# pyHamSys
pyHamSys is a Python package for scientific computing involving Hamiltonian systems
Installation:
```
pip install pyhamsys
```
## 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)](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<sub>A</sub>)exp(*h* X<sub>B</sub>). For `VEFRL`, *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).
- 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 S<sub>6</sub>
- `BM6` (order 6, all purpose) refers to S<sub>10</sub>
- `RKN4b` (order 4) refers to SRKN<sub>6</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).
- `RKN6b` (order 6) refers to SRKN<sub>11</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).
- `RKN6a` (order 6) refers to SRKN<sub>14</sub><sup>*a*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).
- 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<sub>A</sub>)exp(*h* X<sub>B</sub>).
- `ABA864` (order (8,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).
- `ABA1064` (order (10,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).
All purpose integrators are for any splitting of the Hamiltonian *H*=∑<sub>*k*</sub> *A*<sub>*k*</sub> in any order of the functions *A*<sub>*k*</sub>. 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
### 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.
### 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](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 `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.
---
## 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*<sub>0</sub>) = *y*<sub>0</sub>
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*<sub>0</sub>) = *y*<sub>0</sub>.
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*=∑<sub>*k*</sub> *A*<sub>*k*</sub> leads to facilitated expressions for the operators exp(*h* X<sub>*k*</sub>) where X<sub>*k*</sub> = {*A*<sub>*k*</sub> , ·}.
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<sub>*n*</sub>)...exp(*h* X<sub>1</sub>) *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<sub>*k*</sub> is the Liouville operator associated with the function *A*<sub>*k*</sub>, i.e., for Hamiltonian flows X<sub>*k*</sub> = {*A*<sub>*k*</sub> , ·} 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* X<sub>1</sub>)...exp(*h* X<sub>*n*</sub>) *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 (*t*<sub>0</sub>, *t*<sub>f</sub>). The solver starts with *t*=*t*<sub>0</sub> and integrates until it reaches *t*=*t*<sub>f</sub>. Both *t*<sub>0</sub> and *t*<sub>f</sub> 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](https://pypi.org/project/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//2, 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<sub>*k*</sub>) 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*<sub>f</sub> corresponds to an integer number of step sizes. The step size used in the computation is recorded in the solution as `sol.step`.
### 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 information: <cristel.chandre@cnrs.fr>
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"description": "# pyHamSys\npyHamSys is a Python package for scientific computing involving Hamiltonian systems\n\n\n\n\nInstallation: \n```\npip install pyhamsys\n```\n\n## Symplectic Integrators\npyHamSys includes a class SymplecticIntegrator containing the following symplectic splitting integrators:\n\n- `Verlet` (order 2, all purpose), also referred to as Strang or St\u00f6rmer-Verlet splitting \n- From [Forest, Ruth, Physica D 43, 105 (1990)](https://doi.org/10.1016/0167-2789(90)90019-L): \n - `FR` (order 4, all purpose)\n- From [Yoshida, Phys. Lett. A 150, 262 (1990)](https://doi.org/10.1016/0375-9601(90)90092-3):\n - `Yo#`: # should be replaced by an even integer, e.g., `Yo6` for 6th order symplectic integrator (all purpose)\n - `Yos6`: (order 6, all purpose) optimized symplectic integrator (solution A from Table 1)\n- From [McLachlan, SIAM J. Sci. Comp. 16, 151 (1995)](https://doi.org/10.1137/0916010):\n - `M2` (order 2, all purpose)\n - `M4` (order 4, all purpose)\n- From [Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002)](https://doi.org/10.1016/S0010-4655(02)00451-4): \n - `EFRL` (order 4) optimized for *H* = *A* + *B*\n - `PEFRL` and `VEFRL` (order 4) optimized for *H* = *A*(*p*) + *B*(*q*). For `PEFRL`, *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>). For `VEFRL`, *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).\n- From [Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002)](https://doi.org/10.1016/S0377-0427(01)00492-7):\n - `BM4` (order 4, all purpose) refers to S<sub>6</sub> \n - `BM6` (order 6, all purpose) refers to S<sub>10</sub>\n - `RKN4b` (order 4) refers to SRKN<sub>6</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).\n - `RKN6b` (order 6) refers to SRKN<sub>11</sub><sup>*b*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>B</sub>)exp(*h* X<sub>A</sub>).\n - `RKN6a` (order 6) refers to SRKN<sub>14</sub><sup>*a*</sup> optimized for *H* = *A*(*p*) + *B*(*q*). Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).\n- From [Blanes, Casas, Farr\u00e9s, Laskar, Makazaga, Murua, Appl. Numer. Math. 68, 58 (2013)](http://dx.doi.org/10.1016/j.apnum.2013.01.003):\n - `ABA104` (order (10,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).\n - `ABA864` (order (8,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).\n - `ABA1064` (order (10,6,4)) optimized for *H* = *A* + ε *B*. Here *chi* should be exp(*h* X<sub>A</sub>)exp(*h* X<sub>B</sub>).\n \nAll purpose integrators are for any splitting of the Hamiltonian *H*=∑<sub>*k*</sub> *A*<sub>*k*</sub> in any order of the functions *A*<sub>*k*</sub>. 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`). \n\n----\n## HamSys class\n\n### Parameters\n- `ndof` : number of degrees of freedom of the Hamiltonian system\n \t'ndof' should be an integer or half an integer. Half integers denote an explicit time dependence.\n\n### Attributes\n- `hamiltonian` : callable \n\tA function of (*t*, *y*) which returns the Hamiltonian *H*(*t*,*y*) where *y* is the state vector.\n- `y_dot` : callable \n \tA 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*).\n- `k_dot` : callable \n\tA function of (*t*, *y*) which returns {*k*,*H*(*t*,*y*)} = -∂*H*/∂*t* where *k* is canonically conjugate to *t* and *H* is the Hamiltonian.\n\n### Functions\n- `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`.\n\n\tDetermine 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.\n\n\t#### Parameters\n\t- `hamiltonian` : callable\n\t\tFunction *H*(*q*, *p*, *t*) –the Hamiltonian expressed in symbolic variables–, expressed using [SymPy](https://www.sympy.org/en/index.html) functions.\n\t- `output` : bool, optional\n\t\tIf True, displays the equations of motion. Default is False.\n\t\n\tThe 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.\n\n- `compute_energy` : callable\n \tA 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). \n\t#### Parameters\n\t- `sol` : OdeSolution \n \t\tSolution provided by `solve_ivp_sympext`. \n \t- `maxerror` : bool, optional \n \t\tDefault is True.\n\n---\n## solve_ivp_symp and solve_ivp_sympext\n\nThe 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: \n\t d*y* / d*t* = {*y*, *H*(*t*, *y*)} \n\t *y*(*t*<sub>0</sub>) = *y*<sub>0</sub> \nHere *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*<sub>0</sub>) = *y*<sub>0</sub>. \n\nThe 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*=∑<sub>*k*</sub> *A*<sub>*k*</sub> leads to facilitated expressions for the operators exp(*h* X<sub>*k*</sub>) where X<sub>*k*</sub> = {*A*<sub>*k*</sub> , ·}.\n\nThe 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*). \n\n### Parameters: \n\n - `chi` (for `solve_ivp_symp`) : callable \n\tFunction of (*h*, *t*, *y*) returning exp(*h* X<sub>*n*</sub>)...exp(*h* X<sub>1</sub>) *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<sub>*k*</sub> is the Liouville operator associated with the function *A*<sub>*k*</sub>, i.e., for Hamiltonian flows X<sub>*k*</sub> = {*A*<sub>*k*</sub> , ·} where {· , ·} is the Poisson bracket.\n\t`chi` must return an array of the same shape as `y`.\n - `chi_star` (for `solve_ivp_symp`) : callable \n\tFunction of (*h*, *t*, *y*) returning exp(*h* X<sub>1</sub>)...exp(*h* X<sub>*n*</sub>) *y* at time *t*.\n\t`chi_star` must return an array of the same shape as `y`.\n - `hs` (for `solve_ivp_sympext`) : element of class HamSys \n\tThe 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. \n - `t_span` : 2-member sequence \n\tInterval of integration (*t*<sub>0</sub>, *t*<sub>f</sub>). The solver starts with *t*=*t*<sub>0</sub> and integrates until it reaches *t*=*t*<sub>f</sub>. Both *t*<sub>0</sub> and *t*<sub>f</sub> must be floats or values interpretable by the float conversion function.\t\n - `y0` : array_like, shape (n,) \n\tInitial state.\n - `step` : float \n\tStep size.\n - `t_eval` : array_like or None, optional \n\tTimes 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.\n - `method` : string, optional \n \tIntegration methods are listed on [pyhamsys](https://pypi.org/project/pyhamsys/). \n\t'BM4' is the default.\n - `omega` (for `solve_ivp_sympext`) : float, optional \n \tCoupling parameter in the extended phase space (see [3]). Default = 10.\n - `command` : function of (*t*, *y*) \n\tFunction to be run at each step size (e.g., plotting an observable associated with the state vector *y*, or register specific events).\n - `check_energy` (for `solve_ivp_sympext`) : bool, optional \n\tIf True, the attribute `hamiltonian` of `hs` should be defined. Default is False. \n\n### Returns: \n Bunch object with the following fields defined:\n - `t` : ndarray, shape (n_points,) \n\tTime points.\n - `y` : ndarray, shape (n, n_points) \n\tValues of the solution `y` at `t`.\n - `k` (for `solve_ivp_sympext`) : ndarray, shape (n//2, n_points)\n \tValues 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).\n - `err` (for `solve_ivp_sympext`) : float\n \tError in the computation of the total energy. Only for `solve_ivp_sympext` and if `check_energy` is True.\n - `step` : step size used in the computation.\n\n### Remarks: \n - Use `solve_ivp_symp` is the Hamiltonian can be split and if each partial operator exp(*h* X<sub>*k*</sub>) can be easily expressed/computed. Otherwise use `solve_ivp_sympext` if your coordinates are canonical. \n - 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*<sub>f</sub> corresponds to an integer number of step sizes. The step size used in the computation is recorded in the solution as `sol.step`. \n\n### References: \n - [1] Hairer, Lubich, Wanner, 2003, *Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations* (Springer) \n - [2] McLachlan, *Tuning symplectic integrators is easy and worthwhile*, Commun. Comput. Phys. 31, 987 (2022); [arxiv:2104.10269](https://arxiv.org/abs/2104.10269) \n - [3] Tao, M., *Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance*, Phys. Rev. E 94, 043303 (2016)\n\n### Example\n\n```python\n>>> import numpy as xp\n>>> import sympy as sp\n>>> import matplotlib.pyplot as plt\n>>> from pyhamsys import HamSys, solve_ivp_sympext\n>>> hs = HamSys()\n>>> hamiltonian = lambda q, p, t: p**2 / 2 - sp.cos(q)\n>>> hs.compute_vector_field(hamiltonian, output=True)\n>>> sol = solve_ivp_sympext(hs, (0, 20), xp.asarray([3, 0]), step=1e-1, check_energy=True)\n>>> print(f\"Error in energy : {sol.err}\")\n>>> plt.plot(sol.y[0], sol.y[1])\n>>> plt.show()\n```\n---\nFor more information: <cristel.chandre@cnrs.fr>\n",
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