The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d＇Alembert-Lagrange princi...The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d＇Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal trans- formations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.展开更多
A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplec...A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplectic-energy-first integrators for mechanico-electrical systems are derived. To do this, the time step adaptation is employed. The discrete variational principle and the Euler-Lagrange equation are derived for the systems. By using this discrete algorithm it is shown that mechanico-electrical systems are not symplectic and their energies are not conserved unless they are Lagrange mechanico-electrical systems. A practical example is presented to illustrate these results.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11272287 and 11472247)the Program for Changjiang Scholars and Innovative Research Team in University of China(Grant No.IRT13097)
文摘The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is estab- lished, and the fractional Lagrange equations are obtained by virtue of the d＇Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal trans- formations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10672143 and 60575055)the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciencesthe Natural Science Foundation of Henan Province Government, China (Grant No 0511022200)
文摘A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplectic-energy-first integrators for mechanico-electrical systems are derived. To do this, the time step adaptation is employed. The discrete variational principle and the Euler-Lagrange equation are derived for the systems. By using this discrete algorithm it is shown that mechanico-electrical systems are not symplectic and their energies are not conserved unless they are Lagrange mechanico-electrical systems. A practical example is presented to illustrate these results.
