pyhamsys 0.43

Some tools for Hamiltonian systems

pyHamSys

pyHamSys is a Python package for scientific computing involving Hamiltonian systems

Installation within a Python virtual environment:

python3 -m pip install pyhamsys

For more information on creating a Python virtual environment, click here.

Symplectic Integrators

pyHamSys includes a class SymplecticIntegrator containing the following symplectic splitting integrators:

• Verlet (order 2, all purpose), also referred to as Strang or Störmer-Verlet splitting
• From Forest, Ruth, Physica D 43, 105 (1990):
  • FR (order 4, all purpose)
• From Yoshida, Phys. Lett. A 150, 262 (1990):
  • Yo#: # should be replaced by an even integer, e.g., Yo6 for 6th order symplectic integrator (all purpose)
  • Yos6: (order 6, all purpose) optimized symplectic integrator (solution A from Table 1)
• From McLachlan, SIAM J. Sci. Comp. 16, 151 (1995):
  • M2 (order 2, all purpose)
  • M4 (order 4, all purpose)
• From Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002):
  • EFRL (order 4) optimized for H = A + B
  • PEFRL and VEFRL (order 4) optimized for H = A(p) + B(q). For PEFRL, chi should be exp(h X_A)exp(h X_B). For VEFRL, chi should be exp(h X_B)exp(h X_A).
• From Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002):
  • BM4 (order 4, all purpose) refers to S₆
  • BM6 (order 6, all purpose) refers to S₁₀
  • RKN4b (order 4) refers to SRKN₆^b optimized for H = A(p) + B(q). Here chi should be exp(h X_B)exp(h X_A).
  • RKN6b (order 6) refers to SRKN₁₁^b optimized for H = A(p) + B(q). Here chi should be exp(h X_B)exp(h X_A).
  • RKN6a (order 6) refers to SRKN₁₄^a optimized for H = A(p) + B(q). Here chi should be exp(h X_A)exp(h X_B).
• From Blanes, Casas, Farrés, Laskar, Makazaga, Murua, Appl. Numer. Math. 68, 58 (2013):
  • ABA104 (order (10,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).
  • ABA864 (order (8,6,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).
  • ABA1064 (order (10,6,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).

All purpose integrators are for any splitting of the Hamiltonian H=∑_k A_k in any order of the functions A_k. Otherwise, the order of the operators is specified for each integrator. These integrators are used in the functions solve_ivp_symp and solve_ivp_sympext by specifying the entry method (default is BM4).

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HamSys class

Parameters

• ndof : number of degrees of freedom of the Hamiltonian system 'ndof' should be an integer or half an integer. Half integers denote an explicit time dependence.
• complex : bolean
  If False, the dynamical variables (q, p) are real and canonically conjugate. If True, the dynamical variables are (ψ, ψ^*) where $\psi=(q + i p)/\sqrt{2}$. Default is False.

Attributes

• hamiltonian : callable
  A function of (t, y) which returns the Hamiltonian H(t,y) where y is the state vector.
• y_dot : callable
  A function of (t, y) which returns {y,H(t,y)} where y is the state vector and H is the Hamiltonian. In canonical coordinates (used, e.g., in solve_ivp_sympext) where y = (q, p), this function returns (∂H/∂p, -∂H/∂q).
• k_dot : callable
  A function of (t, y) which returns {k,H(t,y)} = -∂H/∂t where k is canonically conjugate to t and H is the Hamiltonian.

Functions

• compute_vector_field : from a callable function (Hamiltonian in canonical coordinates) written with symbolic variables (SymPy), computes the vector fields, y_dot and k_dot.

  Determine Hamilton's equations of motion from a given scalar function –the Hamiltonian– H(q, p, t) where q and p are respectively positions and momenta.

  Parameters

  • hamiltonian : callable Function H(q, p, t) –the Hamiltonian expressed in symbolic variables–, expressed using SymPy functions.
  • output : bool, optional If True, displays the equations of motion. Default is False.

  The function compute_vector_field determines the HamSys function attributes y_dot and k_dot to be used in solve_ivp_sympext. The derivatives are computed symbolically using SymPy.

• compute_energy : callable A function of sol –a solution provided by solve_ivp_sympext– and maxerror, a boolean indicating whether the maximum error in total energy is given (if True) or all the values of the total energy (if False).

  Parameters

  • sol : OdeSolution
    Solution provided by solve_ivp_sympext.
  • maxerror : bool, optional
    Default is True.

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solve_ivp_symp and solve_ivp_sympext

The functions solve_ivp_symp and solve_ivp_sympext solve an initial value problem for a Hamiltonian system using an element of the class SymplecticIntegrator, an explicit symplectic splitting scheme (see [1]). These functions numerically integrate a system of ordinary differential equations given an initial value:
  dy / dt = {y, H(t, y)}
  y(t₀) = y₀
Here t is a 1-D independent variable (time), y(t) is an N-D vector-valued function (state). A Hamiltonian H(t, y) and a Poisson bracket {. , .} determine the differential equations. The goal is to find y(t) approximately satisfying the differential equations, given an initial value y(t₀) = y₀.

The function solve_ivp_symp solves an initial value problem using an explicit symplectic integration. The Hamiltonian flow is defined by two functions chi and chi_star of (h, t, y) (see [2]). This function works for any set of coordinates, canonical or non-canonical, provided that the splitting H=∑_k A_k leads to facilitated expressions for the operators exp(h X_k) where X_k = {A_k , ·}.

The function solve_ivp_sympext solves an initial value problem using an explicit symplectic approximation obtained by an extension in phase space (see [3]). This symplectic approximation works for canonical Poisson brackets, and the state vector should be of the form y = (q, p).

Parameters:

• chi (for solve_ivp_symp) : callable
  Function of (h, t, y) returning exp(h X_n)...exp(h X₁) y at time t. If the selected integrator is not all purpose, refer to the list above for the specific ordering of the operators. The operator X_k is the Liouville operator associated with the function A_k, i.e., for Hamiltonian flows X_k = {A_k , ·} where {· , ·} is the Poisson bracket. chi must return an array of the same shape as y.
• chi_star (for solve_ivp_symp) : callable
  Function of (h, t, y) returning exp(h X₁)...exp(h X_n) y at time t. chi_star must return an array of the same shape as y.
• hs (for solve_ivp_sympext) : element of class HamSys
  The attributes y_dot of hs should be defined. If check_energy is True. It the Hamiltonian system has an explicit time dependence (i.e., the parameter ndof of hs is half an integer), the attribute k_dot of hs should be specified.
• t_span : 2-member sequence
  Interval of integration (t₀, t_f). The solver starts with t=t₀ and integrates until it reaches t=t_f. Both t₀ and t_f must be floats or values interpretable by the float conversion function.
• y0 : array_like, shape (n,)
  Initial state.
• step : float
  Step size.
• t_eval : array_like or None, optional
  Times at which to store the computed solution, must be sorted and equally spaced, and lie within t_span. If None (default), use points selected by the solver.
• method : string, optional
  Integration methods are listed on pyhamsys.
  'BM4' is the default.
• omega (for solve_ivp_sympext) : float, optional
  Coupling parameter in the extended phase space (see [3]). Default = 10.
• command : function of (t, y)
  Function to be run at each step size (e.g., plotting an observable associated with the state vector y, or register specific events).
• check_energy (for solve_ivp_sympext) : bool, optional
  If True, the attribute hamiltonian of hs should be defined. Default is False.

Returns:

  Bunch object with the following fields defined:

• t : ndarray, shape (n_points,)
  Time points.
• y : ndarray, shape (n, n_points)
  Values of the solution y at t.
• k (for solve_ivp_sympext) : ndarray, shape (n_points,) Values of k at t. Only for solve_ivp_sympext and if check_energy is True for a Hamiltonian system with an explicit time dependence (i.e., the parameter ndof of hs is half an integer).
• err (for solve_ivp_sympext) : float Error in the computation of the total energy. Only for solve_ivp_sympext and if check_energy is True.
• step : step size used in the computation.

Remarks:

• Use solve_ivp_symp is the Hamiltonian can be split and if each partial operator exp(h X_k) can be easily expressed/computed. Otherwise use solve_ivp_sympext if your coordinates are canonical.
• If t_eval is a linearly spaced list or array, or if t_eval is None (default), the step size is slightly readjusted so that the output times contain the values in t_eval, or the final time t_f corresponds to an integer number of step sizes. The step size used in the computation is recorded in the solution as sol.step.
• For integrating multiple trajectories at the same time, extend phase space and define a state vector y = (y₁, y₂,...y_N) where N is the number of trajectories. The Hamiltonian is given by $H(t,\mathbf{y})=\sum_{i=1}^N h(t, y_i)$.

References:

• [1] Hairer, Lubich, Wanner, 2003, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer)
• [2] McLachlan, Tuning symplectic integrators is easy and worthwhile, Commun. Comput. Phys. 31, 987 (2022); arxiv:2104.10269
• [3] Tao, M., Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance, Phys. Rev. E 94, 043303 (2016)

Example

import numpy as xp
import sympy as sp
import matplotlib.pyplot as plt
from pyhamsys import HamSys, solve_ivp_sympext
hs = HamSys()
hamiltonian = lambda q, p, t: p**2 / 2 - sp.cos(q)
hs.compute_vector_field(hamiltonian, output=True)
sol = solve_ivp_sympext(hs, (0, 20), xp.asarray([3, 0]), step=1e-1, check_energy=True)
print(f"Error in energy : {sol.err}")
plt.plot(sol.y[0], sol.y[1])
plt.show()

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