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A special Lie symmetry and Hojman conserved quantity of the Appell equations for a Chetaev nonholonomic system are studied. The differential equations of motion and Appell equations of the Chetaev nonholonomic system are established. Under the special Lie symmetry group transformations in which the time is invariable, the determining equation of the special Lie symmetry of the Appell equations for a Chetaev nonholonomic system is given, and the expression of the Hojman conserved quantity is deduced directly from the Lie symmetry. Finally, an example is given to illustrate the application of the results.

For a weakly nonholonomic system, the Lie symmetry and approximate Hojman conserved quantity of Appell equations are studied. Based on the Appell equations for a weakly nonholonomic system under special infinitesimal transformations of a group in which the time is invariable, the definition of the Lie symmetry of the weakly nonholonomic system and its first-degree approximate holonomic system are given. With the aid of the structure equation that the gauge function satisfies, the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are derived. Finally, an example is given to study the exact and approximate Hojman conserved quantity of the system.

Lie symmetry and conserved quantity deduced from Lie symmetry of Appell equations in a dynamical system of relative motion with Chetaev-type nonholonomic constraints are studied. The differential equations of motion of the Appell equation for the system, the definition and criterion of Lie symmetry, the condition and the expression of generalized Hojman conserved quantity deduced from Lie symmetry for the system are obtained. The condition and the expression of Hojman conserved quantity deduced from special Lie symmetry for the system under invariable time are further obtained. An example is given to illustrate the application of the results.

A type of structural equation, new exact and approximate conserved quantity which are deduced from Mei symmetry of Lagrange equations for a weakly nonholonomic system, are investigated. First, Lagrange equations of weakly nonholonomic system are established. Next, under the infinitesimal transformations of Lie groups, the definition and the criterion of Mei symmetry for Lagrange equations in weakly nonholonomic systems and its first-degree approximate holonomic system are given. And then, the expressions of new structural equation and new exact and approximate conserved quantities of Mei symmetry for Lagrange equations in weakly nonholonomic systems are obtained. Finelly, an example is given to study the question of the exact and the approximate new conserved quantities.