Consider a second order gradient-like system with linear damping $$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Suppose $f\in C^2(\mathbb{R}^n)$ and the solution $x:[0,\infty)\rightarrow \mathbb{R}^n$ is bounded, i.e., $\lVert x(t)\rVert\leq c$ for all $t\in [0,\infty)$, where $c$ is a constant.

If we change the so called "gravity constant" (see this paper) from $1$ to any positive constant $g$, i.e., $$\ddot{x}+\dot{x}+g\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Is the solution to this new equation still bounded?

Consider a second order gradient-like system with linear damping $$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)>-\infty$. The solution $x:[0,\infty)\rightarrow \mathbb{R}^n$ is bounded, i.e., $\lVert x(t)\rVert\leq c$ for all $t\in [0,\infty)$, where $c$ is a constant.

If we change the so called "gravity constant" (see this paper) from $1$ to any positive constant $g$, i.e., $$\ddot{x}+\dot{x}+g\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Is the solution to this new equation still bounded?