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USSC3002 Oscillations and Waves Lecture 9 Euler-Lagrange Equations

USSC3002 Oscillations and Waves Lecture 9 Euler-Lagrange Equations. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749. 1. CHANGE OF VARIABLES.

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USSC3002 Oscillations and Waves Lecture 9 Euler-Lagrange Equations

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  1. USSC3002 Oscillations and Waves Lecture 9 Euler-Lagrange Equations Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

  2. CHANGE OF VARIABLES The diagram below show dependence of variables that characterize the configuration and the state ( = configuration & velocity) of a dynamical system that consists of a single particle moving in a plane: Coordinates Configuration Velocity Orthonormal Polar Computation Based on the Chain Rule  2

  3. PARTIAL DERIVATIVES AND DIFFERENTIALS Observe that and depend linearly on and so it is convenient to express this using matrices and the differentials depend linearly on Observe that the coefficients NOT linear !!! 3

  4. GENERALIZED FORCES Definition The work differential is hence the power exerted by force on a particle moving with velocity is Lemma where  4

  5. GENERALIZED MOMENTA Definion Kinetic energy and generalized momenta Lemma 5

  6. ORTHONORMAL EQUATIONS OF MOTION In orthonormal coordinates Newton’s 2nd Law  We MIGHT be led to guess that in polar coordinates BUT THESE EQUATIONS ARE WRONG !!!!!!!!!!!!! 6

  7. POLAR EQUATIONS OF MOTION substituting these three equations into the first gives Question 1. Show that 7

  8. GENERALIZED EQUATIONS OF MOTION N particles  n = 3N degrees of freedom generalized coordinates&velocities generalized forces generalized momenta EOM Question 2. Derive these EOM. 8

  9. TWO PARTICLES ON A STRING For a system consisting of two particles on a spring Question 3. How can this problem be approached ? 9

  10. TWO PARTICLES ON A STRING Question 4. Discuss these generalized coordinates ? Question 5. What is the potential energy ? Question 6. What happens as 10

  11. CONSTRAINED MOTION ON INCLINED PLANE Since the net force is parallel to the plane hence the EOM are Question 7. At the atomic scale is 11

  12. CONSTRAINED MOTION ON CIRCLE Since the net force has zero in the radial direction hence the EOM are Question 8. What happens for 12

  13. CONSTRAINED MOTION N particles  n = 3N configuration variable are constrained (by (n-m) constraints) so that they are parametrized by m generalized coordinates Then there are m EOM Question 9. Can m = 1 and n = 6 E 23 ? 13

  14. THE EULER-LAGRANGE EOM Forces are conservative if there exists a function such that Lemma For conservative forces Corollary Then we obtain Lagrange’s EOM where where is the Lagrangian function. 14

  15. TUTORIAL 9 1. Derive the Euler-Lagrange EOM for a spherical pendulum (generalized coordinates theta and phi) 2. Derive the Euler-Lagrange EOM for a falling rod. 3. Derive the Euler-Lagrange EOM for a rod of uniform linear density whose ends a supported by springs of reference length h and stiffness k_1 and k_2. 4. Derive the Euler-Lagrange EOM for the AtwoodMachine: a string of length d_1 passes over a light fixed pulley, supporting a mass m_1 at one end and a pulley of mass m_2 at the other. Over the second pulley passes a string of length d_2 supporting a mass m_3 at one end and m_4 on the other. Show that mass m_1 stays in equilibrium iff 15

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