Table of Contents
- Introduction
- What Is the Calculus of Variations?
- Functionals and Their Extremization
- The Euler-Lagrange Equation
- Derivation of the Euler-Lagrange Equation
- Boundary Conditions
- Examples of Variational Problems
- Variational Principles in Physics
- Lagrangian Mechanics and the Principle of Least Action
- Constraints and the Lagrange Multipliers
- Variations with Several Functions and Higher Derivatives
- Hamilton’s Principle
- Noether’s Theorem and Symmetries
- Applications in Physics and Engineering
- Conclusion
1. Introduction
The calculus of variations is a mathematical method used to find the function (or functions) that makes a given quantity — usually an integral — stationary (minimum, maximum, or saddle point). It is the foundation of classical mechanics, optics, and many areas of physics and engineering.
2. What Is the Calculus of Variations?
Unlike traditional calculus, which finds the extrema of functions, the calculus of variations seeks the extrema of functionals — mappings from a space of functions to the real numbers.
A functional typically has the form:
\[
J[y] = \int_{a}^{b} L(x, y(x), y'(x)) \, dx
\]
Our goal: find a function \( y(x) \) such that \( J[y] \) is minimized (or maximized).
3. Functionals and Their Extremization
Given a functional \( J[y] \), we consider small variations of \( y \):
\[
y(x) \to y(x) + \epsilon \eta(x)
\]
where \( \eta(x) \) is an arbitrary differentiable function vanishing at the endpoints: \( \eta(a) = \eta(b) = 0 \), and \( \epsilon \) is small.
4. The Euler-Lagrange Equation
The central result of the calculus of variations is the Euler-Lagrange equation:
\[
\frac{\partial L}{\partial y} – \frac{d}{dx} \left( \frac{\partial L}{\partial y’} \right) = 0
\]
Any function \( y(x) \) that extremizes the functional must satisfy this differential equation.
5. Derivation of the Euler-Lagrange Equation
Start with:
\[
J[y + \epsilon \eta] = \int_a^b L(x, y + \epsilon \eta, y’ + \epsilon \eta’) dx
\]
Differentiate with respect to \( \epsilon \), set \( \epsilon = 0 \), and use integration by parts. The vanishing of the first variation \( \delta J \) leads directly to the Euler-Lagrange equation.
6. Boundary Conditions
- Fixed endpoints: \( y(a) \) and \( y(b) \) fixed → standard Euler-Lagrange
- Free endpoints: Leads to natural boundary conditions:
\[
\left. \frac{\partial L}{\partial y’} \right|_{x=a}^{x=b} = 0
\]
7. Examples of Variational Problems
- Shortest path between two points: yields a straight line
- Brachistochrone problem: find the curve of fastest descent
- Catenary: shape of a hanging chain under gravity
Each problem uses the Euler-Lagrange equation to derive the solution function.
8. Variational Principles in Physics
Many physical laws arise from variational principles. These include:
- Fermat’s principle (optics)
- Hamilton’s principle (mechanics)
- Least action principle (field theory)
9. Lagrangian Mechanics and the Principle of Least Action
In Lagrangian mechanics, the action is:
\[
S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt
\]
Where \( L = T – V \) is the Lagrangian. The path taken by a system between two configurations is the one for which \( S \) is stationary.
10. Constraints and the Lagrange Multipliers
When constraints \( f_i(x, y, y’) = 0 \) are present, we use Lagrange multipliers:
\[
J[y] = \int_a^b \left( L + \lambda f \right) dx
\]
The resulting Euler-Lagrange equations now include terms involving \( \lambda \).
11. Variations with Several Functions and Higher Derivatives
For multiple functions \( y_i(x) \), we get a system of Euler-Lagrange equations:
\[
\frac{\partial L}{\partial y_i} – \frac{d}{dx} \left( \frac{\partial L}{\partial y_i’} \right) = 0
\]
For higher derivatives (e.g., \( y”(x) \)), the equation generalizes accordingly.
12. Hamilton’s Principle
Hamilton’s principle states:
The actual path taken by a physical system between two times is the one that minimizes the action.
This forms the bridge to Hamiltonian mechanics and quantum field theory.
13. Noether’s Theorem and Symmetries
Noether’s theorem connects symmetries of the action to conserved quantities:
- Time symmetry → energy conservation
- Space symmetry → momentum conservation
- Rotational symmetry → angular momentum conservation
It is a profound result rooted in the calculus of variations.
14. Applications in Physics and Engineering
- Deriving equations of motion in classical mechanics
- Field equations in electromagnetism and general relativity
- Optimal control in engineering
- Shape optimization in mechanical structures
15. Conclusion
The calculus of variations is a powerful tool that unifies physics, mathematics, and engineering. From shortest paths to fundamental laws, it provides the framework to derive governing equations from extremal principles.
Understanding variational methods is essential for theoretical physicists, applied mathematicians, and engineers alike.
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