The Newtonian equations of motion are *second-order* in time;
thus both positions and velocities are needed to specify the dynamical
state of a classical system.

The simplest general Newtonian system is a single particle of mass
`m` moving along the `x` axis under the influence of a
position-dependent force `f(x)`. Such a system has the
equation of motion

2 d x (1) m --- = f(x) . 2 dt

Any second-order equation of the form above can be transformed into
a pair of first-order equations by introducing a new variable, the
velocity `v` along the `x` axis. The transformed
equations are

dx dv (2) -- = v , m -- = f(x) . dt dtThe variables

Starting from a given point `(X,V)` at time `t = 0`,
the solution to Eq. 2 may be expressed by the parametric equations

x = x(t) , v = v(t) .These equations describe a

If `f(x)` vanishes for some particular `X`, the phase
curve `passing through' `(X,0)` consists of a single point,
known as an *equilibrium position*.

For an autonomous system with one degree of freedom, it is
possible to find a function `U(x)` such that

dU (3) f(x) = - -- ; dxthe function

1 2 (4) T = - m v , 2and the sum is the

E(x(t),v(t)) = constant.Thus the phase curves of a system are contours of the total energy.

Classical mechanics provides an algorithm for obtaining the
equations of motion in an arbitrary coordinate system. This algorithm
involves defining a function called the *lagrangian* which
depends on the *generalized coordinates*, the *generalized
velocities*, and possibly the time itself. The equations of motion
are then obtained from the principle of least action.

For simplicity, this formalism will be illustrated using `x`
as the generalized coordinate and `v` as the generalized
velocity. The lagrangian is then

1 2 (5) L(x,v,t) = T - U = - m v - U(x,t) . 2The trajectory

d dL dL (6) -- -- - -- = 0 , dt dv dxwhere the derivatives taken with respect to

d dU -- (m v) + -- = 0 , dt dxwhich is just Newton's law of motion, Eq. 1.

An equivalent formulation may be developed by defining the generalized momentum

dL (7) p = -- = m v . dvIn terms of

1 2 (8) H(x,p) = pv - L(x,v,t) = - m v + U(x,t) 2 1 2 = -- p + U(x,t) , 2mwhere

dp dH dx dH (9) -- = - -- , -- = -- , dt dx dt dpand substituting Eq. 8 into Eq. 9 yields

dp dU dx p -- = - -- , -- = - , dt dx dt mwhich is identical in content to Eq. 2.

Much of the above generalizes to systems with many degrees of
freedom. If there are `n` degrees of freedom, then the scalar
variable `x` in Eq. 1 becomes a vector with `n`
components, and `f(x)` is a vector-valued function. The phase
space now has a total of `2n` dimensions.

If `n = 1` it is always possible to define a potential
energy corresponding to any given `f(x)`; let

/ x | (10) U(x) = - | f(y) dy , | / Zwhere

As an example of a system with `n > 1` degrees of freedom,
consider the motion of a particle in a spherically-symmetric
potential, `U(x) = U(|x|)`. The orbit of the particle remains
in a plane perpendicular to the angular momentum vector, so this
system has, in effect, `n = 2` degrees of freedom, and the
associated phase space has `4` dimensions.

Visualizing a space of `4` dimensions is hard, but with the
right choice of coordinates the number of dimensions can be reduced.
In a polar coordinate system `(r,theta)`, where `r =
|x|`, the hamiltonian does not depend on `theta`, and the
corresponding generalized momentum `J = m r v_t` is conserved.
Thus in the `3`-dimensional space `(r,v_r,v_t)` the
orbit of the particle lies in the intersection of a surface of
constant energy and a surface of constant angular momentum. The
former, defined by

1 2 2 (11) - m (v_r + v_t ) + U(r) = constant , 2is a figure of rotation about the

2 d(theta) (12) m r v_t = m r -------- = constant , dtis a hyperbola in the

This system may be reduced to one degree of freedom by defining the
*effective potential*,

2 J (13) V(r) = U(r) + ------ ; 2 2 m rthe equation of motion for

2 d r dV --- = - -- . 2 dr dtBecause

A system of `N` particles, each moving in an
`n`-dimensional space, has a total of `Nn` degrees of
freedom, and the associated phase space has `2Nn` dimensions.
Let the particles have masses `m_i`, positions `x_i`,
and velocities `v_i`, where the index `i` ranges from
`1` to `N`. We will generally be interested in systems
for which the total potential energy `U(x_1, ..., x_N)` may be
expressed as a sum of pair-wise interactions:

-- \ (14) U(x) = | m_i m_j u(|x_i - x_j|) , ~ / -- i, j < iwhere

-- dx_i dv_i \ d (15) ---- = v_i , ---- = - | m_j ---- u(|x_i - x_j|) , dt dt / dx_i -- jwhere the sum runs over all particles except particle

Due date: 1/26/95

3. Draw phase curves for the potential energy graphs plotted here.

4. Suppose that initially a set of points are distributed in phase space within the circle

2 2 1 x + (v - 1) < - . 4Draw the image of these points under the phase flow of (a) the inverse pendulum,

5. Give an example of a nonconservative force field.

- Arnold, V.I. 1978,
*Mathematical Methods of Classical Mechanics*. - Binney, J. & Tremaine, S. 1987,
*Galactic Dynamics*, Appendix 1.D.

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: January 17, 1995