human
Please write code following the instructions in jupyter notebook style.
assistant
#Here is code to and the other steps required.

f, L, w = Function('f'), Function('L'), Function('w') # abstract, Lagrangian and path functions, respectively
t, q, q_point = symbols(r't q \dot{q}') # symbols for the Leibniz notation

Lagrangian_eq = Eq(Derivative(Derivative(L(t, q, q_point),q_point,evaluate=False),
                              t, evaluate=False) - L(t,q,q_point).diff(q),0)
Lagrangian_eq # compact Lagrangian equation, implicit indeed

Lagrangian_eq = Eq(Derivative(Subs(L(t, q, q_point).diff(q_point), [q,q_point], [w(t),w(t).diff(t)]),t) - 
                   Subs(L(t,q,q_point).diff(q), [q,q_point], [w(t),w(t).diff(t)]),0)
Lagrangian_eq

Lagrangian_eq.doit() # a complex explosion by automatic computation

def diff_positional(f, i):
    a = IndexedBase('a')
    der = f.diff(f.args[i])
    def D(*args):
        #return Lambda([a[i] for i in range(len(f.args))], 
        return der.subs({f.args[i]:args[i] for i in range(len(f.args))}, simultaneous=True)
    return D # function D is a function of the meta-language, not in the object language

diff_positional(L(t, q, q_point), 2)(t,w(t),w(t).diff(t)) # :/

def diff_positional(f, i):
    return Function(r'\partial_{{{}}}{}'.format(i, str(f))) # the \partial is just a symbol, it hasn't meaning

(Derivative(diff_positional(L, 2)(t,w(t), w(t).diff(t)), t) - 
 diff_positional(L, 1)(t,w(t), w(t).diff(t))) # :(

Lagrangian_eq = Eq(Derivative(L(t,w(t), w(t).diff(t)).fdiff(argindex=3),t, evaluate=False) - 
                   L(t,w(t), w(t).diff(t)).fdiff(argindex=2),
                   0, evaluate=False)
Lagrangian_eq # :) , although it doesn't use "positional derivative" operator explicitly

def Derivative_eq(f, argindex=1):
    d = Dummy()
    args = [d if i+1 == argindex else a for i,a in enumerate(f.args)]
    lhs = f.fdiff(argindex)
    rhs = Subs(f.func(*args).diff(d), d, f.args[argindex-1])
    return Eq(lhs, rhs, evaluate=False)

Derivative_eq(f(t,q,q_point),2) # applicative Derivative operator

def Gamma(w):
    return Lambda([t], (t, w(t), w(t).diff(t))) # state-space function

Gamma(w),...