~Sam~ said:Homework Statement
Compute the Euler-Lagrange for:
∫y(y')2+y2sinx dx
Homework Equations
[itex]\frac{∂L}{∂y}[/itex]-[itex]\frac{d}{dx}[/itex] ([itex]\frac{∂L}{∂y'}[/itex])
The Attempt at a Solution
Usual computation by hand gives me y'2+2ysin(x) - 2yy'', but Mathematica says it's -y'2-2yy''. Am I doing something wrong?
Ray Vickson said:I get ##L_y - D_x L_{y'} = 2 y \sin(x) - y'^2 - 2 y y''##.
~Sam~ said:Hmmm I looked it over again and I get y'2+2ysin(x)-2(yy''+y'2)
For the first part I get: y'2+2ysinx
Second partial: 2yy'
Then derivative of thatis : 2(yy''+y'2)
Ahh so it's the same as you got ty. Why there is no sin in mathematica solution T.T
Ray Vickson said:I don't use or have access to Mathematica (I use Maple instead), so I have no idea what instructions you entered and what Mathematica did with them.
~Sam~ said:Regardless, I assume that's 2ysin(x)−y′2−2yy′ correct Euler-Lagrange then?
Euler-Lagrange equations are a set of differential equations used to find the stationary points of a functional. They are commonly used in the fields of mathematics, physics, and engineering.
Euler-Lagrange equations are significant because they allow us to find the minimum or maximum value of a functional. This is useful in many applications, such as finding the shortest path between two points or determining the path of a particle that minimizes energy.
Euler-Lagrange equations are derived from the calculus of variations, which is a branch of mathematics that deals with finding the extremum of a functional. The equations involve taking derivatives of the functional with respect to the variables and setting them equal to zero.
Euler-Lagrange equations have a wide range of applications in physics, engineering, and economics. They are used in the study of mechanics, electromagnetism, fluid dynamics, and control theory, among others.
Euler-Lagrange equations have limitations in cases where the functional is not differentiable or the boundary conditions are not well-defined. They also do not provide a global minimum or maximum, only local extrema. In some cases, other methods such as the Hamiltonian formalism may be more suitable.