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Consider an action integral $$I = \int_{t_0}^{t_f}L(\eta(\theta(t)),\dot{\eta}(\theta(t)),t)\,dt\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(0)$$ with boundary conditions $\dot{\theta}(t_0) = \dot{\theta}(t_f)=0$ for some strictly-monotonic quantity $\theta(t)$ satisfying $\dot{\theta}(t) > 0$ in the open interval $t_0 < t < t_f$.

Suppose, I want to find $\eta(\theta)$ minimizing $I$. To this end, I'm rewriting the action integral as

$$ I = \int_{\theta_0}^{\theta_f} \tilde{L}(\eta(\theta),\eta'(\theta),\theta) d\theta = \int_{\theta_0}^{\theta_f} \dfrac{L(\eta(\theta),\eta'(\theta)\omega(\theta)(\theta),\theta)}{\omega(\theta)}d\theta\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$

with $\omega(\theta) = \dot{\theta}(t)$, $\theta_0 = \theta(t_0)$, $\theta_f = \theta(t_f)$, and $\dot{\eta}(\theta)=\eta'(\theta)\dot{\theta}$. Applying EL equations defines the boundary-value problem

$$\dfrac{d}{d \theta} \dfrac{\partial \tilde{L}}{\partial \eta'} - \dfrac{\partial \tilde{L}}{\partial \eta} = 0$$ with prime superscript notation defining the partial derivative with respect to the new variable of integration as $q' = \partial q/\partial \theta$ for some $q$ and BCs $\eta(\theta_0) = \eta_0$ and $\eta(\theta_f) = \eta_f$. Notice the difference in prime and dot notations with $\dot{q}(\theta(t))$ being time derivative of $q$ which can be defined as $\dot{q} = q' \omega$ with $q' = \partial q/\partial \theta$.

I am stuck at this final EL and how to approach it as it's unbounded at the BCs due to the presence of $\omega$ term at the denominator of Eqn (1) with $\omega(\theta_0)=\omega(\theta_f)=0$ despite the fact that the original Lagrangian $L$ is defined everywhere. How are we supposed to approach this problem regarding both steps of analytical derivation and implementation details to get rid of unboundedness?


Intuitive analogy to the problem : One can consider $\theta(t)$ as a generalized coordinate of a single DoF system. A simple analogy toward $\eta$, on the other hand, can be a Jacobian or general transmission ratio, which is a function of position. The system executes a point to point motion between two points $\theta_0$ and $\theta_f$. Due to bijective nature of the relation between $\theta$ and $t$, all functions of $t$ have a trivial realization in $\theta$ domain. In this regard, to avoid a notation abuse in the post, I do that by introducing $\omega(\theta)$ in Eqn. (1) which maps to velocities from positions instead of $\dot{\theta}$ which equivalently maps to velocities from time

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