A Note on the scale symmetry and Noether current

TL;DR

This paper explores the distinction between symmetry of action and symmetry of equations of motion, focusing on scale symmetry in classical mechanics and field theory, and derives related expressions involving Noether charges.

Contribution

It demonstrates that scale transformations not leaving the action invariant can still preserve equations of motion and relates the action to boundary terms of Noether charges.

Findings

Scale symmetry in equations of motion differs from action invariance.

Action can be expressed as boundary differences of formal Noether charges.

In field theory, the action relates to boundary integrals of Noether currents.

Abstract

Usually we consider the symmetry of action as the symmetry of the theory, however, in the Keplar problem the scaling symmetry existing in equa tion of motion is not the ones for action. It changes the multiplicative c onstant of action and the time boundary. In such a case that the scale tran sformation does not leave the action invariant but keeping the equation inva riant, the following statement is proved. The time integration of Lagrangian is explicitly performed and the action ca n be expressed by the difference of formal (non-conserved) Noether charges a t time boundaries. In field theory the action can be expressed by the bound ary integration of the formal Noether current.