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《应用数学教材丛书·力学和对称性导论(第2版)》可供从事应用数学、力学专业的高年级大学生和研究生使用，也可作为相关领域专家、学者的参考书。 |
| Jerrold E. Marsden is professor of control and dynamical systems at Caltech. He received his B.Sc. at Toronto in 1965 and his Ph.D. from Princeton University in 1968, both in applied mathematics. He has done extensive research in mechanics, with applications to rigid-body systems, fluid mechanics, elasticity theory, plasma physics, as well as to general field theory. His current interests include dynamical systems and control theory and how it re.. << 查看详细 |
| preface . about the authors 1 introduction and overview 1.1 lagrangian and hamiltonian formalisms 1.2 the rigid body 1.3 lie poisson brackets, poisson manifolds, momentum maps 1.4 the heavy top 1.5 incompressible fluids 1.6 the maxwell vlasov system 1.7 nonlinear stability 1.8 bifurcation 1.9 the poincare-melnikov method 1.10 resonances, geometric phases, and control 2 hamiltonian systems on linear symplectic spaces 2.1 introduction 2.2 symplectic forms on vector spaces 2.3 canonical transformations, or symplectic maps 2.4 the general hamilton equations 2.5 when are equations hamiltonian? 2.6 hamiltonian flows .2.7 poisson brackets 2.8 a particle in a rotating hoop 2.9 the poincare-melnikov method 3 an introduction to infinite-dimensional systems 3.1 lagrange's and hamilton's equations for field theory 3.2 examples: hamilton's equations 3.3 examples: poisson brackets and conserved quantities 4 manifolds, vector fields, and differential forms 4.1 manifolds 4.2 differential forms 4.3 the lie derivative 4.4 stokes' theorem 5 hamiltonian systems on symplectic manifolds 5.1 symplectic manifolds 5.2 symplectic transformations 5.3 complex structures and kahler manifolds 5.4 hamiltonian systems 5.5 poisson brackets on symplectic manifolds 6 cotangent bundles 6.1 the linear case 6.2 the nonlinear case 6.3 cotangent lifts 6.4 lifts of actions 6.5 generating functions 6.6 fiber translations and magnetic terms 6.7 a particle in a magnetic field 7 lagrangian mechanics 7.1 hamilton's principle of critical action 7.2 the legendre transform 7.3 euler lagrange equations 7.4 hyperregular lagrangians and hamiltonians 7.5 geodesics 7.6 the kaluza-klein approach to charged particles 7.7 motion in a potential field 7.8 the lagrange-d'alembert principle 7.9 the hamilton-jacobi equation 8 variational principles, constraints, & rotating systems 8.1 a return to variational principles 8.2 the geometry of variational principles .. 8.3 constrained systems 8.4 constrained motion in a potential field 8.5 dirac constraints 8.6 centrifugal and coriolis forces 8.7 the geometric phase for a particle in a hoop 8.8 moving systems 8.9 routh reduction 9 an introduction to lie groups 9.1 basic definitions and properties 9.2 some classical lie groups 9.3 actions of lie groups 10 poisson manifolds 10.1 the definition of poisson manifolds 10.2 hamiltonian vector fields and casimir functions 10.3 properties of hamiltonian flows 10.4 the poisson tensor 10.5 quotients of poisson manifolds 10.6 the schouten bracket 10.7 generalities on lie-poisson structures 11 momentum maps 11.1 canonical actions and their infinitesimal generators 11.2 momentum maps 11.3 an algebraic definition of the momentum map 11.4 conservation of momentum maps 11.5 equivariance of momentum maps 12 computation and properties of momentum maps 12.1 momentum maps on cotangent bundles 12.2 examples of momentum maps 12.3 equivariance and infinitesimal equivariance 12.4 equivariant momentum maps are poisson 12.5 poisson automorphisms 12.6 momentum maps and casimir functions 13 lie-poisson and euler-poincare reduction 13.1 the lie-poisson reduction theorem 13.2 proof of the lie-poisson reduction theorem for gl(n) 13.3 lie-poisson reduction using momentum functions 13.4 reduction and reconstruction of dynamics 13.5 the euler poincare equations 13.6 the lagrange poincare equations 14 coadjoint orbits 14.1 examples of coadjoint orbits 14.2 tangent vectors to coadjoint orbits 14.3 the symplectic structure on coadjoint orbits 14.4 the orbit bracket via restriction of the lie-poisson bracket 14.5 the special linear group of the plane 14.6 the euclidean group of the plane 14.7 the euclidean group of three-space 15 the free rigid body 15.1 material, spatial, and body coordinates 15.2 the lagrangian of the free rigid body 15.3 the lagrangian and hamiltonian in body representation 15.4 kinematics on lie groups 15.5 poinsot's theorem 15.6 euler angles 15.7 the hamiltonian of the free rigid body 15.8 the analytical solution of the free rigid-body problem 15.9 rigid-body stability 15.10 heavy top stability 15.11 the rigid body and the pendulum references index ... |
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