Summaries of earlier lectures

Num
| Date |
Summary |
---|---|---|

14 | 10.Oct | We discussed the first- and second-order necessary conditions for optimality in constrained optimization, using Lagrange Multipliers. We worked a simple example. Subsequently, we started our discussion of the Calculus of Variations. We derived the basic Euler-Lagrange Equation. As an example, we showed that the shortest curve between two points in the plane is a straight line. We considered a constrained version of this problem in which one seeks a shortest curve with a given area below the curve. We showed that the curve, when it exists, is now a circular arc. We solved this problem by extending the Calculus of Variations to include Lagrange multipliers. We further observed that the dual problem -- in which one seeks to maximize the area below a curve of fixed length -- gives rise to essentially the same Euler-Lagrange Equation as the original constrained problem, and thus has the same type of optimizing curve, when an optimum exists. In the context of Calculus of Variations, we observed that some cost functions do not have optimal solutions that are twice differentiable. In the constrained problem mentioned above this can happen when the endpoint and area (or length) conditions are incompatible. In that situation, one may be able to interpret the optimal curve as a circular arc with jump discontinuities at the endpoints. |

15 | 22.Oct | We surveyed various generalizations of the so-called
Simplest Problem in the Calculus of Variations that
we had discussed last time. In particular, we considered
optimization with free boundaries, surface integrals, multiple
dependent variables, and higher order derivatives. We
sketched examples for two of these generalizations, as
follows:
As an example of a problem with a free boundary, we worked through key parts of a brachistochrone example: finding the shape of a ski slope that minimizes horizontal traversal time for a skier whose motion is determined by gravity. We mentioned that when the cost integrand
y'F
_{y'} - F = constantWe also considered the case of a free boundary point
specified merely to lie on a curve given implicitly by an
equation of the form F (at the endpoint).
_{y'} - (g_{y}F)/(g_{x} + g_{y}y') = 0As an example of a problem involving a surface integral, we found the shape of a soap film hanging from a ring, in which the potential energy of stretching counteracts the potential energy of gravity. The film seeks an equilibrium shape at which the net potential energy is a minimum (or, more precisely, at which the potential energy is stationary, meaning differential perturbations in the soap film's shape do not change the potential energy). The basic technique is very similar to what we did originally when computing a cost as an integral over a one-dimensional interval (as for the shortest path problem), except now we compute a cost by integrating over a two-dimensional region. In order to obtain an appropriate Euler-Lagrange Equation, one needs a generalization of integration by parts to higher dimensions. That generalization is Stokes Theorem, which we mentioned but did not discuss in detail. The notes contain a description of a two-dimensional instance of Stokes Theorem known also as Green's Theorem (see pages 33 and 34). |

16 | 24.Oct | Today, we began our discussion of Mechanics with a brief history of the Principle of Least Action. We mentioned generalized coordinates, generalized force, generalized velocity, and generalized momentum. (The phrase "generalized coordinate" is an older traditional term. Roboticists might simply speak of configuration parameters or state variables.) We introduced phase space, described in terms of
(generalized) coordinates and momenta. (Here it is convenient
to view velocities simply as time derivatives of coordinates,
independent of momenta.) We also found it convenient to think
in terms of two different coordinate systems for energy: One
coordinate system has kinetic energy and potential energy as
dimensions. The second coordinate system is obtained from the
first by a 45 degree rotation. In this coordinate system, one
dimension (called There are two common formulations for deriving these
differential equations, obtained by writing the Lagrangian
cost function We explored the meaning of Lagrangian and Hamiltonian
dynamics in the 1D setting of a point mass hanging from a
spring under the influence of gravity. Lagrangian Dynamics
recapitulates Newton's Today's lecture material appears on pages 54, 55, and 57-63 of the Notes on the Calculus of Variations. We also discussed Euler's Theorem in lecture, which appears here. |

17 | 29.Oct | Today, we worked out the Lagrangian dynamics of a
two-link manipulator, based on the Principle of Least
Action. |

18 | 31.Oct | Today we finished our discussion of the Principle of Least Action with a derivation of the wave equation and its solution using separation of variables. There are three interesting aspects to this process: (i) The Principle of Least Action can produce partial differential equations over spaces that contain both spatial and time coordinates, (ii) Some apparently continuous parameters can appear only in quantized form (in this case modeling spatial harmonics), (iii) solution of the differential equation produces a collection of orthonormal functions which one then uses to satisfy boundary conditions. |

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