Variational Derivatives in Save

variational

def variational_derivative(expr,u):
    """Returns the variational derivative of `expr` with respect to `u`.
    
    Variational differentiation is a linear operator so we just need to
    figure out how to differentiate a product of terms.

    Parameters
    ----------
    expr : expression
    u : function

    Returns
    -------
    expression
    """
    # expand the expression in a sum of terms
    expr = expr.expand()
    deriv = sum(_deriv_of_product(term,u) for term in expr.operands())
    return deriv
            

def _deriv_of_product(term,u):
    # for n = 0 to highest derivative order in term
    #     compute d/du_n term: foo = ddun(term,u,n)
    #     accumulate: deriv += (-1)**n * foo.deriv(x,n)
    # return deriv

def ddun(f,n):
    # apply product rule
    if f.nops() == 1:
        # return d/dun derivative of f
    else:
        # for factor in f:
        #     accumulate product rule