1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. REDUCTION THEORY 347 reduction theorem states that reduction of T * G by Gao gives reduced spaces that are symplectic ...

348 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY The three wave interaction. The quadratic resonant three wave equations are ...

LECTURE 1. REDUCTION THEORY 349 The (cubic) Hamiltonian is Hamilton's equations for a Hamiltonian H are and it is straightforwar ...

350 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY obtain the symplectic leaves in this reduction, we use the method of invaria ...

LECTURE 1. REDUCTION THEORY 351 The original equations define a dynamical system in the Poisson reduced space and on the symplec ...

...

Lecture 2. Stability, Underwater Vehicle Dynamics and Phases Some history, background and literature on stability. The energy mo ...

354 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY The idea of the energy-momentum method. The setting of the energy- momentum ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 355 structures found in these papers, we consider the case when P = T*Q for ...

356 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY In the underwater vehicle problem, the phase drifts one gets are interestin ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 357 Extension of the energy-momentum method to nonholonomic systems. The en ...

358 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY is the configuration space for the vehicle using reduction theory for mecha ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 359 This motion is stable modulo translations in any direction provided tha ...

360 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY g pO 3 center of buoyancy center of gravity ~ Figure 2.4. A rising, spinnin ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 361 There is a geometric phase drift a long the group directions; i.e., in ...

362 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY 0.1 0 .1 cf^0 >^0 1 0.^1 0 10 20 30 40 0 10 20 30 40 1 0. 1 N 0 N 0 a ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 363 0.1 ,:::. .0 0.1 1 0 10 20 30 40 0 10 20 30 40 0.1 ~^0 .0 N^0 0.1 1 0 1 ...

364 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY reduction y S l ""'"'" l n by V Figure 2. 7. Reduction by stages. given in L ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 365 with the addition of a buoyancy potential: where we determine A , B , C ...

366 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY 6 (^6 2) 1.5 f ······· l ........ 0 ........ l ! ........ 0.5 0 0.5 1.5 Real ...