Lagrangian Mechanics — Mathematics Knowledge System

Overview

Lagrangian mechanics reformulates classical Newtonian mechanics using generalised coordinates and a scalar function — the Lagrangian — defined as kinetic energy minus potential energy. The equations of motion follow from the principle of stationary action: the physical trajectory minimises (or makes stationary) the action integral. This approach is coordinate-independent, naturally handles constraints, and forms the mathematical foundation for quantum field theory and general relativity.

Intuition

Instead of tracking forces at every point (Newton), Lagrangian mechanics encodes all dynamics in a single scalar function L = T − V. Nature picks the path between two configurations that makes the action ∫L dt stationary — the principle of least action. Constraints are handled automatically through the choice of generalised coordinates, making curved geometries and coupled systems tractable.

Formal Definition

Definition

Given a mechanical system with n degrees of freedom described by generalised coordinates q = (q₁, …, qₙ), the Lagrangian L and action S are defined, and the Euler–Lagrange equations give the equations of motion.

L(q, \dot{q}, t) = T(q, \dot{q}, t) - V(q, t)

Lagrangian (kinetic minus potential energy)

S[q] = \int_{t_1}^{t_2} L(q(t), \dot{q}(t), t)\, dt

Action functional

\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0, \quad i = 1, \ldots, n

Euler-Lagrange equations of motion

p_i = \frac{\partial L}{\partial \dot{q}_i}

Generalised momentum conjugate to qᵢ

Notation

Notation	Meaning
$q_i$	Generalised coordinate
$\dot{q}_i$	Generalised velocity (time derivative of qᵢ)
$L$	Lagrangian function
$S$	Action functional
$p_i$	Generalised (conjugate) momentum

Theorems

Theorem 1: Principle of Stationary Action

\delta S = \delta \int_{t_1}^{t_2} L\, dt = 0 \iff \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0

Theorem 2: Conservation of Generalised Momentum

\frac{\partial L}{\partial q_i} = 0 \implies \frac{d p_i}{dt} = 0

Theorem 3: Energy Conservation (Beltrami Identity)

H = \sum_i p_i \dot{q}_i - L = \text{const} \text{ when } \partial L / \partial t = 0

Worked Examples

1
Use angle θ as the generalised coordinate. Kinetic energy T = ½mℓ²θ̇², potential V = −mgℓcos θ.
$L = \tfrac{1}{2} m\ell^2 \dot{\theta}^2 + mg\ell\cos\theta$
2
Compute the partial derivatives.
$\frac{\partial L}{\partial \dot{\theta}} = m\ell^2 \dot{\theta},\quad \frac{\partial L}{\partial \theta} = -mg\ell\sin\theta$
3
Apply the Euler–Lagrange equation.
$m\ell^2 \ddot{\theta} + mg\ell\sin\theta = 0 \implies \ddot{\theta} + \frac{g}{\ell}\sin\theta = 0$

✓ Answer

Pendulum EOM: θ̈ + (g/ℓ) sin θ = 0, recovering the standard result.

Practice Problems

Mediumfree response

Set up the Lagrangian for an Atwood machine (two masses m₁, m₂ connected by an inextensible string over a pulley) and derive the equation of motion.

Common Mistakes

Common Mistake

The Lagrangian L = T − V always holds

L = T − V is valid for conservative systems in Cartesian or generalised coordinates. Non-conservative forces or velocity-dependent potentials (e.g., electromagnetic) require modification: L = T − V + velocity-dependent terms.

Common Mistake

The Euler-Lagrange equations only hold in Cartesian coordinates

The Euler–Lagrange equations hold in any generalised coordinate system — that is precisely their power over Newtonian mechanics.

Quiz

Historical Background

Joseph-Louis Lagrange published his 'Mécanique Analytique' in 1788, unifying all of mechanics in a purely algebraic form free from geometric constructions. Lagrange built on d'Alembert's principle and Euler's earlier calculus-of-variations work. William Rowan Hamilton later recast the theory in terms of a phase-space Hamiltonian (1833–1835), and Emmy Noether's 1918 theorem revealed the deep connection between symmetries and conservation laws within the Lagrangian framework.

1788
Lagrange publishes 'Mécanique Analytique', establishing the Lagrangian formalism
Joseph-Louis Lagrange
1834
Hamilton formulates the principle of stationary action and derives Hamiltonian mechanics
William Rowan Hamilton
1918
Noether proves that every continuous symmetry corresponds to a conserved quantity
Emmy Noether

Summary

The Lagrangian L = T − V encodes all dynamics of a conservative mechanical system.
The Euler–Lagrange equations d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0 are the equations of motion.
The action S = ∫L dt is stationary (principle of least action) along physical trajectories.
Cyclic coordinates lead to conserved momenta; time-independence of L implies energy conservation.
The formalism is coordinate-independent and handles constraints naturally via generalised coordinates.

References

BookGoldstein, H., Poole, C. & Safko, J. — Classical Mechanics, 3rd ed. (2002), Addison-Wesley, Chapter 1–2
BookLandau, L.D. & Lifshitz, E.M. — Mechanics, 3rd ed. (1976), Butterworth-Heinemann, §1–2
BookArnol'd, V.I. — Mathematical Methods of Classical Mechanics, 2nd ed. (1989), Springer, Chapter IV
WebsiteWikipedia — Lagrangian mechanics
WebsiteMathWorld — Lagrangian