1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 387 Since particles in constant magnetic fields move in circles with co ...

388 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY Falling cat problem. This problem is an abstraction of the problem of how a ...

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 389 then gives Wong's equations by the following simple calculations: 8l ...

390 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY u m g l = pendulum length m = pendulum bob mass M =cart mass g = acceleratio ...

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 391 Note that the variable s is "shifted" by the one form T and a term q ...

392 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY Consider a carrier rigid body with a rotor aligned along the third principal ...

LECTURE 4. CONTROL AND STABILIZATION OF BALANCE SYSTEMS 393 where k is a gain parameter. The system retains the 51 symmetry and ...

394 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY place as the gain is increased can be viewed in terms of a modification of t ...

Lecture 5. Variational Integrators For conservative mechanical systems with symmetry, it is of interest to develop numerical sch ...

396 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY and McCracken [1978] and Abraham, Marsden and Ratiu [1988] for details. An ...

LECTURE 5. VARIATIONAL INTEGRATORS 397 is symplectic if 1 - ,\A is invertible for some real ,. To apply this to our situation, r ...

398 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY each of which is symplectic. Thus, the composition FT = GT o H n namely, FT ...

LECTURE 5. VARIATIONAL INTEGRATORS 399 Proposition 5.1. (Ge and Marsden [1988]) If an algorithm for a given Hamilton- ian system ...

400 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY The discrete variational principle. We now discuss discrete variational prin ...

LECTURE 5. VARIATIONAL INTEGRATORS 401 The fiber derivative is analogous to the standard Legendre transform. The coordi- nate ex ...

402 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY sum is defined by N-l § = L lL (vk+1, vk). k=O Extremize § : v N+l -t JR sub ...

LECTURE 5. VARIATIONAL INTEGRATORS 403 the comparison of these methods with energy-momentum methods. For the rigid body, we use ...

404 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY for all oq(t) with oq(a) = oq(b) = o. Abbreviating q, = q+Eoq, and using int ...

LECTURE 5. VARIATIONAL INTEGRATORS 405 issues discussed in lectures 3 and 4 are examples. Other examples are the direct variatio ...

406 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY space CL may b e identified with the initial conditions for the fl.ow; to V ...