1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 367 More realistic models of the fluid dynamics, especially in cases when v ...

368 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY 0.45 0.4 0.35 0.3 e 0.25 0.2 0.15 0.1 0.05 0 0.3 0.2 0.1 ---- 0.7 0.6 0.5 0. ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 369 area=A Figure 2.11. A parallel movement of a vector around a spherical ...

370 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY A method that is more geometric in nature, is based on Montgomery's [1991b] ...

LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 371 Thus, dG = -D is the symplectic form. By Stokes theorem, 1 e + J e + J ...

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Lecture 3. Systems with Rolling Constraints and Locomotion In this lecture we consider the basic mechanics of systems with rolli ...

374 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY z 8 y Figure 3.1. The rolling disk. Consistent with the fact that each Vq is ...

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 375 z x Figure 3.2. The roller racer-Tennessee racer. The roller racer is interestin ...

376 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY the constraints; i.e., the distribution 'D. One can introduce local coordina ...

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 377 for our space Q , where R is the base manifold and 7rQ,R is a submersion and the ...

378 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY structure Q -> R is really a "red herring". The notion of curvature as a ...

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 379 where TJ(t) is a curve in g vanishing at the endpoints. The Euler-Poincare equa ...

380 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY z Figure 3.4. The mechanics of a bicycle (from Koon and Marsden [1998a]). mo ...

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 381 has the ability to build up momentum, which can be traced to the presence of for ...

382 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY adapted from the animal world where similar things happen, but in fact are m ...

LECTURE 3. SYSTEMS WITH ROLLING CONSTRAINTS 383 0.25 0.2 0 .1 5 0 .1 0.05 E .._.. 0 ""-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.5 ·0.4 ...

384 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY example-see, for example, the discussion in Bloch and Marsden [1989]) one ne ...

Lecture 4. Optimal Control and Stabilization of Balance Systems In this lecture we consider two problems from control theory. Th ...

386 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY (0, 0, a) Figure 4.1. An optimal steering problem. point (0, 0, a) after ti ...