Noether charges in particle mechanics

First consider a transformation that leaves time unchanged and shifts generalized coordinates infinitesimally by ε times a specified function ξ(q,t). Its velocity shift is ε times the total time derivative of ξ. Calculate the first-order Lagrangian change using both shifts. If the result is ε times the total time derivative of a function F(q,t), the action changes only at the endpoints. This is a variational symmetry, sometimes called a quasi-symmetry when F is nonzero.

The boundary term matters because it changes the conserved charge. For this fixed-time transformation the charge is the sum of canonical momentum times ξ, minus F. Its normalization depends on how the transformation parameter was defined. For translation, ξ has the components of the chosen translation direction. For rotation, ξ is a cross product with the position. One should write the transformation explicitly before identifying a physical momentum, especially when a gauge potential makes canonical momentum differ from mechanical momentum.

Differentiate the proposed charge. Its terms contain momentum derivatives, derivatives of ξ, and the derivative of F. On an Euler–Lagrange solution, momentum derivatives equal coordinate derivatives of L. The first two groups then reconstruct the first-order change in L, which cancels the boundary derivative. Thus conservation holds on solutions, not for arbitrary trial paths. The theorem identifies a constant along allowed motion; it does not guarantee that solving for all trajectories becomes easy or that every conserved quantity is obvious in the chosen coordinates.

For an exactly translation-invariant Lagrangian, F vanishes and the appropriate component of canonical momentum is constant. Exact rotational invariance gives the corresponding angular-momentum component when the usual particle kinetic term applies. Time translation requires a transformation of the independent time variable, so the fixed-time formula above should not be reused blindly. The direct energy identity is safer: the Lagrangian energy changes at minus the explicit time derivative of L. An autonomous L therefore conserves that energy along its solutions.

A constant-force particle provides a symmetry with a nonzero boundary term. Translating position changes L by a constant proportional to the translation parameter, which is the time derivative of a linear function of time. Ordinary momentum is not conserved because a force acts, but momentum minus force times time is constant. This does not contradict Newton's law: differentiating the charge reproduces that law. The example shows why action symmetry is more flexible than demanding identical numerical L at every translated configuration.

The oscillator lab gives a separate autonomous example in which an energy check can be performed by hand. Set damping ratio to zero and choose a reachable natural frequency such as one. Predict the normalized relation x²+v²/ω₀²=1 for its fixed initial displacement and velocity, then inspect a few stepped readouts. The lab does not discover Noether transformations, calculate boundary terms, or verify general conservation laws. Adding damping removes this conservative energy relation, which is a useful reminder that a symmetry argument must match the actual equation being displayed.