Lagrangian Mechanics and Equations of Motion

Why Do We Need an Alternative to Newton?

Newtonian mechanics is a powerful, yet cumbersome tool when a system has constraints or is described using curvilinear coordinates. Imagine a pendulum: in Cartesian coordinates, we must express the tension force as an unknown and then eliminate it. The Lagrangian approach works directly with the angular coordinate θ, automatically “ignoring” the constraint forces.

Lagrangian mechanics is a reformulation of classical mechanics in the language of functional calculus: instead of forces, we deal with energies, and the equations of motion derive from a single principle—the principle of least action.

Generalized Coordinates

Let’s consider a system of N particles. In 3D space, we would need 3N Cartesian coordinates. But if there are constraints (strings, rigid rods), some coordinates depend on the others. Generalized coordinates q = (q₁, ..., qₙ) are any independent set of parameters that uniquely determine the configuration.

Examples: a pendulum with length l has one degree of freedom—the angle θ. A double pendulum has two angles (θ₁, θ₂). A water molecule—9 coordinates for three atoms, but 6 degrees of freedom after 3 constraints fixing the bond lengths. Generalized coordinates are the “natural” variables of the problem.

The Lagrangian and the Principle of Least Action

The Lagrangian is defined as the difference between kinetic and potential energies:

L(q, q̇, t) = T − V

Here, T is the kinetic energy, V is the potential energy, q̇ are the generalized velocities (derivatives of q with respect to time).

Action is a functional, that is, a number assigned to each trajectory q(t) over the interval [t₁, t₂]:

S[q] = ∫_{t₁}^{t₂} L(q, q̇, t) dt

The principle of least action (Hamilton’s principle): the actual trajectory of the system is the one for which δS = 0, that is, the first variation of the action equals zero. From this principle, by integrating by parts, we obtain the Lagrange equations:

d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0 for each i = 1, ..., n

These are n second-order equations for n functions qᵢ(t). The quantity pᵢ = ∂L/∂q̇ᵢ is called the generalized momentum conjugate to coordinate qᵢ.

Symbol Explanation

• L — Lagrangian, dimension: J (joule), unit of energy
• q̇ᵢ — generalized velocity, derivative dqᵢ/dt with respect to time
• ∂L/∂q̇ᵢ — partial derivative of L with respect to the generalized velocity (generalized momentum pᵢ)
• ∂L/∂qᵢ — derivative of L with respect to the generalized coordinate (generalized force)
• d/dt(...) — total derivative with respect to time, taking into account that q and q̇ depend on t

Fully Worked Example: Mathematical Pendulum

A pendulum—a string of length l with a mass m, angular displacement θ. Cartesian coordinates: x = l sin θ, y = −l cos θ.

Step 1. Kinetic energy: T = (ml²/2) θ̇²

Step 2. Potential energy (zero at the pivot point): V = −mgl cos θ

Step 3. Lagrangian: L = T − V = (ml²/2) θ̇² + mgl cos θ

Step 4. Calculate ∂L/∂θ̇ = ml² θ̇, then d/dt(ml² θ̇) = ml² θ̈

Step 5. ∂L/∂θ = −mgl sin θ

Step 6. Lagrange equation: ml² θ̈ + mgl sin θ = 0, i.e., θ̈ + (g/l) sin θ = 0

This is the famous pendulum equation, obtained without a single mention of the tension forces in the string!

For small oscillations (sin θ ≈ θ): θ̈ + (g/l) θ = 0, whence angular frequency ω = √(g/l) and period T = 2π√(l/g). For l = 1 m, g = 9.81 m/s²: T ≈ 2.006 s.

Noether’s Theorem: Symmetry Generates Conservation Laws

This is, arguably, the most profound result of theoretical physics: if the action S is invariant under a continuous family of transformations qᵢ → qᵢ + ε ξᵢ(q, t), then the quantity

Q = Σᵢ (∂L/∂q̇ᵢ) ξᵢ

is an integral of motion, that is, it is conserved.

Examples: if L does not explicitly depend on time t → energy is conserved. If L does not depend on some coordinate qᵢ (coordinate is “cyclic”) → its conjugate momentum pᵢ is conserved. Invariance under rotations → law of conservation of angular momentum.

Real Application: Satellite in Orbit

The Kepler problem (planet around the Sun) in polar coordinates (r, φ): L = m(ṙ² + r²φ̇²)/2 + GMm/r. The coordinate φ is cyclic (L does not depend on φ), so pφ = mr²φ̇ = const—this is the law of conservation of angular momentum. From this, the second Kepler law (equal areas in equal times) follows immediately—without any force calculations!

Lagrangian Mechanics in Modern Technologies

The Lagrangian approach has become the basis for calculating dynamics in a wide range of technical systems. In robotics, Lagrange’s equations are used to derive dynamic equations of manipulators: the generalized coordinates are the joint angles, kinetic and potential energies are written analytically, and the algorithm generates the control equations automatically. In aerospace engineering, the method allows the consideration of complex constraints and structural flexibility without explicitly calculating support reactions. In particle physics, the principle of least action generalizes to fields: the Lagrangian density ℒ(φ, ∂_μφ) determines the field equations of motion via the Euler–Lagrange equation. Electromagnetic fields, gluons, the Higgs boson—all are described through Lagrangians, whose symmetries specify conservation laws via Noether’s theorem. Symbolic computation systems (Mathematica, Maple) automatically generate Lagrange equations from the given T and V, making the method an indispensable tool in modern engineering and theoretical physics.

Assignment: Double pendulum: mass m₁ on a rod of length l₁, mass m₂ on a rod l₂ attached to m₁. Write T and V in terms of θ₁, θ₂. Derive the Lagrange equations. Linearize for small oscillations and find the normal frequencies for m₁ = m₂ = m, l₁ = l₂ = l.