Conserved quantities of Euler-Lagrange system via complex Lagrangian.

Area of Science:

• Theoretical Physics
• Mathematical Physics
• Classical Mechanics

Background:

• Euler-Lagrange (EL) systems are fundamental in classical mechanics.
• Existing methods identify Noether and Lie conserved quantities.
• Previous studies by Fang et al. and Nucci expanded on conserved quantities for EL systems.

Purpose of the Study:

• To introduce a novel complex Lagrangian technique for deriving Noether-like operators and conserved quantities.
• To demonstrate that previously identified conserved quantities can be obtained using this complex variational formalism.
• To uncover additional conserved quantities beyond those previously reported.

Main Methods:

• Application of complex Lagrangian formalism.
• Derivation of Noether-like operators.
• Identification of associated conserved quantities (first integrals).

Main Results:

• The complex Lagrangian technique successfully generates 10 Noether-like operators and 10 corresponding invariant quantities for EL systems.
• Three specific conserved quantities (Noether, Lie, Mei) identified by Fang et al. are reproduced.
• Several other conserved quantities align with those reported by Nucci, with additional novel ones discovered.

Conclusions:

• The complex variational formalism provides an effective alternative approach for computing invariant quantities in EL systems.
• This methodology expands the number of known conserved quantities for these systems.
• The findings offer new insights into the symmetries and conservation laws of Euler-Lagrange systems.