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Unformatted text preview: ME 201 Advanced Dynamics • Instructor: Igor Mezic, Engineering II 2339, Tel: 8937603, email: [email protected] • Lecture Hours: Tue Thu 9:3010:45 in HSSB 4201 • Office Hours: Tue, Thu 1112:00, 2339. • Homeworks: Weekly. • Midterm exams: 1. • Grading policy: Homeworks 20 %, Midterm 30 %, Final 50 %. 1 201 Advanced Dynamics contents: • Newtons laws and symmetries, Newton, Laplace and principle of determinism, Figure 1: Isaac Newton 16421727 • qualitative analysis of Newtons equations of motion: dynamical systems theory • 1 degree of freedom (DOF) systems, linear algebra and stability of fixed points, parametric resonance, • 2 DOF systems, motion in central fields and application to molec ular dynamics, control of classical dynamical systems, • Lagrangian mechanics: variational principles and EulerLagrange equations, Legendre transform, equivalence of Lagrangian and Hamiltonian mechanics, • volume preservation, chaos and ergodic theory, rigid body motion. 2 Lecture 1 Assumptions of classical mechanics • Space and time are independent. Space is threedimensional, Eu clidean and time is onedimensional. • Galileo’s principle of relativity: There are coordinate systems (called inertial) that possess the following two properties: 1. All the laws of nature at all moments of time are the same in all inertial coordinate systems. 2. All coordinate systems in uniform rectilinear motion with respect to the inertial are themselves inertial. • Newton’s principle of determinacy: The initial state of a mechan ical system (the totality of positions and velocities of its points at some moment of time) uniquely determines all of its motion. Problems arise at very small (quantum mechanics) and very large (rel ativity theory) scale. 3 • Newton’s law for single particle of mass m moving in a three dimnensional Euclidean space with position denoted by x and velocity ˙ x = v : d dt m v = F ( x , ˙x ,t ) . (1) • Assume constant mass: ¨ x = f ( x , ˙x ,t ) . where f is force per mass of the particle. • Knowing initial position x and initial velocity ˙ x , we know x ( t ) , ˙ x ( t ) for all time. • Recall the springmass system: the force F = F d + F s + G ( t ), where F d = c ˙ x , F s = kx and G ( t ) is a timedependent external forcing. We assume c,k are constants determining the linear re sponse of damping force to change in velocity and linear response of spring force to change in position, respectively. Newton gives m ¨ x + c ˙ x + kx = G ( t ) . Provided G ( t ) = 0 ,c = 0 the solutions are simple oscillations x ( t ) = A sin( ωt ) + B cos( ωt ) y ( t ) = C sin( ωt ) + D cos( ωt ) (2) whose phase space is shown in figure 2. In phase space (or state space) we represent motion as curve of velocity vs. position in stead of velocity vs. time or position vs. time. Henri Poincar´ e used this concept to develop geometric dynamical systems theory....
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 Fall '08
 Mezic,I
 Force, Hamiltonian mechanics, Lagrangian mechanics, Dynamical systems, phase space

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