1. Dmitry G Luchinsky, Nonlinear Dynamics Group, A11 (tel:93206), A14 (tel:93079) d.luchinsky@lancaster.ac.uk M332 D.G.Luchinsky Nonlinear Dynamics GroupIntroduction Syllabus: Variational methods. Principle of least action. Hamilton's Principle. Lagrange's equations of motion. Normal modes. Conservation laws. Phase space. Classical mechanics and nonlinear dynamics: introduction http://www.lancs.ac.uk/depts/physics/teaching/py332/phys332.htm

2. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Brief outlook I. Reminder – Newtonian formalism II. Lagrangian & Hamiltonian Formalism III. Two-body problem IV. Symmetries and conservation laws V. Linear oscillations and normal modes VI. Nonlinear oscillations and Chaos

3. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Recommended literature 1.T.W.B. Kibble and F.H. Berkshire, Classical mechanics, fourth ed. (Addison Wesley Logman Ltd., Essex, 1995). 2.J.B. Marion and S.T. Thornton, Classical dynamics of particles and systems, fourth ed. (Saunders College Pub., Fort Worth, 1995). 3.T.L. Chow, Classical mechanics (John Wiley and Sons, New York, 1995). 4.L.D. Landau and E. Lifshits, Mechanics, Vol. 1 of Course of Theoretical Physiscs, 3rd ed. (Pergamon, Oxford, 1978). 5.H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, London, 1982). 6.H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002). 7.V. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1978). 8.J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical Systems, and bifurcations of vector fields (Springer-Verlag, New-York, 1983). 9.R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky, Nonlinear physics: from the pendulum to turbulence and chaos (Harwood Academic, Chur, 1988). 10.M.A. Lieberman and A.J. Lichtenberg, Regular and Chaotic dynamics, (Springer, New York, 1992).

4. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Assumptions of classical mechanics suggested by our observations II. Our time is one dimensional and uniform. III. There exist so-called inertial frames of reference that have the following properties: All laws of Nature are the same in all inertial frames of references at all instants of time; All frames of reference moving relative to an inertial one with a constant velocity V are inertial. I.Our space is 3 dimensional and its geometry is Euclidean, i.e. the position of any point in our space is specified by three real numbers (x,y,z) and the distance between two points is defined as:

5. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Assumptions of classical mechanics suggested by our observations IV. Motion of a mechanical system is uniquely determined by the initial state of a mechanical system, i.e. by defining its coordinates and velocities at some initial moment of time. V. Coordinates and velocities of a mechanical system can be measured simultaneously and there is no limit in principle to the accuracy with which they can be measured.

6. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Newton’s laws 1. mv = constant (in the absence of forces) 2. d(mv)/dt = F 3. F 12 = -F 21

7. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Newton’s laws: Inertial frames x1x1 x 2 > x 1 h g Problem 1

8. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Inertial frames (continued) Galilean transformations: r’ = r+Vt v’ = v+V t’ = t

9. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Main concepts Mass (m g = m i ) Force F (Torque r  F) Momentum p=mv (Angular momentum r  p) Kinetic energy T = mv 2 /2 Potential Energy U = U(r) Mechanical Work W =   Fdr

10. Nonlinear Dynamics GroupIntroduction Classical Mechanics M332 D.G.Luchinsky Main concepts Problem 2