Classical Mechanics Lagrangian and Hamiltonian Cheat Sheet

Key Facts

• The action is $S = \int_{t_1}^{t_2} L(q_i,\dot{q}_i,t)\,dt$, and physical motion satisfies $\delta S = 0$.
• For conservative mechanical systems, the Lagrangian is often $L = T - V$, where $T$ is kinetic energy and $V$ is potential energy.
• The Euler-Lagrange equation for each generalized coordinate is $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$.
• The generalized momentum conjugate to $q_i$ is $p_i = \frac{\partial L}{\partial \dot{q}_i}$.
• The Hamiltonian is the Legendre transform $H(q_i,p_i,t) = \sum_i p_i\dot{q}_i - L(q_i,\dot{q}_i,t)$ after writing velocities in terms of $q_i$ and $p_i$.
• Hamilton's equations are $\dot{q}_i = \frac{\partial H}{\partial p_i}$ and $\dot{p}_i = -\frac{\partial H}{\partial q_i}$.
• If $\frac{\partial L}{\partial q_i} = 0$, then $q_i$ is cyclic and the conjugate momentum $p_i$ is conserved.
• If $\frac{\partial L}{\partial t} = 0$ and standard conditions hold, then the Hamiltonian $H$ is conserved and often equals the total energy $E = T + V$.

Vocabulary

Generalized coordinate

A variable $q_i$ that describes the configuration of a system using coordinates adapted to its constraints.

Lagrangian

The function $L(q_i,\dot{q}_i,t)$, often equal to $T - V$, used to derive equations of motion from the action.

Action

The time integral $S = \int L\,dt$ whose stationary value determines the physical path of a system.

Canonical momentum

The momentum $p_i = \frac{\partial L}{\partial \dot{q}_i}$ conjugate to the generalized coordinate $q_i$.

Hamiltonian

The function $H(q_i,p_i,t) = \sum_i p_i\dot{q}_i - L$ that generates motion in phase space.

Cyclic coordinate

A coordinate $q_i$ that does not appear explicitly in $L$, causing its conjugate momentum $p_i$ to be conserved.