Systems Of Equations
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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical,
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Lagrange Equations, Equations of Motion, Kinetic Energy, Equations of Motion - Explicit Form, Centrifugal and Coriolis Forces, Christoffel Symbols, Mass Matrix, V Matrix, Final Equation of Motion
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Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and...more
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Joint Space Dynamics, Newton-Euler Algorithm, Inertia Tensor, Example, Newton-Euler Equations, Lagrange Equations, Equations of Motion
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May 19, 2009 - Leonard Susskind lectures on a new class of systems, magnetic systems. He goes on to talk about mean field approximations of molecules in multidimensional lattice systems.
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Examples Of Autonomous Linear Dynamical Systems, Finite-State Discrete-Time Markov Chain, Numerical Integration Of Continuous System, High Order Linear Dynamical Systems, Mechanical Systems, Linearization Near Equilibrium Point, Linearization Along Trajectory
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Linear Systems: Basic Definitions, Direct Proportionality As Example, Special Cases Of Linear Systems, Eigenvectors And Eigenvalues, The Spectral Theorem And Finding A Basis Of Eigenvectors, Matrix Multiplication = Only Example Of Finite Dimensional Linear Systems, Integration Against A Kernel Generalizing Matrix Multiplication, Example: The Fourier Transform
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Overview Of Linear Dynamical Systems, Why Study Linear Dynamical Systems?, Examples Of Linear Dynamical Systems, Estimation/Filtering Example, Linear Functions And Examples
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Review Of Last Lecture: LTI Systems And Convolution, Comment On Time Invariant Discrete Systems, The Fourier Transform For LTI Systems; Complex Exponentials As Eigenfunctions, Discussion Of Sine And Cosine V. Complex Exponentials As Eigenfunctions (Generally They Are Not), Discrete Version (Discrete Complex Exponentials Are Eigenvectors), Discrete Results From A Matrix Perspective
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In the absence of a net external torque on an object, angular momentum is conserved. When an object oscillates about an axis of rotation, there is a variable restoring torque acting on the object. A review is given of equations for angular momentum and torque, and the importance of choosing the point of origin. These equations are exercised using an example of a circular orbit.
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Sensitivity Of Linear Equations To Data Error, Low Rank Approximations, Distance To Singularity, Application: Model Simplification, Controllability And State Transfer, State Transfer, Reachability, Reachability For Discrete-Time LDS
