A Geometric Variational Problem for Pseudo-Galilean Particles

A Geometric Variational Problem for Pseudo-Galilean Particles

This study explores the dynamics of particle motion in pseudo-Galilean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>−</mo></mrow></semantics></math&g...

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Bibliographic Details
Main Authors: Ayşe Yılmaz Ceylan, Tunahan Turhan, Gözde Özkan Tükel
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Axioms
Subjects:
calculus of variation
curvature energy
Euler–Lagrange equations
pseudo-Galilean space
torsion energy
Online Access:https://www.mdpi.com/2075-1680/14/7/520
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Summary:This study explores the dynamics of particle motion in pseudo-Galilean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>−</mo></mrow></semantics></math></inline-formula>space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>G</mi><mrow><mn>3</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>G</mi><mrow><mn>3</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula>. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>G</mi><mrow><mn>3</mn></mrow><mn>1</mn></msubsup></semantics></math></inline-formula> and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications.
ISSN:2075-1680

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