## Step 1: Define the symbolic representation of the variables
The variables are ['airborne infantry companies', 'engineer platoons', 'air defense batteries'] which can be represented symbolically as 'x0', 'x1', and 'x2' respectively.

## Step 2: Create a symbolic representation of the problem
The symbolic representation of the variables is:
- 'x0' for 'airborne infantry companies'
- 'x1' for 'engineer platoons'
- 'x2' for 'air defense batteries'

## Step 3: Define the objective function in symbolic notation
The objective function to minimize is: $7x_0^2 + 5x_0x_1 + 5x_0x_2 + 3x_1^2 + x_2^2 + 9x_0 + x_1$

## 4: List the constraints in symbolic notation
The constraints are:
- $12x_0 \leq 120$ (mobility rating of airborne infantry companies)
- $4x_0 \leq 197$ (fuel demand of airborne infantry companies)
- $13x_1 \leq 120$ (mobility rating of engineer platoons)
- $11x_1 \leq 197$ (fuel demand of engineer platoons)
- $10x_2 \leq 120$ (mobility rating of air defense batteries)
- $4x_2 \leq 197$ (fuel demand of air defense batteries)
- $12x_0 + 10x_2 \geq 33$ (total mobility rating from airborne infantry companies and air defense batteries)
- $13x_1 + 10x_2 \geq 36$ (total mobility rating from engineer platoons and air defense batteries)
- $12x_0 + 13x_1 + 10x_2 \geq 36$ (total mobility rating from all units)
- $4x_0 + 4x_2 \geq 57$ (total fuel demand from airborne infantry companies and air defense batteries)
- $4x_0 + 11x_1 + 4x_2 \geq 57$ (total fuel demand from all units)
- $2x_0 - 5x_2 \geq 0$ (relationship between airborne infantry companies and air defense batteries)
- $12^2x_0^2 + 13^2x_1^2 + 10^2x_2^2 \leq 66$ (total mobility rating squared)
- $4x_0 + 4x_2 \leq 146$ (total fuel demand from airborne infantry companies and air defense batteries)

## 5: Consider variable bounds and types
- $x_0$ is an integer (whole number of airborne infantry companies)
- $x_1$ is an integer (nonfractional number of engineer platoons)
- $x_2$ is an integer (nonfractional number of air defense batteries)

## 6: Formulate the problem in Gurobi
We will use Gurobi to solve this optimization problem.

```json
{
    'sym_variables': [('x0', 'airborne infantry companies'), ('x1', 'engineer platoons'), ('x2', 'air defense batteries')],
    'objective_function': '7*x0^2 + 5*x0*x1 + 5*x0*x2 + 3*x1^2 + x2^2 + 9*x0 + x1',
    'constraints': [
        '12*x0 <= 120',
        '4*x0 <= 197',
        '13*x1 <= 120',
        '11*x1 <= 197',
        '10*x2 <= 120',
        '4*x2 <= 197',
        '12*x0 + 10*x2 >= 33',
        '13*x1 + 10*x2 >= 36',
        '12*x0 + 13*x1 + 10*x2 >= 36',
        '4*x0 + 4*x2 >= 57',
        '4*x0 + 11*x1 + 4*x2 >= 57',
        '2*x0 - 5*x2 >= 0',
        '144*x0^2 + 169*x1^2 + 100*x2^2 <= 66',
        '4*x0 + 4*x2 <= 146'
    ]
}
```

## 7: Write the Gurobi code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="airborne_infantry_companies", vtype=gurobi.GRB.INTEGER)
x1 = m.addVar(name="engineer_platoons", vtype=gurobi.GRB.INTEGER)
x2 = m.addVar(name="air_defense_batteries", vtype=gurobi.GRB.INTEGER)

# Objective function
m.setObjective(7*x0**2 + 5*x0*x1 + 5*x0*x2 + 3*x1**2 + x2**2 + 9*x0 + x1, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(12*x0 <= 120)
m.addConstr(4*x0 <= 197)
m.addConstr(13*x1 <= 120)
m.addConstr(11*x1 <= 197)
m.addConstr(10*x2 <= 120)
m.addConstr(4*x2 <= 197)
m.addConstr(12*x0 + 10*x2 >= 33)
m.addConstr(13*x1 + 10*x2 >= 36)
m.addConstr(12*x0 + 13*x1 + 10*x2 >= 36)
m.addConstr(4*x0 + 4*x2 >= 57)
m.addConstr(4*x0 + 11*x1 + 4*x2 >= 57)
m.addConstr(2*x0 - 5*x2 >= 0)
m.addConstr(144*x0**2 + 169*x1**2 + 100*x2**2 <= 66)
m.addConstr(4*x0 + 4*x2 <= 146)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Airborne Infantry Companies: ", x0.varValue)
    print("Engineer Platoons: ", x1.varValue)
    print("Air Defense Batteries: ", x2.varValue)
else:
    print("The model is infeasible")
```