Differential Equations of Motion

Initial Value Problem IVP

We recall that velocity v is defined as the change dx of position x per unit time step dt:

  • v = \frac{dx}{dt} .

We also refer to \frac{dx}{dt} as the derivative of x with respect to t. Similarly, we have

  • \frac{dv}{dt} = a ,

where a is the accelleration. In other words:

  • Derivative of position with respect to time = velocity.
  • Derivative of velocity with respect to time = acceleration.

We will denote derivative with respect to time, alternatively with a dot:

  • \dot x = \frac{dx}{dt} , \quad \dot v=\frac{dv}{dt} ,

where we in the spirit of Leibniz assume the time step to be vanishingly small. Newton used \dot x and Leibniz used \frac{dx}{dt} to denote the derivative of x with respect to time t.

Newton’s age is sometimes referred to as the dot-age (associated with a decline of mathematics in England and to be compared with dotage = a state or period of senile decay marked by decline of mental poise and alertness), while our age may be referred to as the dot-com-age.

The process of computing the derivative \dot x(t) of a function x(t) is called differentiation, or more precisely differentiation with respect to t.

Using (anyway) Newton’s notation we can write the equations of motion, assuming M=1 for simplicity, as differential equations:

  • \dot x = v,\quad \dot v = F ,

which is to be interpreted as

  • \dot x(t) = v(t) ,\quad \dot v(t) = F(t)\quad \mbox{for } t>0 ,
  • x(0) =x^0 , \quad v(0)=v^0 ,

where x(t), v(t) and F(t) are viewed as functions of t, and x^0 and v^0 are given initial values of position and velocity at an initial time t=0.

Measures of Change: Continuity, Derivative

Calculus is the mathematics of change with the derivative being a measure of change, and thus can be viewed as the mathematics of IVPs.

The time derivate \dot x(t) of position x(t) as function of time t, measures the change of position per unit time step. A function x(t) with derivative \dot x(t) is said to be differentiable.

Another basic concept of Calculus related to change is continuity, which is a form of poor cousin of derivative, also measuring change but in a less precise way.

You will below meet the precise definitions of derivative and continuity, as more or less precise measures of change.

We here prepare these basic definitions with a short introductory discussion.

A Basic Example

If the velocity \dot x(t) of position x(t) is constant \dot x(t)=v with v a constant velocity, then the position x(t) changes linearly with time: x(t)=x^0+vt for t>0 with x^0 the position for t=0.

Thus x(t)=x^0 + vt is a linear function of t, since it has the form c_0+c_1t with c_0 and c_1 constants.

If v(t) is not constant, then x(t) will not be linear in t, but if v(t) is almost constant locally, for small changes of t, then x(t) will be almost linear locally. We here meet both the concept of continuity and the concept of differentiability:

  • A function v(t) is continuous if v(t) is locally close to a constant.
  • A function x(t) is differentiable (with derivative \dot x(t)) if x(t) is locally close to a linear function in t.

We shall below meet the concept of continuity as Lipschitz continuity including a quantitativemeasure of the local deviation from a constant.

It is natural to generalize, an essential aspect of mathematics, to:

  • A function x(t) is \emph{two times differentiable} if x(t) is locally close to a quadratic function in t up to a third order term.

This connects to a particle with position x(t) subject to constant acceleration a, in which case the velocity v(t)=\dot x(t)=at+v^0 and the position

  • x(t)=\frac{a}{2}t^2+v^0t+x^0   (*)

is exactly equal to a quadratic function in t. Differentiating \dot x=at with respect to t, we find that the second derivative \ddot x(t)=\frac{d}{dt}\dot x(t)=a, and thus (*) can be written

  • x(t)=x(0)+\dot x(0)t+\frac{\ddot x(0)}{2}t^2

while for a general twice differentiable function x(t)

  • x(t)\approx x(0)+\dot x(0)t+\frac{\ddot x(0)}{2}t^2\quad\mbox{for }\vert t\vert\, \mbox{small},

up to a term of order \vert t\vert ^3 (allowing t to also be negative).

We shall below recover this expression as an example of Taylor’s formula expressing a general function locally as a polynomial with coefficients given by the values of the function and its derivativesat a specific point.

Perspectives

Leibniz manuscript from October 29, 1675, introducing the integral sign \int (in the box):

  • It is useful to denote summation (omnia) by \int
Yes, it has shown to be very useful.

To Think About

  • What did Galileo say about motion?
  • How is the derivative of a function defined? Computed?
  • How does a speedometer on a bike work?
  • How does a distance meter on a bike work?

Watch

  • Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind),men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning.
  • When man reasoneth, he does nothing else but conceive a sum total, from addition of parcels; or conceive a remainder, from subtraction of one sum from another: which, if it be done by words, is conceiving of the consequence of the names of all the parts, to the name of the whole; or from the names of the whole and one part, to the name of the other part. And though in some things, as in numbers, besides adding and subtracting, men name other operations, as multiplying and dividing; yet they are the same: for multiplication is but adding together of things equal; and division, but subtracting of one thing, as often as we can. These operations are not incident to numbers only, but to all manner of things that can be added together, and taken one out of another. For as arithmeticians teach to add and subtract in numbers, so the geometricians teach the same in lines, figures (solid and superficial), angles, proportions, times, degrees of swiftness, force, power, and the like; the logicians teach the same in consequences of words, adding together two names to make an affirmation, and two affirmations to make a syllogism, and many syllogisms to make a demonstration; and from the sum, or conclusion of a syllogism, they subtract one proposition to find the other. (Leviathan, Thomas Hobbes )

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