pyHamSys

pyHamSys is a Python package for scientific computations involving Hamiltonian systems

Symplectic Integrators

PyHamSys includes a class SymplecticIntegrator containing the following integrators:

Names of integrators include:

• 'Verlet' (order 2)
• 'Forest-Ruth' (order 4) from Forest, Ruth, Physica D 43, 105 (1990)
• 'EFRL', 'PEFRL' or 'VEFRL' (order 4) from Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002). Optimized for H = A + B; 'PEFRL' and 'VEFRL' are for splitting H = A(p) + B(q)
• 'BM4' (order 4) refers to S₆ from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)
• 'BM6' (order 6) refers to S₁₀ from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)
• 'RKN4b' (order 4) refers to SRKN₆^b from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002) for splitting H = A(p) + B(q)
• 'RKN6b' (order 6) refers to SRKN₁₁^b from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002) for splitting H = A(p) + B(q)
• 'RKN6a' (order 6) refers to SRKN₁₄^a from Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002) for splitting H = A(p) + B(q)
• 'ABA104' (order (10,4)) from S. Blanes, F. Casas, A. Farrés, J. Laskar, J. Makazaga, A. Murua, Appl. Numer. Math. 68, 58 (2013) for splitting H = A + ε B
• 'ABA864' (order (8,6,4)) from S. Blanes, F. Casas, A. Farrés, J. Laskar, J. Makazaga, A. Murua, Appl. Numer. Math. 68, 58 (2013) for splitting H = A + ε B
• 'ABA1064' (order (10,6,4)) from S. Blanes, F. Casas, A. Farrés, J. Laskar, J. Makazaga, A. Murua, Appl. Numer. Math. 68, 58 (2013) for splitting H = A + ε B

Usage: integrator = SymplecticIntegrator(name, step, order) where name is one of the names listed above, step is the time step of the integrator (float), and order is the order of the splitting, so 1 or -1 depending on the order AB or BA of the splitting.

The function integrator._integrate integrates the Hamiltonian flow by one step.

The function integrator.integrate integrates the Hamiltonian flow from the initial conditions specified by the initial state vector y using integrator, one of the selected symplectic splitting integrators. It returns the value of y at times defines by the float, list or array times.

Parameters:

• chi : function of (h, y), y being the state vector. Function returning exp(h X_n)...exp(h X₁) y.
• chi_star : function of (h, y). Function returning exp(h X₁)...exp(h X__n) y.
• y : initial state vector (numpy array)
• times : times at which the values of the state vector are computed
• command : function of (t, y). Function to be run at each time step (e.g., plotting an observable associated with the state vector, or register specific events).
• autonomous : boolean. If autonomous is False, the state vector y should be of the form y = [t, x], where the first coordinate is time.

Returns:

• If times is a float of integer, the output is a tuple (t, y or x) where y is the value of the state vector and y = [t, x] if autonomous is False.
• If times is a list or array, returns the times and values of y or x at times.
• If times is a list or array with a single element, returns the times and values of y or x at all computed times.

References:

• McLachlan, R.I, 2022, Tuning symplectic integrators is easy and worthwhile, arxiv:2104.10269

Installation:

pip install pyhamsys