For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls $$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$ and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$. (a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that $$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$ (b) Similarly calculate $\gamma_{\theta,0}(t)$.
For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls
$$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$
and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$.
(a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that
$$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$
(b) Similarly calculate $\gamma_{\theta,0}(t)$.