Sympy函数导数和方程组

Question

I'm working with nonlinear systems of equations. These systems are generally a nonlinear vector differential equation. I now want to use functions and derive them with respect to time and to their time-derivatives, and find equilibrium points by solving the nonlinear equations 0=rhs(eqs). Similar things are needed to calculate the Euler-Lagrange equations, where you need the derivative of L wrt. diff(x,t).

Now my question is, how do I implement this in Sympy? My main 2 problems are, that deriving a Symbol f wrt. t diff(f,t) , I get 0. I can see, that with

x = Symbol('x',real=True);
diff(x.subs(x,x(t)),t) # because diff(x,t) => 0

and

diff(x**2, x)

does kind of work.

However, with

x = Fuction('x')(t);
diff(x,t);

I get this to work, but I cannot differentiate wrt. the funtion x itself, like

diff(x**2,x) -DOES NOT WORK.

Since I need these things, especially not only for scalars, but for vectors (using jacobian) all the time, I really want this to be a clean and functional workflow. Which kind of type should I initiate my mathematical functions in Sympy in order to avoid strange substitutions? It only gets worse for matricies, where I cannot get

eqns = Matrix([f1-5, f2+1]);
variabs = Matrix([f1,f2]);
nonlinsolve(eqns,variabs);

to work as expected, since it only allows symbols as input. Is there an easy conversion here? Like eqns.tolist() - which doesn't work either?

EDIT: I just found this question, which was answered towards using expressions and matricies. I want to be able to solve sets of nonlinear equations, build the jacobian of a vector wrt. another vector and derive wrt. functions as stated above. Can anyone point me into a direction to start a concise workflow for this purpose? I guess the most complex task is calculating the Lie-derivative wrt. a vector or list of functions, the rest should be straight forward.

Edit 2:

def substi(expr,variables): 
   return expr.subs( {w:w(t)} )

would automate the subsitution, such that substi(vector_expr,varlist_vector).diff(t) is not all 0.

Answer 1

The following defines x to be a function of t

import sympy as s
t = s.Symbol('t')    

x = s.Function('x')(t)

This should solve your problem of diff(x,t) being evaluated as 0 . But I think you will still run into problems later on in your calculations. I also work with calculus of variations and Euler-Lagrange equations. In these calculations, x' needs to be treated as independent of x . So, it is generally better to use two entirely different variables for x and x' so as not to confuse Sympy with the relationship between those two variables. After we are done with the calculations in Sympy and we go back to our pen and paper we can substitute x' for the second variable.

Answer 2

Yes, one has to insert an argument in a function before taking its derivative. But after that, differentiation with respect to x(t) works for me in SymPy 1.1.1, and I can also differentiate with respect to its derivative. Example of Euler-Lagrange equation derivation:

t = Symbol("t")
x = Function("x")(t)
L = x**2 + diff(x, t)**2    # Lagrangian
EL = -diff(diff(L, diff(x, t)), t) + diff(L, x)

Now EL is 2*x(t) - 2*Derivative(x(t), t, t) as expected.

That said, there is a build-in method for Euler-Lagrange :

EL = euler_equations(L)

would yield the same result, except presented as a differential equation with right-hand side 0: [Eq(2*x(t) - 2*Derivative(x(t), t, t), 0)]