[Paper Review] Symmetries of the equations of motion that are not shared by the Lagrangian

G. F. Torres del Castillo, Alhan Moreno-Ruiz|arXiv (Cornell University)|May 23, 2017

Geotechnical and Geomechanical Engineering6 references4 citations

TL;DR

This paper demonstrates that equations of motion can remain form-invariant under certain transformations even when the corresponding Lagrangian is not invariant under those same transformations. The key contribution is proving the converse of the standard Noetherian logic: symmetry of equations of motion does not imply invariance of the Lagrangian, with explicit examples in both mechanical systems and field theories, including a non-Lorentz-invariant Lagrangian for source-free Maxwell equations that still yields Lorentz-covariant equations of motion.

ABSTRACT

We show that if a Lagrangian is invariant under a transformation (with the invariance defined in the standard manner), then the equations of motion obtained from it maintain their form under the transformation. We also show that the converse is not true, giving examples of equations of motion that are form-invariant under a transformation, but these equations can be derived from a Lagrangian that is not invariant under such transformation. The conclusions are valid for discrete or continuous systems.

Motivation & Objective

• To clarify the relationship between symmetries of equations of motion and symmetries of the corresponding Lagrangians, especially in light of widespread misconceptions in textbooks.
• To demonstrate that form-invariance of equations of motion under a transformation does not require invariance of the Lagrangian under the same transformation.
• To provide explicit counterexamples where equations of motion are symmetric but the Lagrangian is not, in both discrete and continuous systems.
• To show that even in relativistic field theories, such as Maxwell's equations, a Lagrangian can fail to be invariant under Lorentz transformations while still yielding Lorentz-covariant equations of motion.

Proposed method

• Defining invariance of equations of motion under a coordinate transformation (q′, t′) as the preservation of the functional form of the equations in the new variables.
• Using the standard Euler-Lagrange formalism to derive equations of motion from a Lagrangian L(qi, q̇i, t), treating variables as independent.
• Introducing a transformed Lagrangian L′ that accounts for time reparametrization via d t / d t′, ensuring equivalence of the resulting equations of motion.
• Applying the condition for invariance of a Lagrangian under a transformation: L′(q′i, q̇′i, t′) = L(qi, q̇i, t) × (d t / d t′), which preserves the dynamics.
• Analyzing infinitesimal Lorentz transformations to test invariance of a non-standard Lagrangian density for source-free Maxwell fields.
• Using tensor calculus and Levi-Civita symbol identities to show that the variation of the Lagrangian under a boost cannot be written as a total divergence, proving non-invariance.

Experimental results

Research questions

• RQ1 Can equations of motion be form-invariant under a transformation even when the underlying Lagrangian is not invariant under that transformation?
• RQ2 Is the converse of the standard Noether theorem result true: does symmetry of equations of motion imply invariance of the Lagrangian?
• RQ3 Can a Lagrangian that is not invariant under Lorentz transformations still yield Lorentz-covariant equations of motion for the electromagnetic field?
• RQ4 Are there field-theoretic examples where a gauge-invariant Lagrangian density exists that is not invariant under spacetime symmetries like Lorentz boosts?

Key findings

• The equations of motion derived from a Lagrangian are always form-invariant under any transformation that leaves the Lagrangian invariant, confirming the standard textbook claim.
• The converse does not hold: equations of motion can be form-invariant under a transformation even when the Lagrangian is not invariant under that transformation.
• An explicit example is given of a Lagrangian for the source-free Maxwell equations that is not invariant under Lorentz boosts, yet yields equations of motion that are Lorentz-covariant.
• The non-invariance of the Lagrangian under Lorentz transformations is proven by showing that its variation cannot be expressed as a total divergence, violating the necessary condition for invariance.
• The paper provides a non-Lagrangian formulation of Maxwell's equations using only E and B fields, which is gauge-independent and avoids potential issues with potential-based formulations.
• The result implies that the search for fundamental symmetries in field theories should not be restricted to Lagrangians alone, as equations of motion may exhibit symmetries absent in the Lagrangian.