celeries.mmr.MMR.fp_modes

MMR.fp_modes(v_fp, delta)

Compute the eigenvalues and eigenmodes around a fixed point.

This method allows to perform a symplectic change of coordinates from the variables v = (y, phi, x, dL) to new variables \(\mathbf{u} = (u, \tilde{u})\): \(v = Q \mathbf{u}\).

The eigenmodes are provided with all “angles” \(u\) first and then all associated “actions” \(\tilde{u}\). The eigenvalues (and corresponding eigenmodes) are sorted from the larger to the smaller absolute value. For elliptical fixed points, the change of variables is such that: \(\tilde{u} = -i \bar{u}\). The frequency (imaginary part of the eignevalue) corresponding to \(u_i\) can be positive or negative depending on the fixed point being a local minimum or maximum of the Hamiltonian in the direction of the corresponding eigenmode.

Parameters:

v_fp(4*npla-4,) ndarray

Coordinates of the fixed point.

deltafloat

Value of delta (normalized by Gamma).

Returns:

eig_val(4*npla-4,) ndarray

Eigenvalues.

eig_vec(4*npla-4, 4*npla-4) ndarray

Matrix Q of the symplectic change of variables.