If I could turn back time

Classical Mechanics Level 3

Newton's law states F = m r ¨ ( t ) F = m\ddot{r}(t) . Solving this equation for a given force field yields a curve r ( t ) r(t) that governs the motion of particles in that field.

The physics in a field is said to obey time-reversal symmetry if whenever r ( t ) r(t) is a permissible motion, r ( − t ) r(-t) is as well.

Of the following forces, which does not obey time-reversal symmetry?

  • Gravitational F = − G M m r ( t ) 2 r ^ \displaystyle F = -G\frac{Mm}{r(t)^2}\hat{r}
  • Electric F = q E ⃗ \displaystyle F = q\vec{E}
  • Magnetic F = q v ⃗ ( t ) × B ⃗ \displaystyle F = q\vec{v}(t)\times\vec{B}

Assumptions

  • Take the fields in question to be constant, and time-independent.
Electric Gravitational Magnetic

Josh Silverman by Josh Silverman

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3 solutions

Steven Chase
Dec 9, 2016

The "poor man's solution" is that the Coulomb force and the gravity force are exactly the same structurally (at least when looking at fields generated by point particles). By process of elimination, the magnetic force has to be the odd one out. Not sure how to address the more general case of induced E-fields and G-fields though.

2 Helpful 0 Interesting 1 Brilliant 0 Confused
Rohan Joshi
Feb 7, 2021

Magnetic force depends on velocity, more precisely the cross product of velocity with the magnetic field. If time is reversed, so is velocity, leading to the negative of the original cross product. Thus the force will point in the opposite direction

0 Helpful 0 Interesting 1 Brilliant 0 Confused

Bonus question: what does this time reversal symmetry have to do with the field being conservative or non conservative?

Rohan Joshi - 4 months, 1 week ago
Swapnil Saha
Jul 20, 2015

As the magnetic force is always perpendicular to the velocity of the particle; it can not change the magnetude ; just can change the ditection of the velocity.That's why we can not use F=ma to get the magnetude of a.

1 Helpful 0 Interesting 0 Brilliant 1 Confused

I am having a hard time understanding how this solution addresses the question. What does the force being perpendicular to the velocity of the particle have to do with time reversal symmetry?

Josh Silverman Staff - 5 years, 10 months ago

@Josh Silverman check my solution

Rohan Joshi - 4 months, 1 week ago

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