Consider a dynamical system with the Lagrangian function L(q, \(\dot{q}\)) = T - U, where the kinetic energy
T = a(q)\(\dot{q}\)2 ≥ 0
and the potential energy U ∶= U(q) with a(q) > 0.
Which of the following statements are true?
1
The associated Lagrange's equation is \(\frac{d}{d t} \frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q}\)
2
The associated Lagrange's equation is \(\frac{d}{d t} \frac{\partial L}{\partial q}=\frac{\partial L}{\partial \dot{q}}\)
3
The point (q0, \(\dot{q}\)0) is an equilibrium position of the dynamical system if and only if \(\dot{q}_0=0 \quad \text { and }\left.\quad \frac{\partial U}{\partial q}\right|_{q=q_0}=0\)
4
The point (q0, \(\dot{q}\)0) is an equilibrium position of the dynamical system if and only if \(\dot{q}_0=0 \quad \text { and }\left.\quad \frac{\partial U}{\partial q}\right|_{q=q_0}>0\)