Conformal Invariance And Conserved Quantities For Mechanical Systems

Applying symmetries to seek conserved quantities of all kinds of constrained mechanical is a modern developmental direction in analytical dynamics, and possesses important theoretical and practical significance in mathematical sciences. Studying a new symmetry of mechanical system will break through original some types of symmetries in theory of symmetries and conserved quantities of a mechanical system. It provides a new rationale to reveal inherent properties and laws of a mechanical system. In this paper, we study a new symmetry called conformal invariance and conserved quantities of a mechanical system. conformal invariance and conserved quantities of mechanical systems. Firstly, conformal invariance for Nambu mechanical systems is studied. From the dynamical equation, the definition and determining equation of conformal invariance are given. Using the relationship between conformal invariance and Lie symmetry, conformal factor is dedued. We aslo give the Kai conserved quantities, Hojman conserved quantities,I type Hojman conserved quantities andâ…¡type Hojman conserved quantities. Secondly, conformal invariance for higher-order holonomic mechanical systems is studied. From the dynamical equation, the definition and determining equation of conformal invariance are given. Using the relationship between conformal invariance and Lie symmetry, conformal factor is dedued. We give the Noether conserved quantities yet. Thirdly, conformal invariance for higher-order nonholonomic mechanical systems is studied. From the dynamical equation, the definition and determining equation of conformal invariance is given. Using the relationship between conformal invariance and Lie symmetry, conformal factor is dedued. We give the Noether conserved quantities. Fourthly, conformal invariance for Vacoo mechanical systems is studied. From the dynamical equation, the definition and determining equation of conformal invariance are given. Using the relationship between conformal invariance and Lie symmetry, conformal factor is dedued. We give the Noether conserved quantities and Hojman conserved quantities yet. Lastly, the researches in this paper are summarized and theoritical researches on conformal invariance and conserved quantities for mechanical systems are prospected.