Student Friendly Introduction to QFT · Problem 10
Interactions: The Underlying Theory
Physics — QFT · lagrangian-symmetry-spring-particle
Problem
For a particle attached to a spring confined to move in one dimension, the potential energy . Use this to find the Lagrangian of this system. Is this Lagrangian symmetric with respect to translations of ? Is momentum conserved in the direction? Find the equation of motion for the system using the Lagrangian approach. What does the momentum equal? Is this Lagrangian symmetric in ? Is momentum conserved in the direction? Does this make sense physically?Intuitive explanation
The Lagrangian is the kinetic energy minus potential energy, , and it encodes all the dynamics of a system. Symmetries of the Lagrangian directly tell you which quantities are conserved via Noether's theorem: if doesn't change when you shift a coordinate, momentum in that direction is conserved. Here you'll see that the spring potential breaks translation symmetry in (the direction of motion), so momentum isn't conserved—but (a direction the particle can't move in) has no forces, so translation symmetry and momentum conservation hold there, which is physically sensible.
Intuitive explanation
The Lagrangian is the kinetic energy minus potential energy, , and it encodes all the dynamics of a system. Symmetries of the Lagrangian directly tell you which quantities are conserved via Noether's theorem: if doesn't change when you shift a coordinate, momentum in that direction is conserved. Here you'll see that the spring potential breaks translation symmetry in (the direction of motion), so momentum isn't conserved—but (a direction the particle can't move in) has no forces, so translation symmetry and momentum conservation hold there, which is physically sensible.