How can a potential depending on velocities give equal and opposite forces?

In summary, on page 10 of Goldstein's classical mechanics, it is stated that if the potential V_{jj} is also dependent on the difference of some other pair of vectors associated with the particles, such as their velocities or intrinsic "spin" angular momenta, then the forces between the particles would still be equal and opposite, but may not necessarily lie along the direction between them. This means that if the potential is dependent on velocities, for example, the forces derived may not fully comply with the law of action and reaction. However, if the potential is dependent on differences of these vectors, a weaker version of the law can still be satisfied.
  • #1
sadness
7
0
On page 10 of Goldstein's classical mechanics, it was said:

"If [tex]V_{jj}[/tex] were also a function of the difference of some other pair of vectors associated with the particles, such as their velocities or (to step into the domain of modern physics) their intrinsic "spin" angular momenta, then the forces would still be equal and opposite, but would not necessarily lie along the direction between the particles."

What does this mean? IMO if a potential is dependent on velocities, the forces derived typically will not respect any aspects of the law of action and reaction. Why would they still be equal and opposite?
 
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  • #2
sadness said:
On page 10 of Goldstein's classical mechanics, it was said:

"If [tex]V_{jj}[/tex] were also a function of the difference of some other pair of vectors associated with the particles, such as their velocities or (to step into the domain of modern physics) their intrinsic "spin" angular momenta, then the forces would still be equal and opposite, but would not necessarily lie along the direction between the particles."

What does this mean? IMO if a potential is dependent on velocities, the forces derived typically will not respect any aspects of the law of action and reaction. Why would they still be equal and opposite?

The point is that the book precises [tex]V_{jj}[/tex] being function of the difference of some vector associated to the particles, that is not generally of velocities for instance as you say, but for differences of them in such a way that (1.33) still holds but not (1.34), when now the dependence is not just of the relative position and the forces not central, as mentioned in page 7. Then this weak action-reaction law is satisfied, but not the "strong".
 
  • #3
Rebel said:
The point is that the book precises [tex]V_{jj}[/tex] being function of the difference of some vector associated to the particles, that is not generally of velocities for instance as you say, but for differences of them in such a way that (1.33) still holds but not (1.34), when now the dependence is not just of the relative position and the forces not central, as mentioned in page 7. Then this weak action-reaction law is satisfied, but not the "strong".

Thanks. I understood what the authors meant now.
 

1. How does a potential depend on velocities?

The potential energy of a system can depend on the velocities of the particles within the system if there are forces acting on the particles that are velocity-dependent. This means that the potential energy can change depending on the speed and direction of the particles' motion.

2. What does it mean for two forces to be equal and opposite?

When two forces are equal and opposite, it means that they have the same magnitude but are acting in opposite directions. This is known as Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction.

3. How can a potential give equal and opposite forces?

If a potential energy function is dependent on the velocities of particles, then it can produce equal and opposite forces through the process of differentiation. By taking the derivative of the potential energy function with respect to the position of the particles, we can determine the force acting on the particles. This force will be equal and opposite for two particles in the same system.

4. Why is it important for potential energies to give equal and opposite forces?

In order for a system to be in equilibrium, the forces acting on the particles must be balanced. This means that the sum of all forces in the system should be equal to zero. By having equal and opposite forces, the system is able to maintain this equilibrium and remain stable.

5. Can the potential energy of a system change depending on the velocities of particles?

Yes, the potential energy of a system can change depending on the velocities of particles if there are velocity-dependent forces acting on the particles. These forces can result in changes to the potential energy, which in turn affects the motion of the particles. This is why potential energies are often used to describe the behavior of systems in physics and chemistry.

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