# Differential Equations of Motion

# Initial Value Problem IVP

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

- .

We also refer to as the *derivative* of with respect to . Similarly, we have

- ,

where 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:

- ,

where we in the spirit of Leibniz assume the time step to be vanishingly small. Newton used and Leibniz used to denote the derivative of with respect to time .

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 of a function is called *differentiation*, or more precisely *differentiation with respect to* .

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

- ,

which is to be interpreted as

- ,
- ,

where , and are viewed as functions of , and and are given *initial values* of position and velocity at an *initial time* .

# 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 of position as function of time , measures the change of position per unit time step. A function with derivative 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 of position is constant with a constant velocity, then the position changes linearly with time: for with the position for .

Thus is a linear function of , since it has the form with and constants.

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

- A function is continuous if is locally close to a constant.
- A function is differentiable (with derivative ) if is locally close to a linear function in .

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 is \emph{two times differentiable} if is locally close to a quadratic function in up to a third order term.

This connects to a particle with position subject to constant acceleration , in which case the velocity and the position

- (*)

is exactly equal to a quadratic function in . Differentiating with respect to , we find that the *second derivative* , and thus (*) can be written

while for a general twice differentiable function

- ,

up to a term of order (allowing 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

- Return of Descartes
- Soul as Simulation of Body
- Zeno’s Paradox of Particle Motion
- Slinky as Resolution of Zeno’s Paradox

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

*It is useful to denote summation (omnia) by .*

# 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

- Newton, Leibniz or Gore?
- Leibniz Calculus rap
- Importance of Calculus,
- Leibniz Monadology
- Quinton about Leibniz
- Feynman on the Relation of Mathematics and Physics}

*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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