The Hamiltonian for a system described by the generalized coordinate x and generalised momentum p is
\(H=α x^2 p+\frac{p^2}{2(1+2 β x)}+\frac{1}{2} ω^2 x^2\)
where α, β and ω are constants. The corresponding Lagrangian is
1
\(\frac{1}{2}\left(\dot{x}-α x^2\right)^2(1+2 β x)-\frac{1}{2} ω^2 x^2\)
2
\(\frac{1}{2(1+2 β x)} \dot{x}^2-\frac{1}{2} ω^2 x^2-α x^2 \dot{x}\)
3
\(\frac{1}{2}\left(\dot{x}^2-α^2 x\right)^2(1+2 β x)-\frac{1}{2} ω^2 x^2\)
4
\(\frac{1}{2(1+2 β x)} \dot{x}^2-\frac{1}{2} ω^2 x^2+α x^2 \dot{x}\)