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 x and v may be visualized as the coordinates of a two-dimensional phase space. At each point (x,v) of phase space, Eq. 2 defines a vector, known as the phase flow.
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 phase curve passing through the point (X,V) at t = 0. The standard theorems for the existence and uniqueness of solutions to ordinary differential equations imply that there is one and only phase curve passing through each point.
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 U(x) is called the potential energy. Likewise, the kinetic energy is
1 2 (4) T = - m v , 2and the sum is the total energy E(x,v) = T + U of the system. Using the chain rule, it is easy to show that the total energy is conserved along any solution of Eq. 2:
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 x(t) which extremizes the action is given by the Euler-Lagrange equation
d dL dL (6) -- -- - -- = 0 , dt dv dxwhere the derivatives taken with respect to x and v are understood to be partial derivatives. Substituting Eq. 5 into Eq. 6 yields
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 p, the hamiltonian function is
1 2 (8) H(x,p) = pv - L(x,v,t) = - m v + U(x,t) 2 1 2 = -- p + U(x,t) , 2mwhere v has been eliminated in favor of p on the second line. Note that numerically the hamiltonian is equal to the total energy. The equations of motion are then
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 Z is an arbitrary point at which U = 0. But if n > 1 this integral may depend on the exact path taken from Z to x; if it does depend on the path, a unique potential energy cannot be defined. The condition for the integral in Eq. 7 to be path-independent is that f(x) be the gradient of some function of x; this function is, of course, just -U(x). Such a system is called conservative.
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 r axis, while the latter, defined by
2 d(theta) (12) m r v_t = m r -------- = constant , dtis a hyperbola in the (r,v_t) plane.
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 r is then
2 d r dV --- = - -- . 2 dr dtBecause V(r) diverges as r -> 0 the particle is energetically prohibited from coming too close to the origin, and shuttles back and forth between turning points r_min and r_max.
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 u(r) is the two-particle interaction potential. The equations of motion are then
-- 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 i.
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, f(x) = x, and (b) the nonlinear pendulum, f(x) = - sin x.
5. Give an example of a nonconservative force field.
Last modified: January 17, 1995