}$ as functions of $q$ and $p$: If possible, solve the expressions from Hamilton's equations to find $\dot{q}_i$ as functions of $q_i$ and $p_i$.

4. Perform the inverse Legendre Transformation: Compute the Lagrangian by inverting the Legendre transformation: $$L = \sum_i p_i \dot{q}_i - H$$ where $H$ and $\dot{q}_i$ are expressed in terms of $p_i$ and $q_i$, recovering the Lagrangian description.