Minimize doesn't work for a given function

Questions
1

I’m trying to minimize the function V(phi) using the code below

a =.
f =.
th =.
mu =.
md =.
V[phi_] := -2 * (mu Cos[phi + th/2] + md Cos[phi - th/2])
Minimize[V[phi], phi]

But the output is just the command itself!

Out: Minimize[-2 (md Cos[phi - th/2] + mu Cos[phi + th/2]), phi]

What am I doing wrong?

2

I’m not sure what you expect when you don’t have any constraints on the other symbols used in V. No matter what value you try for phi, md and mu could dominate and make the result arbitrarily large or small.

3

Right. How do I add the constraints? I tried to use

`$Assumptions =  th >= 0 && mu >= 0 && md >= 0`

but without much success…

4

This is very strange. Even

Minimize[{Cos[x + a/2], {0 <= x <= 2\[Pi], 0 < a < 1}}, x]

not work.
There are similar question on MSE: [1] [2]
The answers suggest add a to variables of minimization:

Minimize[{Cos[a/2 + x], {0 <= x <= 2\[Pi], 0 < a < 1}}, {a, x}]
Out: {-1, {a -> 9/34, x -> 1/68 (-9 + 68 \[Pi])}}

But this is bad cheating, cause we need parametric solution!
In docs there is example with fully parametric solution for ax^2+b x + c.
I’m going to post revisited question on MSE and query for Support team.

5

Well, we can use mathematical analysis.
If we want to find \min of -2 \cdot f then we need to find \max of f.
So it is root of first derivative with negative second derivative.
I rewrote your expression a little for simplicity, so we try this:

Clear[a, b, t, V];
V[x_] := a Cos[x + t/2] + b Cos[x - t/2];

And if we run next code, in 15-20 sec (on average laptop) we get big dump,
but with meaningful and useful results:

goal = D[V[x], x];
secDeriv = D[V[x], {x, 2}];
Reduce[goal == 0 && secDeriv < 0 && 0 <= x <= 2 \[Pi], x, Reals]
6

Thanks man!! It’s so weird that you cannot just use Minimize/Maximize

7

Well, I like Mathematica, in many ways it’s cool thing! So I report this issue as bug,
looks like problem in trig functions.

8

Parametric minimization seems to be supported only in the algebraic case. You can algebrize your trigonometric problem, though:

subst = {Cos[phi] -> x, Sin[phi] -> y};
TrigExpand[V[phi]] /. subst
Minimize[{%, x^2 + y^2 == 1}, {x, y}]
% /. Map[Reverse, subst]
% // Simplify
9

Very smart idea!