Glossary

13.3 Calculus of variations and Euler-Lagrange equations

3 min read•july 22, 2024

The calculus of variations is a powerful mathematical tool for finding optimal functions. It's used to solve problems like finding the shortest path between two points or the shape of a hanging chain.

At its core, the calculus of variations uses the to find stationary points of functionals. This leads to solutions for various optimization problems in physics, engineering, and economics.

Calculus of Variations

Euler-Lagrange equations derivation

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  • Provide necessary conditions for a function to be a of a (mapping from vector space to real numbers)
  • Consider functional J[y]=∫x1x2F(x,y(x),y′(x))dxJ[y] = \int_{x_1}^{x_2} F(x, y(x), y'(x)) dx and assume y(x)y(x) is a stationary point
  • Introduce small variation ϵη(x)\epsilon \eta(x) around y(x)y(x) and expand J[y+ϵη]J[y + \epsilon \eta] using Taylor series
  • Set first-order term to zero, integrate by parts, and apply fundamental lemma of calculus of variations
  • Resulting Euler- equation: ∂F∂y−ddx∂F∂y′=0\frac{\partial F}{\partial y} - \frac{d}{dx} \frac{\partial F}{\partial y'} = 0
  • Express balance between change in FF with respect to yy and change in FF with respect to y′y'
  • Solutions make functional J[y]J[y] stationary (minimum, maximum, or saddle points)

Variational problem solutions

  • Identify functional J[y]=∫x1x2F(x,y(x),y′(x))dxJ[y] = \int_{x_1}^{x_2} F(x, y(x), y'(x)) dx to be minimized or maximized
  • Write Euler-Lagrange equation: ∂F∂y−ddx∂F∂y′=0\frac{\partial F}{\partial y} - \frac{d}{dx} \frac{\partial F}{\partial y'} = 0
  • Solve differential equation for y(x)y(x) subject to boundary conditions
  • Verify solution satisfies sufficient conditions for optimality (if applicable)
  • Examples:
    • : Find shortest curve connecting two points on a surface
    • : Find curve of fastest descent between two points under gravity
    • : Maximize or minimize quantity subject to perimeter or area constraint

Advanced Topics in Variational Calculus

Legendre transform in variations

  • Convert between (generalized coordinates and velocities) and (generalized coordinates and momenta) formulations
  • definition: Given f(x)f(x), transform g(p)=sup⁡x(px−f(x))g(p) = \sup_x (px - f(x))
  • Relates Lagrangian L(q,q˙,t)L(q, \dot{q}, t) and Hamiltonian H(q,p,t)H(q, p, t)
  • Define p=∂L∂q˙p = \frac{\partial L}{\partial \dot{q}}
  • Hamiltonian given by Legendre transform of Lagrangian: H(q,p,t)=pq˙−L(q,q˙,t)H(q, p, t) = p\dot{q} - L(q, \dot{q}, t)
  • : q˙=∂H∂p\dot{q} = \frac{\partial H}{\partial p} and p˙=−∂H∂q\dot{p} = -\frac{\partial H}{\partial q}
  • Advantages: Symmetric treatment of coordinates and momenta, rich mathematical structure (symplectic geometry, Poisson brackets)

Optimality conditions for variations

  • Ensure solution to Euler-Lagrange equations is minimum or maximum of functional
  • δ2J[y]\delta^2 J[y]: Second-order term in Taylor expansion of J[y+ϵη]J[y + \epsilon \eta]
    • For minimum (maximum), must be non-negative (non-positive) for all admissible variations η(x)\eta(x)
  • : Necessary condition for to be non-negative (non-positive)
    • For minimum: ∂2F∂y′2≥0\frac{\partial^2 F}{\partial y'^2} \geq 0
    • For maximum: ∂2F∂y′2≤0\frac{\partial^2 F}{\partial y'^2} \leq 0
  • : Strengthened requiring absence of conjugate points along solution curve
    • Conjugate points: Points where solution to Jacobi equation (linearized Euler-Lagrange equation) vanishes
    • Presence indicates solution is not strict minimum or maximum
  • : Apply when solution curve has corners (discontinuities in first derivative)
    • At corner: F−y′∂F∂y′F - y' \frac{\partial F}{\partial y'} and ∂F∂y′\frac{\partial F}{\partial y'} must be continuous

Key Terms to Review (44)

Action integral: The action integral is a mathematical expression that encapsulates the dynamics of a physical system by integrating the Lagrangian function over time. This quantity is fundamental in deriving the equations of motion through the principle of least action, which states that the actual path taken by a system is the one that minimizes the action integral. It connects closely with variational principles, allowing for a systematic way to derive governing equations using calculus techniques.
Bounded functional: A bounded functional is a linear functional that maps elements from a vector space to the field of scalars and satisfies a condition of boundedness, meaning there exists a constant such that the absolute value of the functional's output is less than or equal to this constant times the norm of the input element. This concept is crucial in understanding dual spaces and continuity in the context of functional analysis, particularly when dealing with variational problems and Euler-Lagrange equations.
Brachistochrone: The brachistochrone is the curve of fastest descent between two points under the influence of gravity, which is a classic problem in the calculus of variations. This concept exemplifies how to find the optimal path for an object to travel in the least time possible, highlighting the principles of minimizing functionals and the use of differential equations.
Brachistochrone problem: The brachistochrone problem is a classic question in the calculus of variations that seeks to determine the shape of a curve down which a bead will slide, under the influence of gravity, from one point to another in the least time possible. This problem illustrates the application of the Euler-Lagrange equations, which are crucial for finding functions that minimize or maximize certain quantities, thus connecting it to the broader framework of optimization in physics and engineering.
Convex functional: A convex functional is a mapping from a vector space to the real numbers that satisfies the property of convexity, meaning that for any two points in its domain, the functional's value at any point on the line segment connecting these points is less than or equal to the weighted average of its values at those two points. This concept plays a crucial role in variational principles and calculus of variations, as convex functionals often lead to well-defined optimization problems and solutions, helping identify extremal functions that minimize or maximize the functional's value.
Convexity: Convexity refers to a property of a set or a function where a line segment connecting any two points within the set or on the curve lies entirely within the set or above the curve, respectively. This concept is crucial in optimization and variational problems, as it ensures that local minima are also global minima, simplifying the process of finding optimal solutions.
Direct Method: The direct method is a technique used in the calculus of variations to find extrema of functionals by analyzing the properties of the functional directly, rather than through indirect means. This method often involves proving the existence of a minimizer by establishing lower bounds for the functional and demonstrating the continuity of minimizing sequences, leading to the derivation of necessary conditions for optimality, such as those found in the Euler-Lagrange equations.
Euler-Lagrange Equation: The Euler-Lagrange equation is a fundamental equation in the calculus of variations that provides a necessary condition for a functional to have an extremum. It relates the derivatives of a function to its Lagrangian, which encapsulates the dynamics of the system under consideration. This equation plays a crucial role in solving variational problems, allowing us to determine the path or function that minimizes or maximizes a given functional.
Extremal: An extremal is a function or curve that makes a functional achieve its maximum or minimum value in the context of calculus of variations. This concept is crucial for understanding how to determine optimal solutions when dealing with functionals, and it plays a key role in deriving the Euler-Lagrange equations, which provide the necessary conditions for a function to be extremal.
Extremal function: An extremal function is a specific function that minimizes or maximizes a given functional in the calculus of variations. These functions are critical for identifying optimal solutions to problems where one seeks to find the best shape, path, or configuration that satisfies certain constraints. The concept of extremal functions is closely tied to Euler-Lagrange equations, which provide necessary conditions for these functions to be optimal.
First Variation: The first variation refers to the linear approximation of the change in a functional due to small changes in the function being considered. This concept is central in the calculus of variations, where it helps in determining extremal functions that minimize or maximize a given functional. By analyzing the first variation, one can derive necessary conditions for optimality, often leading to the Euler-Lagrange equations that are essential for solving variational problems.
Functional: In mathematics, particularly in functional analysis, a functional is a specific type of function that maps vectors from a vector space to the field of scalars, typically real or complex numbers. Functionals are crucial in various applications, especially in variational principles, where they are used to express quantities that need to be optimized, such as energy or cost. Understanding functionals helps in the formulation of problems in calculus of variations and optimal control theory, where finding extrema or optimal solutions often involves analyzing these mappings.
Generalized momentum: Generalized momentum is a concept in physics that extends the idea of momentum to more complex systems described by generalized coordinates. It is defined as the partial derivative of the Lagrangian with respect to the generalized velocity, allowing for a deeper understanding of dynamics in systems with constraints and varying degrees of freedom.
Geodesics: Geodesics are the shortest paths between points in a given space, often found in the context of curved spaces or manifolds. They play a crucial role in calculus of variations, where the goal is to determine the path that minimizes or maximizes a functional, similar to how geodesics represent the shortest distance between two points.
Geometric Interpretation: Geometric interpretation refers to the visualization of mathematical concepts through geometric figures and shapes, making complex ideas more tangible and intuitive. It connects abstract mathematical constructs with visual elements, allowing for better comprehension of topics like curves, surfaces, and spaces, particularly when analyzing functions and their properties in variational calculus.
Hamilton's equations of motion: Hamilton's equations of motion are a set of first-order differential equations that describe the evolution of a dynamical system in classical mechanics. These equations are derived from the principle of least action and are intimately connected to the calculus of variations, where they provide a powerful alternative to Newton's laws of motion for analyzing systems with many degrees of freedom.
Hamiltonian: The Hamiltonian is a function that represents the total energy of a physical system in classical mechanics, expressed as the sum of its kinetic and potential energies. It plays a critical role in the formulation of the equations of motion and is fundamental in the transition from Lagrangian to Hamiltonian mechanics, allowing for powerful methods in solving physical problems.
Isoperimetric Problem: The isoperimetric problem is a classic question in the field of calculus of variations that seeks to determine the shape of the largest area that can be enclosed by a given perimeter. This problem connects various mathematical concepts, including optimization and geometric analysis, as it addresses how to maximize area while minimizing boundary length. The solutions to this problem lead to profound insights into the nature of shapes and their properties, often revealing that circles yield optimal results in these contexts.
Isoperimetric problems: Isoperimetric problems are mathematical questions that seek to determine the shape that has the maximum or minimum area (or volume) for a given perimeter (or surface area). These problems are closely tied to the calculus of variations, as they involve finding a function or a shape that minimizes or maximizes a certain quantity while satisfying constraints, often leading to the application of Euler-Lagrange equations.
Jacobi's Condition: Jacobi's Condition is a criterion used in the calculus of variations to determine whether a given extremal function is indeed a minimum or maximum of a functional. This condition involves the second variation of the functional, assessing whether it is positive or negative, which helps to identify the nature of the extremum. The significance of Jacobi's Condition lies in its ability to provide necessary conditions for optimality in variational problems, particularly when dealing with Euler-Lagrange equations.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an influential mathematician and physicist known for his contributions to the fields of calculus and mechanics, particularly through the formulation of variational principles. His work laid the foundation for modern calculus of variations and the development of Euler-Lagrange equations, which are crucial for solving extremum problems in various applications.
Lagrange: Lagrange refers to Joseph-Louis Lagrange, an influential mathematician known for his contributions to various fields including mechanics, number theory, and calculus of variations. In the context of calculus of variations, Lagrange's work is pivotal, particularly through the formulation of the Euler-Lagrange equations, which provide a method for finding functions that minimize or maximize functionals.
Lagrangian: The Lagrangian is a function that summarizes the dynamics of a system in classical mechanics, defined as the difference between kinetic and potential energy. It serves as the foundation for deriving equations of motion through the Euler-Lagrange equations, which arise from the principle of least action. In optimal control theory, the Lagrangian is crucial in formulating problems where one seeks to optimize a certain performance index, often leading to insights about control strategies and system behavior.
Least action principle: The least action principle is a fundamental concept in physics and mathematics stating that the path taken by a system between two states is the one for which the action is minimized. This principle connects deeply with calculus of variations, leading to the formulation of Euler-Lagrange equations, which provide a method to determine the function that minimizes or maximizes a certain functional associated with physical systems.
Legendre Condition: The Legendre Condition is a criterion used in the calculus of variations that helps determine whether a given extremal point is a local minimum or maximum for a functional. Specifically, it involves examining the second derivative of the Lagrangian function with respect to the first derivative of the function being optimized. This condition is crucial when working with Euler-Lagrange equations as it provides necessary conditions for identifying the nature of extremals.
Legendre condition: The Legendre condition is a criterion used in the calculus of variations to determine whether a functional has a local minimum. Specifically, it involves the second derivative of the Lagrangian with respect to the velocity variable and ensures that this second derivative is positive, indicating that the functional is convex in that direction. This condition helps identify optimal solutions and is essential for deriving the Euler-Lagrange equations.
Legendre Transform: The Legendre transform is a mathematical operation that transforms a function into another function by interchanging the roles of its variables, particularly in the context of convex functions. This transformation is crucial in various fields, especially in physics and optimization, as it allows the conversion of problems involving one variable into equivalent problems involving a dual variable. By expressing the original function in terms of its slope, the Legendre transform provides insights into the underlying structure and relationships of the functions involved.
Leonhard Euler: Leonhard Euler was an influential Swiss mathematician and physicist known for his extensive contributions to various fields including calculus, graph theory, topology, and mechanics. His work laid the groundwork for modern mathematics and introduced key concepts such as the Euler-Lagrange equations, which are fundamental in variational calculus and extremum problems.
Minimization Problem: A minimization problem is a type of optimization problem where the objective is to find the minimum value of a function, often subject to certain constraints. These problems are central to variational principles, where one seeks to minimize an integral or functional, and they relate closely to the calculus of variations, where techniques are developed to find functions that minimize given functionals.
Neumann boundary conditions: Neumann boundary conditions are a type of boundary condition used in partial differential equations that specify the values of the derivative of a function at the boundary of its domain. These conditions are essential for solving problems in various fields, including physics and engineering, where they often represent phenomena like heat flow or fluid dynamics at the edges of a physical system. They play a crucial role in the calculus of variations and the formulation of Euler-Lagrange equations, impacting how solutions to variational problems are characterized.
Noether's Theorem: Noether's Theorem is a fundamental principle in theoretical physics and mathematics that establishes a deep connection between symmetries and conservation laws. Specifically, it states that for every continuous symmetry of the action of a physical system, there corresponds a conservation law. This relationship is especially important in the context of the calculus of variations and Euler-Lagrange equations, where the action is minimized to derive equations of motion.
Optimal Control: Optimal control refers to the mathematical strategy for finding the best possible control inputs for a dynamic system over time, minimizing or maximizing a certain objective, typically expressed as a cost function. This concept connects closely with the calculus of variations, where the goal is to determine a function that optimizes a performance index subject to constraints.
Optimality conditions: Optimality conditions refer to a set of criteria that determine when a particular solution is optimal for a functional, especially in the context of calculus of variations. These conditions help identify functions that minimize or maximize a given functional, often leading to the application of Euler-Lagrange equations, which provide necessary conditions for optimality in variational problems.
Principle of Stationary Action: The principle of stationary action states that the path taken by a system between two states is the one for which the action is stationary (usually a minimum). This principle forms the foundation of the calculus of variations, which leads to the formulation of the Euler-Lagrange equations, providing a powerful method for deriving equations of motion in physics and other fields.
Second Variation: The second variation is a concept in the calculus of variations that deals with the change in a functional when the perturbation of the function used to compute it is taken into account. It helps assess the nature of the extrema (maximum or minimum) of functionals by examining how small changes in the function impact the functional's value, providing insights into stability and optimization.
Second variation: The second variation refers to a mathematical concept used in the calculus of variations to analyze the stability of a function that extremizes a functional. It is computed as the second derivative of the functional's value with respect to variations in the function itself and plays a crucial role in determining whether a critical point found using the first variation is indeed a minimum, maximum, or a saddle point.
Stationary point: A stationary point is a point on a function where the derivative is zero, indicating a potential local minimum, maximum, or saddle point. In the context of calculus of variations and Euler-Lagrange equations, stationary points represent critical points of functional that are candidates for minimizing or maximizing the functional's value.
Sufficient conditions for extremum: Sufficient conditions for extremum refer to a set of criteria that guarantee a function attains a local minimum or maximum at a given point. In the context of calculus of variations and the Euler-Lagrange equations, these conditions help determine when a functional has an extremum based on its derivatives and the nature of the function involved. Understanding these conditions is essential for solving problems related to optimizing functionals, particularly in variational problems.
Trajectory optimization: Trajectory optimization is a mathematical approach used to determine the optimal path or trajectory that a dynamic system should follow to achieve a specific goal while minimizing or maximizing a given performance criterion. This concept is closely linked to finding solutions that adhere to the governing equations of motion and constraints imposed by the system, and it plays a crucial role in fields like robotics, aerospace, and control theory.
Variational Method: The variational method is a mathematical technique used to find extrema (minimum or maximum values) of functionals, which are often integral expressions that depend on functions and their derivatives. This method is closely related to the calculus of variations, where one derives equations that a function must satisfy in order to minimize or maximize a given functional, typically leading to the Euler-Lagrange equations. It plays a crucial role in physics and engineering, particularly in problems involving optimal control and in determining the states of physical systems.
Variational problem: A variational problem is a mathematical question that seeks to find a function or a set of functions that minimize or maximize a certain quantity, often expressed as an integral. These problems are essential in various fields, as they involve finding optimal solutions subject to specific constraints. The solutions typically rely on techniques from calculus of variations and are closely connected to Euler-Lagrange equations, which provide necessary conditions for optimality. Furthermore, variational problems are foundational in applications like optimal control theory, where they help determine the best way to control dynamic systems.
Weierstrass-Erdmann Corner Conditions: The Weierstrass-Erdmann Corner Conditions are specific criteria used in the calculus of variations to ensure the optimality of a functional when the function has discontinuities in its derivatives. These conditions provide necessary criteria that must be satisfied at points where the path being optimized has corners or sharp changes in direction. They are essential for formulating the Euler-Lagrange equations correctly when considering such discontinuities, thus allowing for more generalized solutions in variational problems.
Weierstrass-Erdmann corner conditions: The Weierstrass-Erdmann corner conditions are criteria used in the calculus of variations to ensure that a curve, or path, is optimal at points where it has corners or discontinuities in its derivatives. These conditions specify the necessary behavior of the first derivative of the extremal function at the corners, helping to identify if a path truly minimizes or maximizes a functional despite having abrupt changes in direction.
Weierstrass' Theorem: Weierstrass' Theorem states that any continuous function defined on a closed and bounded interval attains both its maximum and minimum values within that interval. This important result connects to the concepts of calculus of variations by establishing conditions under which optimal solutions exist, particularly when dealing with functionals that depend on continuous functions.
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