https://github.com/cchandre/pyhamsys

pyHamSys is a Python package for scientific computations involving Hamiltonian systems
https://github.com/cchandre/pyhamsys
hamiltonian hamiltonian-dynamics hamiltonian-systems numpy ode-solver python3 symplectic-integration symplectic-integrators
Last synced: 3 months ago
JSON representation
pyHamSys is a Python package for scientific computations involving Hamiltonian systems
Host: GitHub
URL: https://github.com/cchandre/pyhamsys
Owner: cchandre
License: bsd-2-clause
Created: 2023-04-05T14:48:44.000Z (about 2 years ago)
Default Branch: main
Last Pushed: 2023-10-24T14:19:40.000Z (over 1 year ago)
Last Synced: 2024-03-25T06:11:30.571Z (about 1 year ago)
Topics: hamiltonian, hamiltonian-dynamics, hamiltonian-systems, numpy, ode-solver, python3, symplectic-integration, symplectic-integrators
Language: Python
Homepage:
Size: 181 KB
Stars: 0
Watchers: 1
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project

README

        # pyHamSys

pyHamSys (short for Python Hamiltonian Systems) is a Python library for scientific computing involving Hamiltonian systems. It provides tools to model, analyze, and simulate Hamiltonian systems. In particular, the library offers a streamlined and user-friendly environment for implementing and running symplectic-split integrators. These specialized numerical methods are designed to preserve the geometric structure of Hamiltonian systems, ensuring accurate and stable simulations of their dynamics over long time periods. 

![PyPI](https://img.shields.io/pypi/v/pyhamsys)

![License](https://img.shields.io/badge/license-BSD-lightgray)

## Installation 

Installation within a Python virtual environment: 

```

python3 -m pip install pyhamsys

```

For more information on creating a Python virtual environment, click [here](https://realpython.com/python-virtual-environments-a-primer/).

## Symplectic Integrators

pyHamSys features a dedicated class, `SymplecticIntegrator`, which provides a comprehensive implementation of symplectic-split integrators. These integrators are designed specifically for the numerical integration of Hamiltonian systems, ensuring the preservation of the symplectic structure of phase space—a key property that underpins the stability and accuracy of long-term simulations of such systems.

Symplectic-split integrators decompose the Hamiltonian into subcomponents that are individually solvable, allowing for efficient and accurate integration. This decomposition is particularly effective for complex or high-dimensional systems, as it minimizes numerical drift and conserves critical invariants like energy over extended time intervals.

The `SymplecticIntegrator` class offers a variety of splitting methods, enabling users to select the most appropriate scheme for their specific Hamiltonian system and computational requirements. Each integrator is implemented to handle both autonomous and non-autonomous systems, supporting applications in classical mechanics, molecular dynamics, astrophysics, and quantum mechanics.

- `Verlet` (order 2, all purpose), also referred to as Strang or Störmer-Verlet splitting 

- From [Forest, Ruth, Physica D 43, 105 (1990)](https://doi.org/10.1016/0167-2789(90)90019-L): 

    - `FR` (order 4, all purpose)

- From [Yoshida, Phys. Lett. A 150, 262 (1990)](https://doi.org/10.1016/0375-9601(90)90092-3):

    - `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)](https://doi.org/10.1137/0916010):

    - `M2` (order 2, all purpose)

    - `M4` (order 4, all purpose)

- From [Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)](https://doi.org/10.1016/S0010-4655(02)00451-4): 

    - `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)](https://doi.org/10.1016/S0377-0427(01)00492-7):

    - `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)](http://dx.doi.org/10.1016/j.apnum.2013.01.003):

    - `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`). 

----

## HamSys class   

The `HamSys` class provides a robust framework for defining and integrating Hamiltonian systems. It allows users to specify the number of degrees of freedom, coordinate representations, and key attributes like the Hamiltonian and associated equations of motion.

### Parameters

- `ndof` : integer or half-integer, optional   

       	The number of degrees of freedom in the Hamiltonian system. Half integers denote an explicit time dependence. Default is 1.  

- `complex` : bool, optional  

	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.

### Parameters and Attributes

- `y_dot` : callable, optional   

  	A function of (*t*, *y*) which returns {*y*,*H*(*t*,*y*)} where *y* is the state vector and *H* is the Hamiltonian. In (real) canonical coordinates (used, e.g., in `solve_ivp_sympext`) where *y* = (*q*, *p*), this function returns (∂*H*/∂*p*, -∂*H*/∂*q*). In complex coordinate ψ, this function returns -i ∂*H*/∂ψ^*. For practical implementation, the state vector *y* should be represented as a one-dimensional array with a shape of (*n*,), where *n* denotes the total number of dynamical variables in the system. This ensures compatibility with numerical solvers and facilitates efficient computation of the system's evolution.  

- `k_dot` : callable, optional   

	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.

- `hamiltonian` : callable, optional   

	A function of (*t*, *y*) which returns the Hamiltonian *H*(*t*,*y*) where *y* is the state vector.

### Functions

- `compute_vector_field` : from a callable function (Hamiltonian in canonical coordinates) written with symbolic variables ([SymPy](https://www.sympy.org/en/index.html)), 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. However, it is preferrable to code explicitly and optimize `y_dot` and `k_dot`.

	#### Parameters

	- `hamiltonian` : callable   

		Function *H*(*q*, *p*, *t*) –the Hamiltonian expressed in symbolic variables–, expressed using [SymPy](https://www.sympy.org/en/index.html) 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 a boolean `maxerror`. If `maxerror`is `True` the function `compute_energy` returns the maximum error in total energy; otherwise, it returns all the values of the total energy. 

	#### Parameters

	- `sol` : OdeSolution  

   		Solution provided by `solve_ivp_sympext`. 

 	- `maxerror` : bool, optional  

    		Default is True.

---

## 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:  

	  d*y* / d*t* = {*y*, *H*(*t*, *y*)}  

	  *y*(*t*₀) = *y*₀  

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*(*t*₀) = *y*₀. 

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* , &centerdot;}.

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* , &centerdot;} where {&centerdot; , &centerdot;} 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 attribute `y_dot` of `hs` should be defined. If `check_energy` is True, the attribute `hamiltonian`  and if the Hamiltonian system has an explicit time dependence (i.e., the parameter `ndof` of `hs`  is a half-integer), the attribute `k_dot` of `hs` should be specified. 

  - `t_span` : 2-member sequence  

	Interval of integration (*t*₀, *t*_f). The solver starts with *t*=*t*₀ and integrates until it reaches *t*=*t*_f. Both *t*₀ and *t*_f must be floats or values interpretable by the float conversion function.	

  - `y0` : array_like  

	Initial state. For `solve_ivp_sympext`, the vector `y0` should be with shape (n,).

  - `step` : float   

	Step size.

  - `t_eval` : array_like or None, optional  

	Times at which to store the computed solution, must be sorted, and lie within `t_span`. If None (default), use points selected by the solver.

  - `method` : string, optional  

 	Integration methods are listed on [pyhamsys](https://pypi.org/project/pyhamsys/). Default is 'BM4'.

  - `omega` (for `solve_ivp_sympext`) : float, optional  

   	Coupling parameter in the extended phase space (see [3]). Default is 10.

  - `command` : void function of (*t*, *y*), optional    

	Void function to be run at each step size (e.g., plotting an observable associated with the state vector *y*, modify global or mutable variables, 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 y0.shape + (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` if the Hamiltonian can be split and if each partial operator exp(*h* X_*k*) can be easily and explicitly expressed/computed. Otherwise use `solve_ivp_sympext` if your coordinates are canonical, i.e., in $(q,p)$ or $(\psi,\psi^*)$ variables.  

  - The step size is slightly readjusted so that 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](https://arxiv.org/abs/2104.10269)  

  - [3] Tao, M., *Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance*, Phys. Rev. E 94, 043303 (2016)

### Example

```python

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()

```

For more examples, see the folder [Examples](https://github.com/cchandre/pyhamsys/tree/main/Examples)

---

This work has been carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM).

---

For more information:
ecosyste.ms

Data

Tools

Indexes

Applications

Experiments

Awesome

https://github.com/cchandre/pyhamsys

Awesome Lists containing this project

README
