# Space Symmetry and Conservation of Linear Momentum

I am trying to understand Noether's Theorem which links Space Symmetry and Conservation of Linear Momentum from an intuitive perspective.

1. Let's say we have car rolling down a frictionless surface that has a constant slope. Let's say the force of gravity is uniform over the entire surface. If we shift the car along this surface, then we have space symmetry. However, linear momentum is not conserved since the car gets faster. However, if we consider the car-earth system, then linear momentum is conserved.

2. Let's say we have car rolling down a frictionless surface that has a varying slope. If we shift the car along this surface, then we don't have space symmetry. Linear momentum is also not conserved. However, if we consider the car-earth system, then linear momentum is conserved.

I'm a bit confused about how exactly to apply Noether's Theorem.

1. In case 1, we have space symmetry, but we have to consider the car-earth system before linear momentum is conserved.

2. In case 2, space symmetry is not conserved, but linear momentum is conserved in the car-earth system.

Where am going wrong?

• Have you considered that the conservation of momentum requires that momentum is not changing with time (equivalently, that there is not a net force acting on a body)? In this case, momentum is not conserved for the car since gravity is acting on it. Noether’s theorem applies to the symmetries of the action (time integral of the Lagrangian), so this means that your system does not have this symmetry in this case. For this symmetry to exist, you need to actually satisfy Newton’s third law, which will give you a Lagrangian that satisfies this symmetry. Commented Aug 10, 2023 at 18:53