pyoculus.problems.base_problem.BaseProblem Class Reference

Abstract class that used to drive calculation. More...

Inheritance diagram for pyoculus.problems.base_problem.BaseProblem:

Public Member Functions
	__init__ (self)
	f (self, t, y, *args)
	Returns ODE RHS.
	f_tangent (self, t, y, *args)
	Returns ODE RHS, with tangent.
	convert_coords (self, coord1)
	Converts coordinates (for example \(s,\theta,\zeta\) to \(R,Z,\varphi\))

Public Attributes
int	problem_size = 2
str	poincare_plot_type = "yx"
str	poincare_plot_xlabel = "y"
str	poincare_plot_ylabel = "x"

Detailed Description

Abstract class that used to drive calculation.

This is an abstract class, should never be used as an instance.

All problems should be a derived type of BaseProblem. They should contain the ODEs to solve, as specified by the Hamiltonian system of that problem.

Hamilton's equation

1.5D

If the Hamiltonian system is 1.5 dimensional, the Hamiltonian takes the form \(H=H(p,q,t)\). The Hamilton's equations are given by

\[ \frac{dq}{dt} = \frac{\partial H}{\partial p}, \quad \frac{dp}{dt} = -\frac{\partial H}{\partial q}. \quad\]

2D

If the Hamiltonian system is two-dimensional, the Hamiltonian takes the form \(H=H(p_1,q_1,p_2,q_2)\). The Hamilton's equations are given by

\[ \frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}, \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}, \quad\]

in which \(i=1,2\).

The Hamiltonian \(H\) itself is a constant of motion, i.e. \(dH/dt=0\).

Given \( p_1, q_1, q_2\), we can invert \(H=H(p_1,q_1,p_2,q_2)=\mbox{const} \) to get \(p_2 = p_2(p_1, q_1, q_2)\).

If further \(q_2 \) is monotonically increasing with \(t\), one can replace \(t\) by \(q_2\) and get

\[ \frac{dq_1}{dq_2} = \left.\frac{\frac{\partial H}{\partial p_1}}{\frac{\partial H}{\partial p_2}} \right|_{p_2(p_1, q_1, q_2)}, \]

\[ \frac{dp_1}{dq_2} = \left.\frac{-\frac{\partial H}{\partial q_1} }{\frac{\partial H}{\partial p_2}}\right|_{p_2(p_1, q_1, q_2)}, \]

after computing which the value of \(p_2\) is substituted into these equations.

General ODEs with two variables

The Hamilton's equations, after converted can be combined and written as

\[ \frac{d y_1}{dt} = F_1(t, y_1, y_2),\]

\[ \frac{d y_2}{dt} = F_2(t, y_1, y_2).\]

The tangent of the above ODEs are given by

\[ \frac{d}{dt} \begin{bmatrix}\Delta y_1\\ \Delta y_2 \end{bmatrix} = \begin{bmatrix} \partial_{y_1}F_1 & \partial_{y_2}F_1 \\ \partial_{y_1}F_2 & \partial_{y_2}F_2 \end{bmatrix} \cdot \begin{bmatrix}\Delta y_1\\ \Delta y_2 \end{bmatrix}, \]

where the partial derivatives are evaluated at \(y_1(t), y_2(t)\) as the solution of the original ODEs.

Implementing a Problem class

All problem classes should inherit the BaseProblem class.

class SomeProblem(BaseProblem):
    def __init__(self, params):
        """some codes to initialize the problem"""
    def f(self, t, y, *args):
        """some codes compute the RHS (F) of the ODEs given t and y"""
    def f_tangent(self, t, y, *args):
        """some codes compute the ODEs and the tangent the ODEs given t and y, y contains two tangent vectors"""
    def convert_coords(self, coord1):
        """some codes compute the coordinate transformation"""

The member function f should be implemented in any case. The tangent and convert_coords functions are optional for some solvers.

Constructor & Destructor Documentation

__init__()

pyoculus.problems.base_problem.BaseProblem.__init__

(

self

)

Reimplemented in pyoculus.problems.cartesian_bfield.CartesianBfield, pyoculus.problems.cylindrical_bfield.CylindricalBfield, pyoculus.problems.cylindrical_problem.CylindricalProblem, pyoculus.problems.m3dc1_bfield.M3DC1Bfield, pyoculus.problems.qfm_bfield.QFMBfield, pyoculus.problems.simsgeo_biot_savart.SimsgeoBiotSavart, pyoculus.problems.spec_bfield.SPECBfield, pyoculus.problems.toroidal_bfield.ToroidalBfield, pyoculus.problems.toroidal_problem.ToroidalProblem, and pyoculus.problems.two_waves.TwoWaves.

Member Function Documentation

convert_coords()

pyoculus.problems.base_problem.BaseProblem.convert_coords	(		self,
			coord1 )

Converts coordinates (for example \(s,\theta,\zeta\) to \(R,Z,\varphi\))

Parameters

coords1

the coordinates to convert

Returns

the converted coordinates

Reimplemented in pyoculus.problems.qfm_bfield.QFMBfield, pyoculus.problems.spec_bfield.SPECBfield, pyoculus.problems.toroidal_problem.ToroidalProblem, and pyoculus.problems.two_waves.TwoWaves.

f()

pyoculus.problems.base_problem.BaseProblem.f	(		self,
			t,
			y,
		*	args )

Returns ODE RHS.

Parameters

t	time in ODE
y	variables in ODE
*args	parameter for the ODE

Returns

the RHS of the ODE

Reimplemented in pyoculus.problems.cylindrical_problem.CylindricalProblem, pyoculus.problems.toroidal_bfield.ToroidalBfield, and pyoculus.problems.toroidal_problem.ToroidalProblem.

f_tangent()

pyoculus.problems.base_problem.BaseProblem.f_tangent	(		self,
			t,
			y,
		*	args )

Returns ODE RHS, with tangent.

Parameters

t	time in ODE
y	\(\bf y\) variables in ODE and \(\Delta \mathbf{y}_1\), \(\Delta \mathbf{y}_2\)
*args	parameter for the ODE

Returns

the RHS of the ODE, with tangent

Reimplemented in pyoculus.problems.cylindrical_problem.CylindricalProblem, pyoculus.problems.toroidal_bfield.ToroidalBfield, and pyoculus.problems.toroidal_problem.ToroidalProblem.

Member Data Documentation

poincare_plot_type

str pyoculus.problems.base_problem.BaseProblem.poincare_plot_type = "yx"

poincare_plot_xlabel

str pyoculus.problems.base_problem.BaseProblem.poincare_plot_xlabel = "y"

poincare_plot_ylabel

str pyoculus.problems.base_problem.BaseProblem.poincare_plot_ylabel = "x"

problem_size

int pyoculus.problems.base_problem.BaseProblem.problem_size = 2

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The documentation for this class was generated from the following file:

• pyoculus/problems/base_problem.py