# Is there a relativistic condition for Euler's first law of motion?

Euler's first law of motion states that the linear momentum of a body, $p$ (also denoted $G$) is equal to the product of the mass of the body $m$ and the velocity of its center of mass $v_{cm}$.

However, if a force is applied to some part of the body, then the center of mass cannot move instantaneously which seems to imply that Euler's law has a condition for it to be valid.

• Yes, from the question title there is relativistic momentum and it can be expressed as $\gamma mv$. What is the connection between relativistic momentum and classical momentum in the question text? – Mick Aug 30 '18 at 7:40
• @Mick I was thinking of the time delay between the center of mass moving and the application of the force being less than the time it takes for light to travel over this distance. I wasn't thinking of relativistic momentum. – Physiks lover Aug 30 '18 at 20:45

## 2 Answers

The force applied needs a time to reach the center of mass, because the information/perturbation needs to travel (maximum at velocity of light) from atom to atom. But after this, I don't think there's another condition to the law: as soon $\vec{F} = \frac{d\vec{p}}{dt}$ is different from zero, motion occurs (with allowing conditions).

In fact, the center of mass does begin to accelerate instantaneously. There is no need for the information to propagate to the center at either the speed of light or at the speed of sound in the material.

The material next to the point of application of the force will immediately begin to accelerate. The material far from the point of application will not immediately begin to accelerate.

The center of mass is calculated by averaging over all of the material, and so the center of mass will immediately begin to accelerate since it includes the material next to the point of application of the force. No information needs to propagate.