Nonlinear Mechanics - Physics at Oregon State University

Chapter 1
Lagrangian Dynamics
1.1 Introduction
The possibility that deterministic mechanical systems could exhibit the behavior
we now call chaos was first realized by the French mathematician
Henri Poincaré sometime toward the end of the nineteenth century. His
discovery emerged from analytic or classical mechanics, which is still part
of the foundation of physics. To put it a bit facetiously, classical mechanics
deals with those problems that can be “solved,” in the sense that it is possible
to derive equations of motions that describe the positions of the various
parts of a system as functions of time using standard analytic functions.
Nonlinear dynamics treats problems that cannot be so solved, and it is only
in these problems that chaos can appear. The simple pendulum makes a
good example. The differential equation of motion is
¨θ + ω 2 sin θ = 0 (1.1)
The sin is a nonlinear function of θ. If we linearize by setting sin θ ≈ θ,
the solutions are elementary functions, sin ωt and cos ωt. If we keep the sin,
the solutions can only be expressed in terms of elliptic integrals. This is
not a chaotic system, because there is only one degree of freedom, but if we
hang one pendulum from the end of another, the equations of motion are
hopeless to find (even with elliptic integrals) and the resulting motion can
be chaotic. 1
1 I should emphasize the distinction between the differential equations of motion, which
are usually simple (though nonlinear), and the equations that describe the positions of
the elements of the system as functions of time, which are usually non-existent.
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