# I A question about Noether theorem

Tags:
1. Feb 21, 2017

### larsa

How can I derive that the work of a force perpendicular to velocity is always zero from the theorem of Noether?
I have heard that there is a relation between these two but in Google I found nothing.

Thank you very much

2. Feb 21, 2017

### vanhees71

For that you don't need Noether's theorem. The usual work theorem will do. The Newtonian EoM reads
$$m \ddot{\vec{x}}=\vec{F}.$$
Now multiply with $\dot{\vec{x}}$, and you get
$$m \dot{\vec{x}} \cdot \ddot{\vec{x}}=\frac{\mathrm{d}}{\mathrm{d} t} \frac{m}{2} \dot{\vec{x}}^2 = \vec{F} \cdot \dot{\vec{x}}.$$
Now you if $\dot{\vec{x}} \perp \vec{F}$ the right-hand side is 0, and thus the kinetic energy is constant, i.e., the force doesn't do work.

3. Feb 21, 2017

### Staff: Mentor

Yes, but the question has been, how they are related? I came so far to see that Noether says: work as a function of force and position ($\;dW = F(\vec{x}) \cdot \nabla \vec{x}\;$) is invariant under orthogonal coordinate transformations. But how is this related to the fact, that an orthogonal force doesn't add work? (I just don't see the argument.)

4. Feb 21, 2017

### larsa

There must be some relation but i can't imagine any. Thank you for your answer