calculus of variations

Thân thiện

A mathematician uses the calculus of variations to find the optimal curve.

Definition

1. Noun:
   • A branch of mathematics concerned with finding functions that optimize (maximize or minimize) the value of a definite integral: It deals with problems where the quantity to be optimized depends on an entire function, not just a few variables. The goal is to find the function that makes an integral take on an extreme value (either the largest or smallest possible).

Usage Examples

• Noun:
  • The calculus of variations is essential for deriving the equations of motion in Lagrangian mechanics.
  • Solving this optimal path problem requires applying the calculus of variations.
  • Euler and Lagrange made foundational contributions to the calculus of variations.

Advanced Usage

• "The fundamental lemma of the calculus of variations": A crucial theorem used to derive the Euler-Lagrange equation, which is the central differential equation of the field.
  • The proof relies on the fundamental lemma of the calculus of variations.

Variants and Related Words

• Variational calculus: Another name for the calculus of variations.
  • He is an expert in variational calculus.
• Functional: In this context, a function whose argument is itself a function (e.g., the integral to be optimized).
  • The problem is to minimize a specific functional.

Synonyms

• Variational analysis: A broader term sometimes used synonymously, especially in applied contexts.

Related Phrases

• Euler-Lagrange equation: The primary differential equation derived in the calculus of variations whose solutions are the stationary functions.
  • The optimal shape is found by solving the Euler-Lagrange equation.
• Principle of least action: A foundational physics principle formulated using the calculus of variations.
  • The principle of least action states that the path taken by a system is the one that minimizes the action integral.

Học thuật

Noun

1. the calculus of maxima and minima of definite integrals