Conservation of Energy for Conservative Forces

• Total Energy:
  The total mechanical energy $E$ of a system is given by the sum of its kinetic energy $T$ and potential energy $U$:

  $$E=T(\overrightarrow{v}(t))+U(\overrightarrow{r}(t),t)$$

• Time Derivative of Energy:
  Taking the time derivative of the total energy:

  $$\frac{dE}{dt}=\frac{dT}{dt}+\frac{dU}{dt}$$

• Kinetic Energy:
  The kinetic energy $T$ is given by:

  $$T=\frac{1}{2}m{v}^2$$

  The time derivative of $T$ can be computed as follows:

  $$\frac{dT}{dt}=m\overrightarrow{v}\cdot \frac{d\overrightarrow{v}}{dt}=m\overrightarrow{v}\cdot \overrightarrow{a}=\overrightarrow{v}\cdot \overrightarrow{F}$$

• Potential Energy:
  The time derivative of the potential energy $U$ is given by:

  $$\frac{dU}{dt}=\sum _{\alpha }\frac{\partial U}{\partial x^{\alpha }}\frac{d{x}^{\alpha }}{dt}+\frac{\partial U}{\partial t}$$

  In vector notation, this can be expressed as:

  $$\frac{dU}{dt}=\overrightarrow{v}\cdot \mathrm{\nabla }U+\frac{\partial U}{\partial t}$$

• Total Energy Rate of Change:
  Substituting the expressions for $\frac{dT}{dt}$ and $\frac{dU}{dt}$ into the equation for $\frac{dE}{dt}$:

  $$\frac{dE}{dt}=\overrightarrow{v}\cdot \overrightarrow{F}+\overrightarrow{v}\cdot \mathrm{\nabla }U+\frac{\partial U}{\partial t}$$

• For Conservative Forces:
  For conservative forces, the force can be expressed as:

  $$\overrightarrow{F}=-\mathrm{\nabla }U$$

  Substituting this into the energy rate of change gives:

  $$\frac{dE}{dt}=\overrightarrow{v}\cdot (-\mathrm{\nabla }U)+\overrightarrow{v}\cdot \mathrm{\nabla }U+\frac{\partial U}{\partial t}$$

  The first two terms cancel each other out, leading to:

  $$\frac{dE}{dt}=\frac{\partial U}{\partial t}$$

• Constant Potential Energy:
  If $U\equiv U(\overrightarrow{r})$ (i.e., potential energy does not explicitly depend on time), then:

  $$\frac{\partial U}{\partial t}=0$$

  Therefore, we have:

  $$\frac{dE}{dt}=0$$

• Conclusion:
  Since the time derivative of the total energy $E$ is zero, this means that $E$ is a constant:

  $$E=\text{constant}$$

  This shows the conservation of mechanical energy in a system where only conservative forces are acting.