Energy not conserved if mass is variable In classical Mechanics it is possible to prove that the total Energy $E = T + V$ is conserved if the force is conservative. However, if you assume the mass is time-dependent this proof no longer holds:
\begin{align}
   \frac{dE}{dt} = \frac{d}{dt}(\frac{1}{2}m\dot{\vec{r}}^2 + V) = \frac{\dot{m}}{2}\dot{\vec{r}}^2+m\dot{\vec{r}}\cdot \ddot{\vec{r}}+\frac{dV}{dt}
\end{align}
Furthermore, $\frac{dV}{dt} = \dot{\vec{r}} \cdot \nabla V = - \dot{\vec{r}} \cdot\vec{F} = -\dot{\vec{r}} \cdot\left(\dot{m} \dot{\vec{r}} + m \ddot{\vec{r}} \right)$  under the assumption that the mass is time-dependent. Therfore, weg get
\begin{align}
\frac{dE}{dt} = \frac{\dot{m}}{2}\dot{\vec{r}}^2+m\dot{\vec{r}}\cdot \ddot{\vec{r}}  -\dot{\vec{r}} \cdot\left(\dot{m} \dot{\vec{r}} + m \ddot{\vec{r}}  \right) = -\frac{\dot{m}}{2} \dot{\vec{r}}^2
\end{align}
So is there a flaw in this reasoning, or is energy (for conservative forces) only conserved if mass is time-independent?
 A: The correct statement of the law of conservation of energy is that the total energy of an isolated system is conserved. This is valid both for classical and for relativistic mechanics.
In your example, the energy is not conserved as a consequence of the fact that the total mass of the system is not constant. In classical mechanics, the total mass can change only if a system is exchanging matter with the external world, that is, if it is not isolated.
In classical mechanics, the total mass of an isolated system is constant.
I stress the fact that the total mass is constant. An isolated system of many bodies can have individual masses which are time-dependent (for example if there are bodies which collapse one on the other, or if there are explosions). Also in this case the total mass and the total energy are conserved.
A: I try to explain the idea that I got following the computational study of a variable mass system.
Let us consider an isolated system in a certain state of motion. It will have a certain mass and geometry, i.e. a moment of inertia, which quantify its ability to oppose changes in its dynamic state.
Imagine that its mass varies over time for some reason.
The system is isolated and subjected only to conservative forces; for some reason (for example a pendulum made up of a holey sack full of sand) the mass, and therefore the inertia, changes. Then, "the way in which the system reacts to forces" changes, opposing less or more resistance depending on whether the mass decreases or increases, and to remedy this fact, the energy of the system must increase or decrease: in the first case there is an "excess" of work that can be used only by increasing the speed (kinetic energy) and / or amplitude (potential energy); in the other case, on the other hand, work is necessary (which, being the system isolated, must necessarily be provided at the expense of the mechanical energy of the pendulum) in order to maintain the state of motion against the increase in inertia.
Formally, in the cardinal equations of motion, as you can verify, a damping term is proportional, with a minus sign, to the derivative of the mass (in the case of a material point).
If the mass increases (decreases) then the derivative is positive (negative) and the object describes a damping (forcing), like a viscous friction with a variable coefficient.
For example, in a variable mass pendulum, computational work will lead you to the following result:
if the mass decreases then the oscillations increase in amplitude (ergo the energy of the system increases), if instead the inertia increases then the oscillations decrease in amplitude (energy is consumed).
The new term in the equation mathematically denounces the fact that the system is changing its inertial mass and, depending on the case, this requires or provides work: to maintain a motion, even constant in speed, it is necessary to provide external work if the system increases in mass; if instead the system loses mass, this translates into usable energy: in the case of
pendulum pierced the force of gravity decreases the mass and the consequent excess of energy is used by the system to increase the amplitude of the oscillations (if one is in an oscillatory regime, otherwise a chaotic motion is expected).
