# Relation between Potential energy and torque

I know, this may seem as a bit of a weird question. As we know that the relation between force and potential energy i.e: $$F = -\frac{\text{d}U}{\text{d}x}$$ so can we say that relation between torque about an axis and potential energy to be: $$\tau = -\frac{\text{d}U}{\text{d}\theta}$$

Any help would be appreciated.

Short answer is yes, as long as your system is conservative.

Longer answer is that the Euler-Lagrange equations tell you what the equations of motion really are for these systems, so you setup the Lagrangian $$L(r, \theta,\dot{r},\dot{\theta}) = T(\dot{r},\dot{\theta}) - U(r,\theta)$$, where $$T$$ is the kinetic energy and $$U$$ is potential energy. Then calculate:

$${d \over dt} {\partial L \over \partial \dot{\theta}}={\partial L \over \partial \theta}$$.

In most cases, $${\partial L \over \partial \theta}$$ is exactly $$-dU\over d\theta$$,and the left hand side turns out to be $$I\alpha$$.

• But what happens if the force is non conservative? Commented Jun 17, 2021 at 14:18
• "Short answer is yes" yes to what? Did he ask any question? I can't find one. Commented Dec 28, 2021 at 8:38
• "so can we say that relation between torque about an axis and potential energy to be ....?" Commented Jan 20, 2022 at 17:26

In my opinion, the best definition of torque is the work per unit angle which can be done by a force which is acting in a manner that tends to cause a rotation. If motion occurs then τ = +dU/dθ where U is the total energy of the rotating system (not just the potential energy). Your expression refers to the torque coming from an external source, such as a deformed spring which is losing potential energy as it causes a rotation.