# Category:Calculus of Variations

This category contains results about Calculus of Variations.

Definitions specific to this category can be found in Definitions/Calculus of Variations.

**Calculus of variations** is the subfield of analysis concerned maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.

## Pages in category "Calculus of Variations"

The following 86 pages are in this category, out of 86 total.

### A

### C

- Central Field is Field of Functional
- Condition for Differentiable Functional to have Extremum
- Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative
- Conditions for Extremal Embedding in Field of Functional
- Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation
- Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 1
- Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 2
- Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent
- Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p
- Conditions for Integral Functionals to have same Euler's Equations
- Conditions for Limit Function to be Limit Minimizing Function of Functional
- Conditions for Strong Minimum of Functional
- Conditions for Transformation to be Canonical
- Convexity of Function implies Convexity of its Legendre Transform

### D

### E

- Equivalence of Definitions of Conjugate Point
- Equivalence of Definitions of Weierstrass E-Function
- Euler's Equation for Vanishing Variation in Canonical Variables
- Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations
- Euler's Equation/Independent of x
- Euler's Equation/Independent of y
- Euler's Equation/Independent of y'
- Euler's Equation/Integrated wrt Length Element

### G

- General Variation of Integral Functional/Dependent on N Functions
- General Variation of Integral Functional/Dependent on N Functions/Canonical Variables
- General Variation of Integral Functional/Dependent on n Variables
- Geodesic Equation/2d Surface Embedded in 3d Euclidean Space
- Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder

### I

### J

### L

### N

- Necessary and Sufficient Condition for Boundary Conditions to be Self-adjoint
- Necessary and Sufficient Condition for First Order System to be Field for Functional
- Necessary and Sufficient Condition for First Order System to be Field for Second Order System
- Necessary and Sufficient Condition for First Order System to be Mutually Consistent
- Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation
- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite
- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Dependent on N Functions
- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma
- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2
- Necessary Condition for Integral Functional to have Extremum for given function
- Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions
- Necessary Condition for Integral Functional to have Extremum for given function/Dependent on n Variables
- Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on N-th Derivative of Function
- Necessary Condition for Integral Functional to have Extremum for given function/Lemma
- Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point
- Necessary Condition for Integral Functional to have Extremum/Two Variables
- Necessary Condition for Integral Functional to have Extremum/Two Variables/Lemma
- Necessary Condition for Twice Differentiable Functional to have Minimum
- Noether's Theorem (Calculus of Variations)
- Noether's Theorem (Hamiltonian Mechanics)
- Nonnegative Quadratic Functional implies no Interior Conjugate Points

### P

### S

- Simple Variable End Point Problem
- Simple Variable End Point Problem/Endpoints on Curves
- Simplest Variational Problem
- Simplest Variational Problem with Subsidiary Conditions
- Simplest Variational Problem with Subsidiary Conditions for Curve on Surface
- Sturm-Liouville Problem/Unit Weight Function
- Sturm-Liouville Problem/Unit Weight Function/Lemma
- Sufficient Condition for Twice Differentiable Functional to have Minimum
- Sufficient Conditions for Weak Extremum