Treacherous Euler-Lagrange equation(y′)2=2(1−cos⁡(y)) where y is a function of x subjected to boundary conditions y(x)→0...

Patricia Crane

Patricia Crane

Answered

2022-01-18

Treacherous Euler-Lagrange equation
(y)2=2(1cos(y)) where y is a function of x subjected to boundary conditions y(x)0 as x and y(x)2π as x+, how might I find all its solutions?

Answer & Explanation

Karen Robbins

Karen Robbins

Expert

2022-01-19Added 49 answers

2y(x)y(x)=2y(x)siny(x)
which implies y(x)=siny(x). After substitution y(x)=πθ(x) this translates into θ(x)=sinθ(x) which is the pendulum equation. Your boundary condition require that limxθ(x)=π and limx+θ(x)=π. Hence the solution is not periodic.
Becky Harrison

Becky Harrison

Expert

2022-01-20Added 40 answers

Use 1cosy=1(cos2y2sin2y2)=2sin2y2.
alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

1cos(y)=2sin2(y2). Hence, y2=4sin2(y2)y=±2sin(y2)

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