SymPy求解含偏微分的机械连杆符号方程组问题

问题描述 #

我正在开发一款机械连杆的符号求解器。目前SymPy的solve函数可求解大型线性方程组，但在处理含偏微分的方程组时，无法输出有效结果，求解器会混淆求解时机与对象。

最小示例代码 #

# Try to solve Y=Z X=dY(Z)^3/dZ
import sympy as lib_sympy

def bad_derivative_wrong( in_x : lib_sympy.Symbol, in_y : lib_sympy.Symbol, in_z : lib_sympy.Symbol ):
    l_equation = []
    l_equation.append( lib_sympy.Eq( in_y, in_z ) )
    l_equation.append( lib_sympy.Eq( in_x, lib_sympy.Derivative(in_y*in_y*in_y, in_z, evaluate = True) ) )
    solution = lib_sympy.solve( l_equation, (in_x,in_y,), exclude = () )
    return solution

def bad_derivative_unhelpful( in_x : lib_sympy.Symbol, in_y : lib_sympy.Symbol, in_z : lib_sympy.Symbol ):
    l_equation = []
    l_equation.append( lib_sympy.Eq( in_y, in_z ) )
    l_equation.append( lib_sympy.Eq( in_x, lib_sympy.Derivative(in_y*in_y*in_y, in_z, evaluate = False) ) )
    solution = lib_sympy.solve( l_equation, (in_x,in_y,), exclude = () )
    return solution

def good_derivative( in_x : lib_sympy.Symbol, in_y : lib_sympy.Symbol, in_z : lib_sympy.Symbol ):
    l_equation = []
    l_equation.append( lib_sympy.Eq( in_y, in_z ) )
    l_equation.append( lib_sympy.Eq( in_x, lib_sympy.Derivative(in_z*in_z*in_z, in_z, evaluate = True) ) )
    # what happens here is that Derivative has already solved the derivative, it's not a symbol
    solution = lib_sympy.solve( l_equation, (in_x,in_y,), exclude = () )
    # lib_sympy.dsolve
    return solution

if __name__ == '__main__':
    # n_x = lib_sympy.symbols('X', cls=lib_sympy.Function)
    n_x = lib_sympy.symbols('X')
    n_y = lib_sympy.Symbol('Y')
    n_z = lib_sympy.Symbol('Z')
    print("Wrong Derivative: ", bad_derivative_wrong( n_x, n_y, n_z ) )
    print("Unhelpful Derivative: ", bad_derivative_unhelpful( n_x, n_y, n_z ) )
    print("Good Derivative: ", good_derivative( n_x, n_y, n_z ) )

运行输出 #

Wrong Derivative:  {Y: Z, X: 0}
Unhelpful Derivative:  {Y: Z, X: Derivative(Y**3, Z)}
Good Derivative:  {Y: Z, X: 3*Z**2}

技术问询 #

我需要找到一种在方程组中添加偏微分符号的方法，使SymPy求解器能够正确求解，例如处理速度是位置对时间的导数、位置对角度的灵敏度与精度及力相关这类场景。

内容的提问来源于stack exchange，提问作者05032 Mendicant Bias