Classical mechanics and nonlinear dynamics:
introduction
M332
D.G.Luchinsky
Dmitry G Luchinsky,
Nonlinear Dynamics Group,
A11 (tel:93206), A14 (tel:93079)
d.luchinsky@lancaster.ac.uk
Syllabus:
Variational methods.
Principle of least action.
Hamilton's Principle.
Lagrange's equations of motion.
Normal modes.
Conservation laws.
Phase space.
http://www.lancs.ac.uk/depts/physics/teaching/py332/phys332.htm
Introduction
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Brief outlook
I. Reminder –
Newtonian
formalism
II. Lagrangian
& Hamiltonian
Formalism
IV. Symmetries
and conservation
laws
V. Linear
oscillations and
normal modes
Introduction
III. Two-body
problem
VI. Nonlinear
oscillations
and Chaos
Nonlinear Dynamics Group
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).
Introduction
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Assumptions of classical mechanics suggested by our observations
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:
 ( x, y, z ) 
II.
x2  y 2  z 2
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.
Introduction
Nonlinear Dynamics Group
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.
Introduction
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Newton’s laws
1. mv = constant (in the absence of forces)
2. d(mv)/dt = F
3. F12 = -F21
Introduction
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Newton’s laws:
Inertial frames
Problem 1
g
x2 > x 1
x1
h
Introduction
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Inertial frames (continued)
Galilean
transformations:
r’ = r+Vt
v’ = v+V
t’ = t
Euclidean
 ( x, y , z ) 
Introduction
space
x2  y 2  z 2
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Main concepts
Mass
Force
(mg = mi)
F (Torque r  F)
Momentum p=mv (Angular momentum r  p)
Kinetic energy
T = mv2/2
Potential Energy
U = U(r)
Mechanical Work
W =
Introduction

Fdr
Nonlinear Dynamics Group
Classical Mechanics
M332
D.G.Luchinsky
Main concepts
Problem 2
Introduction
Nonlinear Dynamics Group