Variable force in one dimension

• Consider the motion of a particle of mass $m$ moving along the x-axis under the action of a force F.

  Write Newton's second law ($F=ma$) in terms of velocity for a most general force $\textbf{F(x,v,t)}$

  
  

  $$\frac{dv}{dt} =\frac{F(x,v,t)}{m}$$

  $$\frac{dx}{dt} =v$$ (4.1)

  
  
• In principle, the functions $v=v(t)$ and $x=x(t)$ can be found by solving Equation 4.1 for every $F(x,v,t)$ function (i.e., constant acceleration for F = constant, or harmonic oscillating motion for F= -kx).
• However, for more complex forces $F(x,v,t)$ the analytical solution may not always be available.
• In this case, we will consider the numerical solution.

• We want to find the solutions of the Equation 4.1 at the equally spaced times $t_1, t_2, \ldots, t_N$ and $x_i$ and $v_i$.
• Write the velocity and position derivatives in the form of forward-difference derivative approximation. Take $dt=h$ as step length, then:

  $$\frac{v_{i+1}-v_i}{h}=\frac{F(x,v,t)}{m} \longrightarrow \boxed{v_{i+1}=v_i+\frac{F_i}{m}h }$$

  $$\frac{x_{i+1}-x_i}{h}=v_i \longrightarrow \boxed{x_{i+1}=x_i+v_i h}$$

• By given initial velocity $v_1$, initial position $x_1$ at the time $t_1=0$ and the values for $F_i=F(x_i,v_i,t_i)$; the values for $v_i, x_i$ can be calculated for $i=2.3, \ldots$ in a loop.

• When the function is explicitly known, we can emulate the methods of calculus.
• If we are working with experimental data that are displayed in a table of [$x,f(x)$] pairs emulation of calculus is impossible.
  • We must approximate the function behind the data in some way.
• Differentiation with a Computer:
  • Employs the interpolating polynomials to derive formulas for getting derivatives.
  • These can be applied to functions known explicitly as well as those whose values are found in a table.
• Numerical Integration-The Trapezoidal Rule:
  • Approximates, the integrand function with a linear interpolating polynomial to derive a very simple but important formula for numerically integrating functions between given limits.

• We cannot often find the true answer numerically because the analytical value is the limit of the sum of an infinite number of terms.
• We must be satisfied with approximations for both derivatives and integrals but, for most applications, the numerical answer is adequate.
• The derivative of a function, $f(x)$ at $x=a$, is defined as
  $$\frac{df}{dx} \vert_{x=a}=lim_{\Delta x \rightarrow 0 } \frac{ f(a+\Delta x)-f(a)}{\Delta x}$$

• This is called a forward-difference approximation.
• The limit could be approached from the opposite direction, giving a backward-difference approximation.