Classical Mechanics Lagrangian and Hamiltonian Cheat Sheet

Classical mechanics with Lagrangian and Hamiltonian methods gives a powerful way to model motion using energy instead of only forces. This cheat sheet helps students organize the main definitions, equations, and transformations used in analytical mechanics. It is especially useful for systems with constraints, generalized coordinates, or conserved quantities.

The goal is to connect physical meaning with the formulas used to derive equations of motion.

The Lagrangian is usually defined as $L = T - V$ , and the path of a system follows the stationary action principle $\delta S = 0$ . The Euler-Lagrange equation $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$ gives the equations of motion in generalized coordinates. The Hamiltonian is formed using $H = \sum_i p_i\dot{q}_i - L$ , where $p_i = \frac{\partial L}{\partial \dot{q}_i}$ .

Hamilton's equations describe evolution in phase space using $\dot{q}_i = \frac{\partial H}{\partial p_i}$ and $\dot{p}_i = -\frac{\partial H}{\partial q_i}$ .

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.

Common Mistakes to Avoid

Using $L = T + V$ instead of $L = T - V$ for a standard conservative system is wrong because the sign of the potential energy controls the correct force direction.
Treating $q_i$ and $\dot{q}_i$ as dependent during partial differentiation is wrong because $\frac{\partial L}{\partial q_i}$ and $\frac{\partial L}{\partial \dot{q}_i}$ are computed by holding the other variable fixed.
Forgetting the total time derivative in $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)$ is wrong because the derivative must include time dependence through $q_i(t)$ and $\dot{q}_i(t)$ .
Assuming $H$ always equals total energy is wrong because this equality can fail for time-dependent transformations, velocity-dependent potentials, or nonstandard Lagrangians.
Calling a coordinate cyclic just because $\dot{q}_i$ appears in $L$ is wrong because cyclic means $q_i$ itself is absent from $L$ , not its velocity.

Practice Questions

1 For a one-dimensional mass with $L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2$ , use the Euler-Lagrange equation to derive the equation of motion.
2 Given $L = \frac{1}{2}m\dot{x}^2 - mgx$ , find the canonical momentum $p_x$ and construct the Hamiltonian $H$ .
3 For a particle in polar coordinates with $L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) - V(r)$ , identify the cyclic coordinate and its conserved momentum.
4 Explain why generalized coordinates are useful for a pendulum or bead-on-wire system compared with Cartesian coordinates.

Understanding Classical Mechanics Lagrangian and Hamiltonian

Stationary action does not mean that a moving object chooses the shortest path through space. It compares whole histories between fixed starting and ending times. A small imagined change in the path produces no first order change in the action for the real motion.

This idea can seem abstract, but it reproduces familiar results. For a ball moving under gravity, it gives the same acceleration found from Newton's second law.

Its advantage appears when forces are hard to list separately, while the system's energy and geometry are easy to describe. The method treats all allowed motions at once, then selects the physically consistent one.

Generalized coordinates are chosen to match the system, not necessarily to match horizontal, vertical, and depth directions. A pendulum can be described by one angle because its rod fixes the distance from the pivot. A bead sliding on a curved wire needs only one coordinate along the wire.

These choices build constraints into the description from the start. This reduces the number of unknowns and avoids calculating constraint forces that do no useful work. When choosing coordinates, count the independent ways the system can move.

Then express every position, speed, and energy in terms of those coordinates and their rates of change. Careful geometry matters more than advanced calculus at this stage.

Canonical momentum deserves special attention because it is not always ordinary mass times velocity. For a particle described with Cartesian position coordinates and no velocity dependent interactions, the two are the same. In polar coordinates, however, the momentum linked to angle includes the distance from the origin.

It represents angular momentum rather than simple sideways motion. This is why the meaning of momentum depends on the coordinate selected. The Hamiltonian description uses coordinates and their canonical momenta as separate axes in phase space.

One point in phase space specifies a complete instantaneous state. Its motion through phase space tracks how position and momentum change together. This viewpoint is useful in orbital motion, oscillations, statistical physics, and later quantum mechanics.

Conservation laws come from symmetries, which are features of a system that remain unchanged under a transformation. If rotating a system around an axis changes nothing, angular momentum about that axis is conserved. If shifting the entire experiment through time changes nothing, energy is usually conserved.

A cyclic coordinate signals one of these symmetries directly. For example, if a potential does not depend on a pendulum's angle around a vertical axis, the associated angular momentum stays fixed. Students should check units, signs, and hidden time dependence before claiming conservation.

Friction, driving forces, or a moving support can make the energy change. Practice by deriving a simple pendulum or spring equation from energy expressions, then compare the result with the force based equation. This comparison makes the formalism feel like a reliable tool rather than a list of rules.

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Classical Mechanics Lagrangian and Hamiltonian Cheat Sheet

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