# Second Variation: Euler-Lagrange Equation & \delta^2 S

In summary, the Second Variation in the context of Euler-Lagrange Equation is the second derivative of the action functional, used to find critical points and determine the minimum or maximum value. It is used to determine the stability of a critical point and is significant in mathematical physics and the principle of least action. It can be extended to multiple variables and relates to small variations in calculus of variations, both of which are used to find extremal solutions and determine stability.
Let be a functional S so $$\delta S =0$$ give the Euler-Lagrange equation where:

$$S= \int_{a}^{b}dtL(q,\dot q, t)$$

My question is ..How the "second variation" $$\delta ( \delta S )=0= \delta ^{2} S$$ defined??.. in order we could decide if a function q=q(t) is either a maximum or a minimum point of function space.. thanks.

How's the "first variation" defined ?

Daniel.

The first variation of S (are you physicist..i say so because i use to see you in the QM forum ) are the Euler-Lagrange equation defined via the functional derivative

## 1. What is the Second Variation in the context of Euler-Lagrange Equation?

The Second Variation in the context of Euler-Lagrange Equation refers to the second derivative of the action functional, which is used to find the critical points of the functional. It is a necessary condition for a functional to have a minimum or maximum value.

## 2. How is the Second Variation used in the Euler-Lagrange Equation?

The Second Variation is used in the Euler-Lagrange Equation to determine the stability of a critical point of the action functional. It helps in identifying whether the critical point is a minimum, maximum, or a saddle point.

## 3. What is the significance of the Second Variation in mathematical physics?

The Second Variation is significant in mathematical physics as it plays a crucial role in the principle of least action. It allows us to find the extremal solution for a given functional and determine whether it corresponds to a minimum, maximum, or a saddle point.

## 4. Can the Second Variation be extended to multiple variables?

Yes, the Second Variation can be extended to multiple variables in the context of the Euler-Lagrange Equation. It involves taking the second derivative of the action functional with respect to each variable and setting it equal to zero to find the critical points.

## 5. How does the Second Variation relate to the concept of small variations in calculus of variations?

In the calculus of variations, small variations refer to infinitesimal changes in the independent variables, while the Second Variation refers to the second derivative of the action functional. Together, they are used to find the extremal solutions for a given functional and determine its stability.

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