# Finding minimum of the function W under the constraint H = E

See also perturbative solution of the problem.

Here, the functions H and W depend on two variables, x and y.
They include a harmonic part, cubic, and quartic anharmonicity:

H = H0 + H1 + H2, W = W0 + W1 + W2.

Harmonic parts are quadratic functions:

H0(x,y) = x2/2 + y2/2, W0(x,y) = wx2(x-x0)2/2 + wy2(y-y0)2/2.

We call wx and wy "frequencies", and x0 and y0 "displacements".

 Enter the frequencies wx = wy = Enter the displacements x0 = y0 =

Cubic anharmonicity is a cubic polynomial:

H1(x,y) = h30x3 + h21x2y + h12xy2 + h03y3,
W1(x,y) = w30(x-x0)3 + w21(x-x0)2(y-y0) + w12(x-x0)(y-y0)2 + w03(y-y0)3.

 Enter the parameters of the cubic anharmonicity h30 = h21 = h12 = h03 = w30 = w21 = w12 = w03 =

Quartic anharmonicity is a fourth degree polynomial:

H2(x,y) = h40x4 + h31x3y + h22x2y2 + h13xy3 + h04y4,
W2(x,y) = w40(x-x0)4 + w31(x-x0)3(y-y0) + w22(x-x0)2(y-y0)2 + w13(x-x0)(y-y0)3 + w04(y-y0)4.

 Unless H2 = 0 and W2 = 0 enter the parameters of the quartic anharmonicity h40 = h31 = h22 = h13 = h04 = w40 = w31 = w22 = w13 = w04 =

The "energy" is always one, E = 1.

 Now, minimize the function W under the constraint H = E Default response

Download Mathematica programs used for this calculation

A paper related to minimizations of quadratic functions

Results of work at BGU with B. Segev

More on-line calculations