1. **Problem statement:** We want to prove or refute that under the condition $\beta < \alpha$, the functional $\mathcal{F}$ is a Lyapunov function for the GTA flow given by $$\frac{dS}{dt} = -\alpha \nabla_{\mathcal{M}} \mathcal{F}(S) + \beta T_c(S)$$ on a closed constraint manifold $\mathcal{M} \subset \mathcal{S}$. 2. **Recall the definition of a Lyapunov function:** A functional $\mathcal{F}$ is a Lyapunov function for a flow if along trajectories $S(t)$ of the flow, the time derivative $\frac{d}{dt} \mathcal{F}(S(t)) \leq 0$ and equals zero only at equilibrium points. 3. **Compute the time derivative of $\mathcal{F}$ along the flow:** $$\frac{d}{dt} \mathcal{F}(S(t)) = \langle \nabla_{\mathcal{M}} \mathcal{F}(S), \frac{dS}{dt} \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on the tangent space of $\mathcal{M}$. 4. **Substitute the flow expression:** $$\frac{d}{dt} \mathcal{F}(S(t)) = \left\langle \nabla_{\mathcal{M}} \mathcal{F}(S), -\alpha \nabla_{\mathcal{M}} \mathcal{F}(S) + \beta T_c(S) \right\rangle = -\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \|^2 + \beta \langle \nabla_{\mathcal{M}} \mathcal{F}(S), T_c(S) \rangle$$ 5. **Analyze the term $\langle \nabla_{\mathcal{M}} \mathcal{F}(S), T_c(S) \rangle$:** Since $T_c(S)$ is a tangent vector field on $\mathcal{M}$, if $T_c(S)$ is orthogonal to $\nabla_{\mathcal{M}} \mathcal{F}(S)$, then $$\langle \nabla_{\mathcal{M}} \mathcal{F}(S), T_c(S) \rangle = 0$$ which simplifies the derivative to $$\frac{d}{dt} \mathcal{F}(S(t)) = -\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \|^2 \leq 0$$ 6. **If $T_c(S)$ is not orthogonal, use Cauchy-Schwarz inequality:** $$|\langle \nabla_{\mathcal{M}} \mathcal{F}(S), T_c(S) \rangle| \leq \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| \cdot \| T_c(S) \|$$ 7. **Rewrite the derivative bound:** $$\frac{d}{dt} \mathcal{F}(S(t)) \leq -\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \|^2 + \beta \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| \cdot \| T_c(S) \|$$ 8. **Factor out $\| \nabla_{\mathcal{M}} \mathcal{F}(S) \|$:** $$\frac{d}{dt} \mathcal{F}(S(t)) \leq \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| \left(-\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| + \beta \| T_c(S) \| \right)$$ 9. **Condition for negativity:** For $\frac{d}{dt} \mathcal{F}(S(t)) \leq 0$ to hold for all $S$, it suffices that $$-\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| + \beta \| T_c(S) \| \leq 0$$ which implies $$\alpha \| \nabla_{\mathcal{M}} \mathcal{F}(S) \| \geq \beta \| T_c(S) \|$$ 10. **Since $\beta < \alpha$, if $\| T_c(S) \| \leq \| \nabla_{\mathcal{M}} \mathcal{F}(S) \|$ for all $S$, then the inequality holds and $\mathcal{F}$ is a Lyapunov function.** 11. **Conclusion:** Under the condition $\beta < \alpha$ and assuming $\| T_c(S) \| \leq \| \nabla_{\mathcal{M}} \mathcal{F}(S) \|$ or $T_c(S)$ orthogonal to $\nabla_{\mathcal{M}} \mathcal{F}(S)$, the functional $\mathcal{F}$ is a Lyapunov function for the GTA flow on $\mathcal{M}$. Otherwise, the presence of the $\beta T_c(S)$ term may prevent $\mathcal{F}$ from being strictly decreasing, refuting it as a Lyapunov function in general.