fundamental.py¶
Define fundamental function and Hamiltonian.
Derive equations for the geomorphic surface fundamental function \(\mathcal{F}\) and the corresponding Hamiltonian \(\mathcal{H}\).
- class gme.core.fundamental.FundamentalMixin¶
Define fundamental function.
Fundamental function and Hamiltonian equations supplement to equation definition class.
- define_Fstar_eqns() → None¶
Define the fundamental function.
- Okubo_Fstar_eqn¶
\(\dfrac{\sqrt{p_{x}^{2} + p_{z}^{2}}}{F^{*}} = \dfrac{p_{x}^{- \eta} \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{\eta}{2}}} {\varphi{\left(\mathbf{r} \right)}}\)
- Type
Code¶
r"""
Define fundamental function and Hamiltonian.
Derive equations for the geomorphic surface fundamental function
:math:`\mathcal{F}` and the corresponding Hamiltonian
:math:`\mathcal{H}`.
---------------------------------------------------------------------
Requires Python packages/modules:
- :mod:`SymPy <sympy>`
- `GME`_
.. _GMPLib: https://github.com/geomorphysics/GMPLib
.. _GME: https://github.com/geomorphysics/GME
.. _Matrix:
https://docs.sympy.org/latest/modules/matrices/immutablematrices.html
---------------------------------------------------------------------
"""
# Disable these pylint errors because it doesn't understand SymPy syntax
# - notably minus signs in equations flag an error
# pylint: disable=invalid-unary-operand-type, not-callable
# Library
import warnings
import logging
# from typing import Dict, Type, Optional # , Tuple, Any, List
# SymPy
from sympy import Eq, simplify, solve, Abs, sign
# GME
from gme.core.symbols import Fstar, H, px, varphi, varphi_r, varphi_rx, rx, rvec
warnings.filterwarnings("ignore")
__all__ = ["FundamentalMixin"]
class FundamentalMixin:
"""
Define fundamental function.
Fundamental function and Hamiltonian equations supplement
to equation definition class.
"""
# Prerequisites
p_norm_pxpz_eqn: Eq
p_varphi_pxpz_eqn: Eq
varphi_rx_eqn: Eq
# Definitions
Okubo_Fstar_eqn: Eq
Fstar_eqn: Eq
H_eqn: Eq
H_varphi_rx_eqn: Eq
def define_Fstar_eqns(self) -> None:
r"""
Define the fundamental function.
Attributes:
Okubo_Fstar_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\dfrac{\sqrt{p_{x}^{2} + p_{z}^{2}}}{F^{*}}
= \dfrac{p_{x}^{- \eta}
\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{\eta}{2}}}
{\varphi{\left(\mathbf{r} \right)}}`
Fstar_eqn (:class:`~sympy.core.relational.Equality`):
:math:`F^{*} = p_{x}^{\eta}
\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{1}{2}-\frac{\eta}{2}}
\varphi{\left(\mathbf{r} \right)}`
"""
logging.info("gme.core.fundamental.define_Fstar_eqns")
# Note force px >= 0
self.Okubo_Fstar_eqn: Eq = simplify(
Eq(
self.p_norm_pxpz_eqn.rhs / Fstar, self.p_varphi_pxpz_eqn.rhs
).subs({Abs(px): px, sign(px): 1})
)
self.Fstar_eqn: Eq = Eq(
Fstar,
(solve(self.Okubo_Fstar_eqn, Fstar)[0]).subs(
{varphi_rx(rx): varphi}
),
).subs({Abs(px): px, sign(px): 1})
def define_H_eqns(self) -> None:
r"""
Define the Hamiltonian.
Attributes:
H_eqn (:class:`~sympy.core.relational.Equality`):
:math:`H
= \dfrac{p_{x}^{2 \eta}
\left(p_{x}^{2} + p_{z}^{2}\right)^{1 - \eta}
\varphi^{2}{\left(\mathbf{r} \right)}}{2}`
H_varphi_rx_eqn (:class:`~sympy.core.relational.Equality`):
:math:`H
= \dfrac{\varphi_0^{2} p_{x}^{2 \eta} x_{1}^{- 4 \mu}
\left(p_{x}^{2} + p_{z}^{2}\right)^{1 - \eta}
\left(\varepsilon x_{1}^{2 \mu} +
\left(x_{1} - {r}^x\right)^{2 \mu}\right)^{2}}{2}`
"""
logging.info("gme.core.fundamental.define_H_eqns")
self.H_eqn: Eq = Eq(H, simplify(self.Fstar_eqn.rhs ** 2 / 2))
self.H_varphi_rx_eqn: Eq = simplify(
self.H_eqn.subs(varphi_r(rvec), self.varphi_rx_eqn.rhs)
)
#
