By A. Richard Olson, Henry W. Hicock

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Example text

Suppose we know all solutions corresponding to all possible initial conditions, x (t, s, y ) ≡ Φ t,s (y ) . 43) ˜ This two-parameter set of solutions defines a mapping of P onto P, y → x = ˜ ˜ Φ t,s (y ). This mapping is unique, and both it and its inverse are differentiable. ˜ ˜ The set Φ t,s (y ) is called the flow in phase space P. ˜ Consider a˜ system whose initial configuration at time s is y ∈ P. The flow describes how the system will evolve from there under the action˜ of its dynamics. At time t it takes on the configuration x , where t may be later or earlier than ˜ s.

Clearly, such a domain is no longer singly connected. 32 1. 16 One-Dimensional Motion of a Point Particle Let q be the coordinate, p the corresponding momentum, and F (q) the force. We then have q˙ = 1 p; m p˙ = F (q) . 36) This is another way of writing the equation of motion. The first equation repeats the definition of the momentum, F (q) on the right-hand side of the second equation is the force field (in one dimension). The kinetic energy is T = mq˙ 2 /2 = p 2 /2m. The function F (q) is assumed to be continuous.

The physical motion “flows” across the phase space. To illustrate this new picture of mechanical processes we consider two more examples. 1 The Harmonic Oscillator The harmonic oscillator is defined by its force law F (q) = −mω2 q. The applied force is proportional to the elongation and is directed so that it always drives the particle back to the origin. 38) where q0 can be chosen to be zero, without loss of generality. 37) read explicitly F1 = x˙1 = 1 x2 , m x˙2 = −mω2 x1 . 34 1. Elementary Newtonian Mechanics The total energy is conserved and has the form E= x22 1 + mω2 x12 = const .