Source: Lecture Notes, Calculus III (University of Michigan)
Tags: Euler-Lagrange equation, variational calculus, calculus of variations, Lagrangian, action functional, classical mechanics, optimisation
Difficulty: Intermediate | Prerequisites: Multivariable calculus (partial derivatives, integrals), basic differential equations. Familiarity with Newtonian mechanics is helpful but not required.
The Euler-Lagrange equation sits at the heart of variational calculus, which is about finding functions that optimise (minimise or maximise) a given quantity. In Calc III you have already met optimisation of ordinary functions; this extends the idea to entire paths or trajectories. The equation underpins classical mechanics, optimal control, and a surprising amount of modern physics. If you are comfortable with partial derivatives and integration, you have the tools you need.
The Euler-Lagrange equation is a differential equation that tells you the path or trajectory a system must follow to minimise (or make stationary) a quantity called the action. You write down a Lagrangian that captures the system's physics, plug it into the equation, and out come the equations of motion. It replaces force-by-force analysis with a single, elegant optimisation principle.
Functional
A mapping that takes a function as input and returns a scalar (a single number). In simple terms, a function eats numbers and returns numbers; a functional eats entire functions and returns a number.
Action (S)
The integral of the Lagrangian over time, from t₁ to t₂. Think of it as the total "cost" of a particular path through time; the system picks the path whose action is stationary.
Lagrangian (L)
A function L(x, ẋ, t) that encodes the dynamics of the system, typically defined as kinetic energy minus potential energy in mechanics. In simple terms, the Lagrangian is the recipe that tells the Euler-Lagrange equation what the system cares about.
Stationary point (of a functional)
The function at which the functional's value does not change to first order under small perturbations. This is the functional analogue of setting a derivative to zero to find a minimum or maximum.
Variational principle
The idea that the laws of motion can be derived by requiring a certain functional (typically the action) to be stationary. In simple terms, nature "chooses" the path that makes the action stationary.
Partial derivative ∂L/∂x and ∂L/∂ẋ
The derivative of the Lagrangian with respect to position and velocity, respectively, treating the other variables as constants. These are the building blocks you plug into the Euler-Lagrange equation.
Leonhard Euler first developed the equation in 1744
Joseph-Louis Lagrange proposed the underlying variational idea in 1762, arguing that equations of motion could be found by minimising a functional
William Rowan Hamilton wrote the equation in its modern form in 1837
The equation carries both Euler's and Lagrange's names, though Hamilton deserves credit for the notation most courses now use
Start with the action functional: the integral of the Lagrangian L(x, ẋ, t) over a time interval [t₁, t₂]
Require that the action be stationary: any small perturbation of the path x(t) should not change the action to first order
The perturbation must vanish at the endpoints (boundary conditions: δx(t₁) = δx(t₂) = 0)
Apply integration by parts to the term involving δẋ
Because the perturbation is arbitrary, the integrand must vanish identically, giving the Euler-Lagrange equation
For each coordinate xᵢ of the system:
(d/dt)(∂L/∂ẋᵢ) − ∂L/∂xᵢ = 0
This is a second-order ODE (once you expand the time derivative), one for each degree of freedom
The first term, (d/dt)(∂L/∂ẋᵢ), is the total time derivative of the "generalised momentum"
The second term, ∂L/∂xᵢ, acts like a generalised force
Physics: deriving equations of motion for particles, fields, and mechanical systems from a Lagrangian (classical mechanics, electrodynamics, general relativity)
Engineering: optimising system performance, trajectory planning, control theory
Mathematics: proving results in the calculus of variations, such as the second variation formula and existence/uniqueness theorems for minimisers
Action functional
S = ∫ from t₁ to t₂ of L(x, ẋ, t) dt
where L is the Lagrangian, x is the position vector, ẋ is the velocity vector, and t is time.
Euler-Lagrange equation (per coordinate)
(d/dt)(∂L/∂ẋᵢ) − ∂L/∂xᵢ = 0
One such equation exists for every generalised coordinate xᵢ. Together they form a system of second-order ODEs that fully determines the motion.
This is the principle behind how aerospace engineers compute fuel-optimal trajectories for spacecraft: define a Lagrangian that captures fuel cost and constraints, then solve the Euler-Lagrange equation for the best path. It is also why light bends through glass the way it does: Fermat's principle (least time) is a variational problem whose solution follows the same mathematics.
Students often think the Euler-Lagrange equation gives you the minimum of the action. It does not, necessarily. It gives a stationary point, which could be a minimum, maximum, or saddle point. "Stationary" means the first variation vanishes, not that you have found the lowest value.
Students sometimes confuse the Lagrangian with the Hamiltonian. The Lagrangian is L = T − V (kinetic minus potential energy). The Hamiltonian is H = T + V (total energy). They are related but distinct formulations.
A common error is treating ∂L/∂ẋ and dL/dẋ as the same thing. The partial derivative holds x and t constant; the total derivative does not. When you apply (d/dt) to ∂L/∂ẋ, you must use the chain rule.
Students sometimes forget the boundary conditions (δx = 0 at the endpoints). Without them, the integration by parts that produces the equation does not go through.
⚠️ You will almost certainly be asked to derive the Euler-Lagrange equation from the action principle using integration by parts. Practise this until it is automatic.
⚠️ Know the difference between stationary and minimum. If an exam question asks "does this minimise the action?", the answer requires checking the second variation, not just applying the Euler-Lagrange equation.
⚠️ Be ready to apply the equation to a concrete Lagrangian (e.g. L = ½mẋ² − V(x)) and recover Newton's second law. This is a classic exam question.
⚠️ Watch for problems that give you a Lagrangian with explicit time dependence. The same equation applies, but energy is no longer conserved.
True or false: The Euler-Lagrange equation always finds the path that minimises the action. (False: it finds a stationary path.)
Fill in the blank: The action S is defined as the ___ of the Lagrangian over a time interval. (integral)
True or false: The Lagrangian is defined as kinetic energy plus potential energy. (False: kinetic minus potential.)
Fill in the blank: The boundary conditions for deriving the Euler-Lagrange equation require that the perturbation δx is ___ at both endpoints. (zero)
True or false: Euler first developed the equation in 1762. (False: 1744.)
Q: State the Euler-Lagrange equation for a system with Lagrangian L(x, ẋ, t).
A: (d/dt)(∂L/∂ẋᵢ) − ∂L/∂xᵢ = 0, for each generalised coordinate xᵢ.
Q: Given L = ½mẋ² − V(x), derive the equation of motion using the Euler-Lagrange equation.
A: ∂L/∂ẋ = mẋ, so (d/dt)(mẋ) = mẍ. ∂L/∂x = −dV/dx. The Euler-Lagrange equation gives mẍ − (−dV/dx) = 0, i.e. mẍ = −dV/dx, which is Newton's second law F = ma with F = −dV/dx.
Q: What role do the boundary conditions play in the derivation of the Euler-Lagrange equation?
A: They require that the perturbation δx vanishes at both endpoints t₁ and t₂. This eliminates the boundary term produced by integration by parts, leaving only the integral whose integrand must vanish.
Q: Explain the difference between a function and a functional, with one example of each.
A: A function maps numbers to numbers (e.g. f(x) = x²). A functional maps functions to numbers (e.g. S[x] = ∫ L(x, ẋ, t) dt). The action S is a functional because it takes an entire path x(t) and returns a single scalar.
Q: Why is "principle of least action" a slightly misleading name?
A: The Euler-Lagrange equation finds stationary points of the action, not necessarily minima. The action could be at a saddle point or even a maximum. "Principle of stationary action" is more precise.
This connects directly to Hamiltonian mechanics, which reformulates the same physics using the Hamiltonian H instead of the Lagrangian L. If your course covers Hamilton's equations, they are derived from the same variational framework.
The calculus of variations, of which the Euler-Lagrange equation is the central result, also appears in differential geometry (geodesics on curved surfaces) and in optimisation and control theory (Pontryagin's maximum principle is a close relative).
In a broader Calc III context, the derivation exercises partial differentiation, the chain rule for multivariable functions, and integration by parts, so it ties together much of what you have been practising all term.
Euler-Lagrange equation, calculus of variations, variational calculus, Lagrangian mechanics, action principle, principle of least action, principle of stationary action, Hamilton's principle, functional, generalised coordinates, generalised momentum, Lagrangian, Hamiltonian, integration by parts, boundary conditions, classical mechanics, equations of motion, optimisation, second variation, trajectory optimisation