# Newtons third law and conservation of momentum

Whats more basic - newtons third law or conservation of momentum
You can prove newtons third law by conservation of momentum but you can also prove conservation of momentum by newtons third law. What comes first?

## Answers and Replies

Dale
Mentor
2020 Award
Conservation of momentum is more basic. It applies in situations where the notion of force can be difficult to even define. For example, in electromagnetism the momentum of the EM field is well-defined, but the idea of a force acting on the EM field is a little strange.

So just like conservation of energy - conservation of momentum is fundamental - it just happens right? - there is no reason why momentum is conserved right?

AlephZero
Science Advisor
Homework Helper
Dale
Mentor
2020 Award
So just like conservation of energy - conservation of momentum is fundamental - it just happens right? - there is no reason why momentum is conserved right?
According to Noether's theorem (see the link provided by AlephZero above) there is a reason. Noether's theorem applies to any physical theory which can be expressed in terms of a Lagrangian. If the Lagrangian has some differential symmetry (i.e. it does not change under some specific transformation) then there is a quantity which is conserved. This link between symmetry and conservation is so fundamental that the most basic theories are expressed in terms of their symmetries, and everything follows from those.

Energy is conserved because the Lagrangian does not change with small translations in time. Momentum is conserved because the Lagrangian does not change with small translations in space. Angular momentum is conserved because the Lagrangian does not change with small rotations in space. Charge is conserved because the Lagrangian does not change with small changes in potential. Etc.