Symmetries and Noether's theorem

Equipment

• Orbit canvas on the central-potential surface (3D, drag to rotate)
• Slider “Mômen động lượng L” (normalized Lz readout)
• Readout “Thế V(r)=−1/r … Lz = … không đổi”
• Rotational-symmetry canvas: circle illustrating the transformation φ → φ + ε

Procedure

1. Observe the orbit in the central potential

   In section 1, drag the canvas to rotate and see the surface $V(r) = -1/r$ symmetric about the vertical axis through the center. The red orbit is not circular — the radius changes in time — yet the readout reports “Lz = … không đổi” because a central force exerts zero torque.

2. Change the angular momentum L

   Move the “Mômen động lượng L” slider through several values; the number beside it gives the normalized Lz. For each L the orbit and readout update, but the message stands: Lz is conserved along the orbit. This is the statement $p_\varphi = \text{const}$ when $\varphi$ is a cyclic coordinate.

3. Read the symmetry transformation

   In section 2, the circle and red dot illustrate rotation about the z axis; the readout explains “Biến đổi φ → φ + ε không làm L thay đổi”. Rotate the canvas to see that turning around the center only changes the azimuthal angle; since the Lagrangian does not depend on $\varphi$, Noether's theorem gives conserved $p_\varphi = L_z$ — a consequence of $\delta L = 0$.

4. Generalize to other symmetries

   Compare two cases: if the potential depended on $\varphi$ (e.g., a tilted “hill” on the surface), rotation would no longer be a symmetry and $L_z$ would change — the force would exert a torque. Predict before observing: by the same Noether logic, time-translation invariance yields energy $E$ conservation, and spatial-translation invariance yields momentum $\vec P$ conservation.

Simulation

Experiment history

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