Conservative Forces & Conservation of Energy?

Question

I'm trying to relate them, I'm trying to find the key relation that would show how the conservative forces serve conservation of energy. How would they relate?

Also, how are non-conservative forces related to conservation of energy?

Answer 1

Let $T=\frac12mv^2$. Then $$ \mathrm dT= mv\dot v\,\mathrm dt=m\dot vv\,\mathrm dt= F\,\mathrm dx $$ In case of conservative forces, $$ F\,\mathrm dx=-\mathrm dV $$ and thus $$ \mathrm d(T+V)= 0 $$ This misses some physically interesting cases (in particular the Lorentz force) so let's see if we can do better: \begin{align*} \mathrm d(T+V) &= F\,\mathrm dx + \frac{\partial V}{\partial x}\mathrm dx + \frac{\partial V}{\partial v}\mathrm dv \\&= F\,\mathrm dx + \frac{\partial V}{\partial x}\mathrm dx + \mathrm d\left(\frac{\partial V}{\partial v} v\right) - \frac{\mathrm d}{\mathrm dt}\frac{\partial V}{\partial v} v\,\mathrm dt \end{align*} which implies $$ \mathrm d\left(T + V - \frac{\partial V}{\partial v}v\right) = \left( F + \frac{\partial V}{\partial x} - \frac{\mathrm d}{\mathrm dt}\frac{\partial V}{\partial v} \right)\,\mathrm dx $$ You obviously get a conserved quantity if $$ F = -\frac{\partial V}{\partial x} + \frac{\mathrm d}{\mathrm dt}\frac{\partial V}{\partial v} $$ This is the non-standard notion of velocity-dependent conservative forces.

The more conventional approach would be the switch to the Lagrangian description, which essentially goes from $$ \frac{\mathrm d}{\mathrm dt}\frac{\mathrm dT}{\mathrm dv}=\dot p_\text{kinetic}= F_\text{Newton}= -\frac{\partial V}{\partial x} + \frac{\mathrm d}{\mathrm dt}\frac{\partial V}{\partial v} $$ to $$ \frac{\mathrm d}{\mathrm dt}\frac{\partial(T-V)}{\partial v}=\dot p_\text{canonical}= F_\text{Lagrange}= \frac{\partial(T-V)}{\partial x} $$ which agree as $\partial T/\partial x=0$.

While energy is in principle always conserved, we can violate energy conservation if we fail to model relevant parts of the environment, either for convenience or by necessity. Examples would be explicit time dependence in potential or Lagrangian or energy dissipation, eg via friction, a force that cannot be written in terms of a velocity-dependent potential.

Answer 2

Conservative force allows us to add a term to the kinetic energy, so that the sum is always a constant. This term depends on position only.