My question is, how do I apply boundary constraints to a Lagrangian such as:
$90 > \theta_1 > -45$
$\theta_2 > 0$
I am trying to use a constrained double pendulum to simulate an underhand throw. I am trying to use the Lagrangian defined from Wolfram's site equation (9) and add constraints to it through Lagrange multipliers.
My confusion comes from the fact that constraints for Lagrange multipliers are normally written as equalities with multiple variables such as:
$ f(\theta_1, \theta_2) = \theta_1 + 2\theta_2 = 90$
and not as inequalities.
When I looked up inequality optimization, the method I got was to set the inequality as an equality and then solve for the boundaries. This doesn't make to me (in my application) because the system might not be at the boundary through the entire simulation, so setting $\theta_2 = 0$ doesn't makes sense to me.