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.