Origins of Classical Mechanics

Preliminaries

1d Equation of Motion for Constant Acceleration

\begin{equation} x = x_0 + v_0 t + \frac{1}{2}a t^2 \end{equation}

Where did this come from?

You likely spent a fair amount of time working with this equation in your introductory mechanics class. Hopefully you derived it from the basic definitions of acceleration and velocity and didn't just pick it from a list of possible equations when you needed to solve a kinematics problem. In the context of this class, we'll see it as one possible solution to a more general differential equation: \begin{equation} \frac{d^2x}{dt^2} = \frac{F}{m} \end{equation} where $F$ and $m$ are both constant in time. There will be many situations where they are not. Also, you should appreciate that it is just a 2nd order polynomial with an independent variable $t$, or, put another way, a parabola in which the coefficients $a$, $v_0$, and $x_0$ filling the normal $A$, $B$, and $C$ values. Sometimes these more general notions are lost during introductory treatments of mechanics.

We can derive nearly all of kinematics (for cases with constant acceleration) by considering the relationships between derivatives and integrals. Let's begin with the definition of acceleration: $$a = \frac{\Delta v}{\Delta t}$$ If we make the $\Delta v$ and $\Delta t$ infinitesimally small, $dv$ and $dt$, then we can rewrite this as: $$a = \frac{d v}{dt} \Rightarrow dv = a \; dt$$ Now, we can take the indefinite integral of both sides: $$\int dv = \int a \; dt$$ Since $a$ is assumed to be constant, we can remove from the integrand. Performing the indefinite integrals: $$v = a t + C_1$$ where $C_1$ is the constant of integration. To determine the constant $C$, consider the equation when $t = 0$. This is the 'initial condition', thus the velocity at this point will be the initial velocity: $v_0$. We therefore obtain: $$v = v_0 + at$$ by considering just the definition of acceleration and the concept of integration.

We can likewise consider the definition of instantaneous velocity: $$v = \frac{dx}{dt}$$ A similar operation leads to: $$\int dx = \int v \; dt$$ Now, we cannot remove $v$ from this integrand since it is not a constant value. However, we just figured out a relation between velocity and time above, so: $$\int dx = \int \left( v_0 + at \right) \; dt$$ In this case, $v_0$ and $a$ are both constants. So the indefinite integral can be solved: $$x = v_0 t + \frac{1}{2}a t^2 + C_2$$ Again, we have a constant of integration to solve for: $C_2$. Let's again consider $t = 0$, i.e. the initial condition. When $t = 0$, the object will be located at the initial $x$ position, $x_0$. Thus $C_2 = x_0$. Finally, we have an equation for $x$ as a function of time given all the initial conditions of position and velocity: $$x = x_0 + v_0 t + \frac{1}{2}at^2$$ This is our fundamental quadratic equation that describes the motion of a particle undergoing translation with constant acceleration.

What does "Classical" mean?

How much do we know?

Where is this ball located right now? Be precise as possible.

What is its velocity? Be precise as possible.

"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes."

Pierre Simon, Marquis de Laplace. (~1814)

A particle is moving in the x-direction and is confined between two perfectly bouncy (ahem, elastic) walls so that it bounces back and forth between $x=0$ and $x=l$, where $l$ is the distance between the two walls. If there is an uncertainty in the initial velocity measurement, i.e. $\Delta v_0$, find the subsequent uncertainty in position at time $t$ later.

Let a particle start from point A and move towards B. Assuming this is a perfect circle and that collisions with the walls are totally elastic, will it ever come back to point A? If so, how many bounces will it take?

A particle bounces off the inner walls of a circular constraint.

Time

If we know how things move in the forward direction of time, can we also know how the process would happen in reverse?

Review

The Outline

Newtonian Mechanics
Special Relativity
Variational Principle
Lagrangian & Hamiltonian Mechanics
Gravitation
Accelerating Reference Frames
Rigid Body Dynamics
Coupled Oscillators
...and ?

Tools

Everything from 20700 and 20800
Advanced Calculus and other Math
Computers

Intro Physics Review

Largely Mechanics (i.e. 20700) but with some references to electric fields and other 20800 material.

What are Newton's 3 Laws of motion?

What are the implications of the conservation of Energy?

... and Momentum?

... and Angular Momentum?

What goes on the R.H.S.?

$$ \frac{dU}{dx} = \rm{?}$$

Math you'll need

Differential Equations

$$\frac{dv}{dt} = \frac{F}{m}$$

But what if $F$ is messy?

Taylor Series

Vectors

We'll need more complicated vector math to deal with different coordinate systems.

To open the door, which direction should the torque vector point?

Linear Algebra

Later, we will express some of the physics in the context of Linear Algebra.

Computational Physics

Quick, you have 5 minutes to prepare a plot of: $$U(x) = -\frac{1}{x}e^{-x}+\frac{1}{5x^2}$$ What do you do?

Evolution into Modern Times

Tools evolve

Analytical vs. Numerical

Analytic Solutions

Sometimes we can obtain an exact solution that describes the motion.

Numeric Solutions

Many more situations will not have an exact, algebraic solution and will require other methods, i.e. numerical methods.

Both of these approaches are useful for physics. When algebraic solutions are obtainable, then it's often easier to understand the system in more fundamental ways. One can see how it will evolve, what the various influences on the system are (and aren't), and the generally satisfy our desire for some degree of 'beauty' in the math. Nature is reduced to simple equations in these contexts. However, at this point, most of the situations that lead to clean algebraic results have already been discovered and investigated so there's not much new to do there.

Many more systems will not produce exact algebraic solutions though. These need to be approached through other means. Even some very basic sounding situations are not solvable through algebraic means. For example, just asking when a certain planet in an elliptical orbit will be at a certain place, can lead to a transcendental equation that has to be solved algorithmically or through other approximation methods. Or, as well see soon, figuring out how far a projectile will travel with linear air resistance cannot be done without numerical methods.

Thus for this course, we will work with both approaches simultaneously. We will study many classic cases of dynamics that yield exact solutions, and at the same time, learn how to approach them in other ways that will help tackle the other, less well behaved systems.