Functional Derivative CalculatorĀ
Functional Derivative Calculator computes the functional derivative \\( \frac{\delta J}{\delta y} \\) for a functional \\( J[y] = \int_a^b F(x, y, y’) \, dx \\), displaying steps with MathJax.
Functional Derivative Calculator
Functional Derivative Calculator computes the functional derivative \\( \frac{\delta J}{\delta y} \\) for a functional \\( J[y] = \int_a^b F(x, y, y’) \, dx \\). Input the integrand and integration bounds to see the derivative with MathJax-rendered steps. Results are copyable, with sharing and embedding options for calculus of variations students.
Example 1: Arc Length Functional
Functional: \\( J[y] = \int_0^1 \sqrt{1 + (y’)^2} \, dx \\).
Step 1: Define the functional.
\\( F(x, y, y’) = \sqrt{1 + (y’)^2} \\).
Step 2: Compute partial derivatives.
\\( \frac{\partial F}{\partial y} = 0 \\), \\( \frac{\partial F}{\partial y’} = \frac{y’}{\sqrt{1 + (y’)^2}} \\).
Step 3: Compute the total derivative.
\\( \frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = \frac{d}{dx} \left( \frac{y’}{\sqrt{1 + (y’)^2}} \right) = \frac{y” (1 + (y’)^2) – y’ \cdot 2 y’ y”}{[1 + (y’)^2]^{3/2}} = \frac{y”}{[1 + (y’)^2]^{3/2}} \\).
Step 4: Apply Euler-Lagrange equation.
\\( \frac{\delta J}{\delta y} = \frac{\partial F}{\partial y} – \frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = 0 – \frac{y”}{[1 + (y’)^2]^{3/2}} = -\frac{y”}{[1 + (y’)^2]^{3/2}} \\).
Step 5: Conclusion.
The functional derivative is \\( -\frac{y”}{[1 + (y’)^2]^{3/2}} \\).
Example 2: Action Functional
Functional: \\( J[y] = \int_0^1 [y^2 + (y’)^2] \, dx \\).
Step 1: Define the functional.
\\( F(x, y, y’) = y^2 + (y’)^2 \\).
Step 2: Compute partial derivatives.
\\( \frac{\partial F}{\partial y} = 2y \\), \\( \frac{\partial F}{\partial y’} = 2y’ \\).
Step 3: Compute the total derivative.
\\( \frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = \frac{d}{dx} (2y’) = 2y” \\).
Step 4: Apply Euler-Lagrange equation.
\\( \frac{\delta J}{\delta y} = \frac{\partial F}{\partial y} – \frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = 2y – 2y” \\).
Step 5: Conclusion.
The functional derivative is \\( 2y – 2y” \\).