PRUEBA DE COINTEGRACIÓN

Newton’s equations of motion in Cartesian coordinates are 3N-second- order differential equations with respect to the time. By solving for the accelerations, they can be written in the following form

¨x(i)=

F(i)(x,˙x, t)

m(i)

, (1.79)

where (i) denotes a particle numbering index ranging over the N particles of the system. Working in Cartesian coordinates for now, the equations can be treated as a single second-order differential equation for an n-dimensional vector in configuration space¨xj_{= a}j_{(x,˙x, t), where x = [x}1_{,. . . , x}n_{] denotes}

a system vector in an n-dimensional configuration space, and n= 3N for motion in three space dimensions. The solution x(t) represents the time evolution of a system point in this space. Note that knowing x(t0) at some

initial time is not sufficient to completely describe the system. One needs to specify the initial velocities˙x(t0) as well.

1.5.1 Numerical Integration

For both visualization and numerical programming purposes, it is better to recast Newton’s equations as a system of 2n first-order differential equations. This is achieved by treating the momentum and coordinates as independent observables. Newton’s equations become

˙p(i)= F(i)(x, p, t), ˙x(i)= p(i)

m(i)

, (1.80)

which can be rewritten in configuration space coordinates as

˙pi= Fi(x, p, t), ˙xi= vi(x, p, t). (1.81)

If the right-hand side of the equation is not explicitly dependent on the time, the equation is said to be autonomous in the independent variable. Note that raised indices are used to denote position coordinates and lowered indices are used to denote momentum coordinates. To numerically integrate these

equations, the force law has to be rewritten as a function of the positions and momenta instead of as a function of the positions and velocities. The equations require 2n initial conditions for their solution.

A large number of ordinary differential equation (ODE) solvers exist for solving the initial value problem for coupled first-order differential equations of the form

dyi

dt = f

i_{(t, y), (1.82)}

where the index ranges over the 2n coordinates defined by the column vector y= x p  . (1.83)

The n-tuple y= {yi} defines a 2n-dimensional phase space vector, where 2n= 6N for N particles in three-dimensional space, subject to the initial conditions y(t0)= y0. One advantage of using a phase space formulation of

the dynamical problem is that the lines-of-flow for the first-order differential equations can never cross, since the instantaneous direction of the flow ˙y = f (t, y) must be a single-valued function of the time. A point in phase space specified by giving the initial values of position and momentum for all the particles therefore uniquely defines the evolution of the system for all future times.

1.5.1.1 Runge-Kutta Methods

Euler’s method for solving first-order differential equations is to convert the first-order differential equations into a set of first-order difference equations, and to numerically solve these by stepwise iteration

y(t+ h) = y(t) + h f (t, y(t)), (1.84) where h= t is the step size of the interval. Euler’s difference equation solver is an example of a first-order Runge-Kutta method with the error proportional to the step size of the time interval.

Greater accuracy for a given step size is given by using higher-order Runge-Kutta integrators. For example, the explicit Runge-Kutta equation solver RK4 is given by9

y(t+ h) = y(t) +h

6(k1+ 2k2+ 2k3+ k4), (1.85)

9_{A good discussion of Runge-Kutta techniques can be found at Wolfram Mathworld,}

where k1= f (t, y(t)), k3= f  t+h 2, y(t)+ h 2k2  , k2= f  t+ h 2, y(t)+ h 2k1  , k4= f (t + h, y(t) + hk3). (1.86)

The RK4 solver is a fourth-order method, meaning that the error in each step is on the order of h5_{, while the accumulated error is of order h}4_.

1.5.1.2 Controlling Numerical Error

Numerical methods suffer from two special sources of error: – Round-off error

– Truncation error

Round-off error increases with the number of integration steps. Truncation error comes from the finite step size.

Runge-Kutta methods know nothing about the underlying dynamics of the ordinary differential equations (ODE). For example, the algorithms do not automatically preserve constants of the motion. This can lead to artificial computational effects, such as numerical energy dissipation due to trunca- tion errors. The error can grow significantly over long integration times. To reduce this effect, one can rewrite the algorithm to enforce constraints on the constants of the motion.

Adaptive variable step-size methods have been developed to deal with stiff differential equations. These are differential equations where rate of change varies dramatically over the limits of integration. Typically, these algorithms work by automatically adjusting the step size so that the local error per step is kept below some prescribed tolerance limit. Such integrators are included in higher level Mathematical software programs, such as Maple™, Mathematica™, Mathcad™, and Matlab™, or in advanced mathematical libraries for common programming languages.10 _{When using numerical}

integration, it is useful to perform “sanity checks” to determine if the algo- rithms are behaving properly. Tests for convergence, error propagation, and boundary conditions are commonly performed.

Although one can write one’s own numerical integrator, the differential equation solvers provided by commercial mathematical software packages

10_{A good source for numerical routines written for use with Maple, Mathematica,}

Mathcad, and MATLAB is the Holistic Numerical Methods Institute of the Uni- versity of South Florida web site, available at http://numericalmethods.eng.usf.edu/ index.html.

such as Matlab or Mathematica makes the task easier and nearly painless to perform. One can often simply specify a tolerance and let the Runge-Kutta solver do its work, adjusting the step size automatically as needed. Event triggers can be placed on the integration to allow handing of exceptional conditions or to extract interesting events, such as detection of zero-crossings of a plane in order to generate Poincaré sections. Moreover, the results are provided in a form directly useful for plotting.

1.5.1.3 Analytical Solutions

The alternative to numerical integration is to solve the problem using analyti- cal methods. A set of 2n first order differential equations is exactly integrable if can be completely reduced to quadrature, yielding 2n constants of inte- gration. Quadrature is defined as the reduction of an ordinary differential equation to the integral of a function of a single variable with respect to that variable. The results can then be integrated, numerically or analytically. Problems reduced to quadrature are considered to be formally solved. The long-term stability of integrable systems can easily be determined. The con- stants of integration can often be related to the symmetries of the problem, yielding important insights into the dynamics of the physical system.