Deriving the Euler-Lagrange Equation | Variational Calculus

A geodesic is the shortest path between two points on a curved surface, analogous to a straight line being the shortest path on a flat plane. Finding geodesics becomes non-trivial on curved surfaces. However, geodesics are preserved under certain smooth transformations—meaning if you know a geodesic on one surface, you can find it on another by applying the transformation. This principle applies to wrapping paper around cylinders, where the shortest path becomes a spiral. Geodesics are fundamental to Einstein's general theory of relativity, as photons travel along geodesic curves in spacetime.

The Brachistochrone problem, posed by Johann Bernoulli in 1696, asks for the curve along which a particle sliding from point A to point B under gravity takes the minimum time; the solution to this problem is a cycloid curve, which is the path traced by a point on the rim of a rolling wheel, and this result is derived using the calculus of variations and the Euler-Lagrange equation.

The brachistochrone problem, which asks for the path of shortest travel time between two points under gravity, is solved using calculus of variations by minimizing the time functional T = ∫√[1 + (dy/dx)²]/√[2g(H - y)] dx; applying the Euler-Lagrange equation with the Beltrami identity leads to the parametric equations of a cycloid, demonstrating that the optimal path is a cycloid curve rather than a straight line or any other intuitive shape.

For the Brachistochrone problem, f = √((1 + (y')²)/(2gy)). Using the Beltrami identity, we derive the differential equation (1 + (y')²)/(2gy) - (y')²/(2gy)√((1 + (y')²)/(2gy)) = constant. Solving this gives the parametric equations x = a(θ - sinθ) + x₀, y = a(1 - cosθ) + y₀, which describe a cycloid. Geometrically, this represents a point on a circle of radius a rolling along a horizontal line, which is the curve that minimizes descent time under gravity.

The brachistochrone curve (inverted cycloid) is the path of fastest descent between two points under gravity, not the shortest path (which would be a straight line). This counterintuitive result is solved using the calculus of variations, a mathematical field that finds minimum or maximum values of functionals by applying small changes to functions. The Euler-Lagrange theorems, derived from minimizing the 'action' in physics, provide the mathematical framework for solving such optimization problems.

The Principle of Least Action is a fundamental variational principle in physics stating that physical systems follow paths that minimize (or make stationary) a quantity called the action. Fermat's Principle (1662) showed light takes the fastest path between two points, explaining refraction. Hamilton generalized this in 1834, proposing all physical theories derive from minimizing action. The action is the integral of the Lagrangian (L = T - V for Newtonian mechanics). This principle applies universally: to Newtonian mechanics (parabolic trajectories), electromagnetism, general relativity (Hilbert derived Einstein's equations from it), and quantum mechanics (Feynman's path integral). The Lagrangian form changes for each theory, but the variational principle remains the same.

Hamilton's principle states that the actual path taken by a physical system between two points in configuration space is the one for which the action integral is stationary (has zero variation). This means that when you perturb the path slightly, the value of the action remains almost unchanged. The action I is defined as the integral over time of the Lagrangian L along the trajectory. This principle allows us to derive the equations of motion (Euler-Lagrange equations) in a general way applicable across classical mechanics, quantum mechanics, electromagnetism, relativity, and engineering.

William Rowan Hamilton's formulation of mechanics in the 1830s introduced the principle of least (or stationary) action as the fundamental principle underlying all dynamics. According to this principle, the actual path taken by a physical system between two states is the one that minimizes (or makes stationary) a quantity called 'action.' Remarkably, this single principle can derive all aspects of classical mechanics, electrodynamics, and even quantum mechanics. The action itself has properties analogous to waves—it can cancel and enhance different paths, suggesting it might be a real physical entity distributed throughout space and time.

Hamilton's Principle (Principle of Least Action) states that the actual path taken by a mechanical system between two points in configuration space is such that the action (the integral of the Lagrangian over time) takes a stationary value. This principle, while historically attributed to Hamilton in the mid-19th century, has earlier versions dating back to 1740 by Maupertuis. The principle is equivalent to the Euler-Lagrange equations and can be extended to other areas of physics including electromagnetism, quantum mechanics, and general relativity. The action is not necessarily a minimum but can be a maximum or saddle point, making it a principle of stationary action rather than least action.

Hamilton's Principle states that nature seeks to minimize the action of a physical system. The action is a functional that represents the integral of the Lagrangian over time. This principle provides a fundamental framework for deriving the equations of motion for physical systems. The principle applies not only to classical mechanics but also to relativity, electromagnetism, and quantum mechanics, making it one of the most powerful and general principles in physics.

The Lagrange multiplier technique solves constrained optimization problems by finding points where the gradient of the objective function is proportional to the gradient of the constraint function, meaning they are parallel vectors; this occurs when the contour lines of the function being optimized are tangent to the constraint curve, requiring three equations: the two gradient proportionality conditions and the original constraint equation.

Lagrange multipliers solve optimization problems with constraints. To maximize/minimize f(x,y,z) subject to g(x,y,z) = 0, solve the system: ∇f = λ∇g and g = 0. This introduces a new variable λ (the Lagrange multiplier). The solution gives the optimal point(s) on the constraint surface. This technique is used in physics (e.g., finding the hottest point on a space probe given its shape and temperature distribution).

The Lagrange multipliers method solves optimization problems with constraints. For functions of three or more variables, the method introduces a multiplier λ to combine the objective function and constraint. The key identity is df + λ dg = 0, which expands to three equations by equating coefficients of dx, dy, dz to zero. These equations, combined with the original constraint, form a system that identifies critical points where extrema may occur. The method transforms constrained optimization into solving simultaneous equations.

Lagrange multipliers provide a systematic method for solving constrained optimization problems by introducing a multiplier (λ) to incorporate the constraint into the objective function, then finding critical points by setting partial derivatives equal to zero; the resulting λ value indicates how much the objective function changes with a unit change in the constraint's constant.

This section extends variational methods to constrained optimization and higher-order derivatives. The method of Lagrange multipliers handles constraints by forming F + λG and applying the Euler equation. For higher-order problems like y'''' = 0, the general solution contains four arbitrary constants requiring four boundary conditions. The technique of substituting the extremal function back into the original functional determines the extremum value. For example, with y = x²/2 + Cx, the integral ∫(y')² dx evaluates to find the extremum. These advanced techniques address complex variational scenarios requiring sophisticated mathematical tools.

Variational problems can involve derivatives of order k ≥ 2: J(Y) = ∫F(x, Y, Y', ..., Y^(k)) dx. Admissible functions are in C^k([a,b]) with boundary conditions at both endpoints: Y^(j)(a) = A^(j), Y^(j)(b) = B^(j) for j = 0, ..., k-1. The Euler equation is ∑_{j=0}^k (-1)^j d^j/dx^j (∂F/∂Y^(j)) = 0, a 2k-th order differential equation. The fundamental lemma states that if ∫φh = 0 for all admissible h, then φ = 0. For example, with J = ∫₀^(π/2) (y'² + z'² + 2xz - 2yz) dx and boundary conditions y(0) = y(π/2) = 0, z(0) = z(π/2) = 0, the Euler equations are y'' - z + y = 0 and z'' - x + z = 0. Substituting z = y'' gives y'''' - y = 0, with general solution y = C₁e^x + C₂e^(-x) + C₃cos(x) + C₄sin(x). The boundary conditions determine the constants, yielding the admissible extremals.

Higher-order variational problems involve functionals of the form J(x) = ∫_a^b F(t, x(t), x'(t), ..., x^(n)(t)) dt, where x^(n)(t) is the nth derivative of x(t). The admissible functions are those that are n times continuously differentiable and satisfy boundary conditions on x(t) and its derivatives up to order n-1 at the endpoints. The metric is ρ(x, y) = max{|x(t) - y(t)|, ..., |x^(n-1)(t) - y^(n-1)(t)|}.

Functionals assign real numbers to functions, with admissible classes like C¹[x₁,x₂] defining permissible functions. The general form I[y] = ∫F(x,y,y')dx extends to multiple variables I[y₁,...,yₙ] = ∫F(x,y₁,...,yₙ,y₁',...,yₙ')dx. Higher-order problems involve I[y] = ∫F(x,y,y',...,y^(n))dx requiring Cⁿ spaces. Multi-dimensional problems use double integrals I[z] = ∫∫F(x,y,z,zₓ,zᵧ)dxdy. Boundary conditions specify endpoint values or derivatives, ensuring well-posed optimization. This framework enables systematic analysis of optimization problems across diverse applications.

To solve functionals with higher-order derivatives: (1) Identify the highest order derivative, (2) Apply the Euler-Poisson equation with appropriate order, (3) Compute all partial derivatives of f with respect to y, y', y'', ..., y^(n), (4) Substitute into the Euler-Poisson equation, (5) Solve the resulting differential equation. For f = 16y² - (y'')² + x², this gives y'''' - 16y = 0, demonstrating the systematic approach to higher-order variational problems.

Variational problems can involve higher-order derivatives of the function. For a functional depending on y'' (second derivative), the Euler-Lagrange equation becomes a fourth-order differential equation. The general form involves terms with ∂f/∂y, ∂f/∂y', ∂f/∂y'', and their derivatives. Such problems require additional boundary conditions to determine the solution. Variational problems can also involve functions of multiple variables, leading to partial differential equations. The Dirichlet functional J[y] = ∫(y_x² + y_y²)dxdy yields Laplace's equation (Δy = 0), meaning minimizing functions are harmonic.