On mutating a variable to a function

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No.

No, no, no, no, no.

And no.

You didn't "change a variable to a function". You redefined (twice !) the Python variable x.

Initially, you didn't define x, which means that by default, the Python variable x is bound to a Python object representing the symbolic variable $x$. This definition is active for lines 1-5 of your code.

Line 6, you redefine the Python variable x, binding it to a Python object representing an (undefined) symbolic function $x$.

Line 7, you redefine the Python variable x again, which binds it to the symbolic expression sol[0][0].rhs(), which turns out to be 1/2*R/p_x.

This symbolic expression has a .diff method :

Let e such a sumbolic expression ; e.diff(u) will return e.fiunction(u).diff(u), meaning *" take e as the body of a function of u and differentiate it with respect to u".

Therefore, x.diff(R).simplify()) is 1/2*R/p_x.function(R).diff(R).simplify() which evaluates to $\frac{p_x}{2}$.

See a Python tutorial (or the Python documentation) to understand the meaning of = in Python : a variable is just a label attached to a Python object ; this is quite different of a Sage symbolic variable, which is a Python object representing such a (mathematical) variable in the sage.symbolic.expression.Expression class.

The problem is compounded by the (hopefully convenient) shortcut of defining var("a") as a shortcut meaning a = SR.var("a"), meaning :

• Create a sage.symbolic.expression.Expression representing the mathematical variable $a$, then

• bind this object to the Python variable a.

This convenient shortcut is confusing to beginners...

HTH,

This is not a short way but without rhs:

var('x y p_x p_y R λ')
U(x,y)= x*y
L(x,y,λ)= U(x,y)+λ*(R-p_x*x-p_y*y)
sol = solve([L(x,y,λ).diff(x)==0,L(x,y,λ).diff(y)==0,L(x,y,λ).diff(λ)==0],
            (x,y,λ),solution_dict=True)[0]
show(x==sol[x])
show(LatexExpr(r'\frac{dx}{dR} = '),sol[x].diff(R))