Classical Mechanics 1

Newtons Equations, Galilean Spacetime, Calculus of Variations

This is the first section of notes for Classical Mechanics!

The goal for this series is to thoroughly cover classical mechanics at a level that can be appreciated by physics and math people at an undergraduate to graduate level. I’ll hopefully be able to indicate different section of math vs physics but we’ll see how it goes.

I’ll be taking my notes from the classes that I’ve taken, my personal adds, and the following books:

  1. Classical Mechanics by Herbert Goldstein, Charles Poole, John Safko

  2. Classical Dynamics: A Contemporary Approach by Jorge V. José, Eugene J. Saletan

  3. Mathematical Methods of Classical Mechanics by V.I. Arnold

  4. Introduction to Mechanics and Symmetry by Jerold E. Marsden, Tudor S. Ratiu

  5. Mechanics by L.D. Landau, E.M. Lifshitz

  6. Mathematical Aspects of Classical and Celestial Mechanics by Vladmir I. Arnold, Valery V. Kozlov, Anatoly. I Neishtadt

  7. The Variational Principles of Mechanics by Cornelius Lanczos

  8. Introduction to Dynamics by I.C. Percival, D. Richards

  9. Foundations of Mechanics by Ralph Abraham, Jerrold E. Marsden

  10. Geometric Mechanics by Waldyr Muniz Oliva

  11. Classical Mechanics with Calculus of Variants and Optimal Control: An Intuitive Introduction

  12. Modern Classical Mechanics by T.M. Helliwell, V.V. Sahakian

  13. Symplectic Geometry and Integrable System Lecture Notes by Anton Izosimov

  14. Lectures on Symplectic Geometry by Ana Cannas da Silva

  15. Introduction to Symplectic Topology by Dusa McDuff, Dietmar Salamon

  16. Symplectic Invariants and Hamiltonian Dynamics by Helmut Hofer, Eduard Zehnder

  17. Symplectic Geometry and Analytic Mechanics by Paulette Libermann and Charles-Michel Marle

  18. Calculus of Variations by I.M. Gelfand, S.V Fomin

  19. Variational Calculus by Jean-Pierre Bourguignon

  20. Calculus of Variations: With Applications to Physics and Engineering by Robert Weinstock

  21. A First Course in the Calculus of Variations by Mark Knot

  22. Lectures on the Calculus of Variations and Optimal Control Theory by L.C. Young

  23. Symplectic Topology and Floer Homology: Volume 1 by Yong-Geun Oh

Among other resources for background information which I will unfortunately have to assume you know basic Newtonian physics(with calculus), linear algebra, some ODE, and some differential geometry. However, I will offer a review of the topics as needed.

While this is a laundry list of book that I admittedly havent read through completely. These will be ever changing lecture notes that will be continually updated and more books will definitely be added.

Being that this is the introduction section let’s quickly go over Newtonian mechanics as well as some new math, the calculus of variations.

Newton’s formulation of Mechanics begins with his namesake equation

\[\begin{aligned} \vec{F} &=m\vec{a}\\ &=m\frac{d\vec{v}}{dt}\\ &=m\frac{d^2\vec{r}}{dt^2} \end{aligned}\]

Where \(\vec{a}\) is the acceleration, \(\vec{v}\) is the velocity, and \(\vec{r}\) is the position. (I may at times forget the vector notation). We will assume that we live in \(\mathbb{R}^3\) with time \(t\in\mathbb{R}\) being a parameter (i.e living in \(\mathbb{R}\times \mathbb{R}^3\), and for those more mathematically inclined I will later explain this is an affine space \(A^4\)).

We see that Newton’s equations gives us \(3N\) for \(N\) particles, but for the most part we’ll focus on the single particle case. As an example let’s do the simple case of a constant force, gravity.

The force of gravity is given by the following equation

\[\vec{F}=-m\vec{g}\]

with \(g=9.81\) being the graviational constant which acts in the \(z\) direction. Our differential equation then becomes

\[\begin{aligned}m\vec{a}&=-m\vec{g}\\\begin{pmatrix}\frac{d^2x}{dt^2}\\\frac{d^2y}{dt^2}\\\frac{d^2z}{dt^2}\end{pmatrix}&=\begin{pmatrix}0\\0\\-g\end{pmatrix}\end{aligned}\]

Solving these second order differential equations we get the solution

\[\begin{pmatrix}x(t)\\y(t)\\z(t)\end{pmatrix}=\begin{pmatrix}c^{x}_{1}t+c^{x}_{2}\\c^{y}_{1}t+c^{y}_{2}\\-gt^2+c^{z}_{1}t+c^{z}_{2}\end{pmatrix}\]

With \(c_x,c_y,c_z\) being constants determined by initial conditions(i.e the initial velocities and positions).

Cool, awesome even. We won’t get into force diagrams and the like but now we know how idealized objects evolve (hint hint dynamical systems hint hint). As long as we know some initial conditions and assume (or get from an experiment) some reasonable force we can calculate the exact path that this object will take. However, the current form of Newton’s equations aren’t the most convenient. Solving Newton’s equations in it’s most basic form is solving a second order ODE for a force in time, but often physicists are more conserned with forces that are position dependent such as gravity (unfortunate that I did solve this before normally but still). Fortunately we can do this in systematic way.

Let’s first introduce a quantitity known as momentum.

\[\vec{p}=m\vec{v}\]

This quantity ends up being more fundamental due to it being velocity dependent, and let’s us rewrite Newton’s equations as such

\[\vec{F}=\frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}\]

Alright back to solving. So the form of Newton’s equation we want to solve is the following

\[m\frac{d\vec{v}}{dt}=F(\vec{r})\]

As normal let’s integrate to solve this ODE (we’ll generalize the calculation to arbitrary dimension later)

\[m\int_{x_i}^{x_f} \frac{dv}{dt}dx=\int_{x_i}^{x_f} F(x)dx\]

We do a simple subsitution for the LHS of the equation

\[\begin{aligned} u &=x\\ \frac{du}{dt} &=\frac{dx}{dt} \\ du&=vdt\end{aligned}\]

Our integral then becomes

\[\begin{aligned}m\int_{x_i}^{x_f} \frac{dv}{dt}dx&=m\int_{t_i}^{t_f} \frac{dv}{dt}vdt\\&=m\int_{t_i}^{t_f}vdv\\&=\frac{1}{2}mv^{2}_{f}-\frac{1}{2}mv^{2}_{i}=\int_{x_i}^{x_f} F(x)dx\end{aligned}\]

We’re very free to make this a function of \(x\) by making the previous integral an indefnite one by the changing the bounds.

We’ll call the quantity \(\frac{1}{2}mv^2\) the kinetic energy (\(K\)) and refer to the result we derived as the Work-Energy Theorem. I recommend you use this to solve the equation for the harmonic oscillator (\(F=-kx\)) for practice.

Now to do this in the multi-dimensional case. Imagine we have a path \(\gamma\) such that \(\gamma:[a,b]\rightarrow \mathbb{R}^3\), and have a force \(\vec{F}\) that successfuly takes our mass from point \(a\) to \(b\). This curve has some infinitesimal length that we’ll call \(d\vec{r}\) and the force acts on this every point on this path which is represented as a dot product \(\vec{F}\cdot d\vec{r}\). We’ll go through a similar procress as above, i.e.

\[\vec{F}\cdot d\vec{r}=m\frac{d\vec{v}}{dt}\cdot d\vec{r}\]

We simply integrate:

\[\int_{\vec{r_i}}^{\vec{r_f}}F(\vec{r})\cdot d\vec{r}=m\int_{\vec{r_i}}^{\vec{r_f}}\frac{d\vec{v}}{dt}\cdot d\vec{r}\]

Then we do a similar change of variables to rewrite the RHS

\[m\int_{\vec{r_i}}^{\vec{r_f}}\frac{d\vec{v}}{dt}\cdot d\vec{r}=m\int_{t_i}^{t_f}\frac{d\vec{v}}{dt}\cdot \vec{v}dt\]

We can solve this integral by taking note of a convenient vector identity

\[\frac{d}{dt}(\vec{v}\cdot\vec{v})=\frac{d\vec{v}}{dt}\cdot\vec{v}+\vec{v}\cdot\frac{d\vec{v}}{dt}=2\left(\frac{d\vec{v}}{dt}\cdot\vec{v}\right)\] \[\implies \frac{d\vec{v}}{dt}\cdot\vec{v}=\frac{1}{2}\frac{d}{dt}(\vec{v}\cdot\vec{v})\]

Substituting this back into the integral we get

\[\begin{aligned}m\int_{t_i}^{t_f}\frac{d\vec{v}}{dt}\cdot \vec{v}dt &=\frac{m}{2}\int_{t_i}^{t_f}\frac{d}{dt}(\vec{v}\cdot\vec{v})dt\\&=\frac{1}{2}mv_{f}^{2}-\frac{1}{2}mv_{i}^{2}\end{aligned}\] \[\implies \int_{\vec{r_i}}^{\vec{r_f}}F(\vec{r})\cdot d\vec{r}=\frac{1}{2}mv_{f}^{2}-\frac{1}{2}mv_{i}^{2}\]

I definitely abused some notation by leaving out the proper symbol for a line integral but this eludes to something pretty big. On the LHS we have a line integral, but on the right hand side we had an integral that only depended on the endpoints and gives us a scalar result. If you know a bit of vector calc (notes coming to your local site soon), this means we can rewrite the force \(\vec{F}\) as a gradient of a vector field say \(\nabla U\) that we call the potential energy (this also aludes to some very cool geometric features of our theory) and the idea of a path independent process which will remain prevelant in our study of classical mechanics.

This is where physics and math kind of diverge. Because this integral only measures differences in $U$ we can arbitrarily choose the our poential energy function and do things like defining \(U(\infty)=0\) for some forces. So defining the result of the integral as \(U(r_i)-U(r_f)\) or simply making \(F=-\nabla U\) is completely allowed. Gotta love physics making our lives easier.

Our next magic trick will be using the fact we just derived. We rewrite the force as \(\vec{F}=-\nabla U\) and integrate once again

\[\int_{\vec{r_i}}^{\vec{r_f}}F(\vec{r})\cdot d\vec{r}=\int_{\vec{r_i}}^{\vec{r_f}}\nabla U(\vec{r})\cdot d\vec{r}=U(\vec{r_i})-U(\vec{r_f})\]

We then recall the work-energy theorem from above we can equate the two integrals and get

\[K_f-K_i=U_i-U_f\]

With a simple rearrangement we finally get

\[K_f+U_f=K_i+U_i\]

This is known as the Conservation of Energy. If you couldn’t tell, this is a massive. With this we can determine the dynamics of a system from some initial or final conditions, but this does beg the question. We know that the overall energy is conserved for a system, what about over time? Let’s define the total energy \(E\) as \(K+U\), and then we’ll take the time derivative of this

\[\begin{aligned}\frac{dE}{dt}&=\frac{d}{dt}(K+U)=\frac{d}{dt}\left(\frac{1}{2}m(\vec{v}\cdot\vec{v})+U(\vec{r})\right)=\frac{1}{2}m\frac{d}{dt}(\vec{v}\cdot\vec{v})+\frac{\partial U}{\partial \vec{r}}\frac{d\vec{r}}{dt}\\&= m\left(\frac{d\vec{v}}{dt}\cdot\vec{v}\right)+\frac{\partial U}{\partial \vec{r}}\frac{d\vec{r}}{dt}=\vec{v}\left(m\vec{a}+\frac{\partial U}{\partial\vec{r}}\right)=\vec{v}(\vec{F}-\vec{F})=0\end{aligned}\]

So the energy is conserved over time as well!

This should hopefully have been a more chill introduction to basic Newtonian mechanics suitable for a physics or math person to start thinking about generalized dynamics.

Next we’ll be going over the nitty gritty mathematical details of what we just talked about (definitely not enough for a course in the resprective math subjects…..that sounds like a good idea) and also give an introduction to the calculus of variations.

However…we can still have some physical motivation for why we describe the math as such. If you’ve ever dropped anything it should be fairly obvious that it doesn’t matter where you are when it drops, it will drop regardless. If you’ve ever hanged on monkey bars and dropped some ice cream you were holding it still dropped the same way AND the kid running across the way also saw it drop (they’re not moving at close to the speed of light so \(c=\infty\) for now) in the same way. This simple, albeit not the best, experiment on the isotropy of the universe shows us that the universe simply doesn’t have a preffered direction. So things like a center of the universe doesn’t make sense. We will demonstrate this invariance, starting with the addition of velocity.

Assume we have some local coordinate system, \(S\). In this coordinate system things are as they should be, objects in this coordinate system have nicely definied positions. Now assume we have another coordinate system \(S'\) that is moving from \(S\) at a constant velocity of \(v\). Remember the objects in \(S\) are also in \(S'\). From the perspective of $S’$ if the objects are still in the \(S\) coordinate system they are moving backward, i.e.

\[\begin{pmatrix}x'\\ y'\\ z'\\ t'\end{pmatrix}\rightarrow \begin{pmatrix}x-vt\\ y'\\ z'\\ t'\end{pmatrix}\]

Or from the perspective of \(S\) things are moving forward,

\[\begin{pmatrix}x\\ y\\ z\\ t\end{pmatrix}\rightarrow \begin{pmatrix}x'+vt\\ y\\ z\\ t\end{pmatrix}\]

Ok great, but you might ask why we assumed that the velocity has to be constant? Let’s work the relative velocitie and acceleration in the \(x\)-direction.

\[x=x'+vt\implies \frac{dx}{dt}=\frac{dx'}{dt}+v\implies \frac{d^2x}{dt^2}=\frac{d^2x'}{dt^2}\]

We see here that constant velocity doesn’t change Newton’s equations at all! The same thing can be shown in the \(S'\) coordinate system (we’ll now simlpy call this a frame) the sign for the velocity will just change. The transformation we just did is called a \(\textit{Galilean transformation}\) and is our first introduction into the notion of symmetries(more on this later).

We can formalize this idea in mathematics with the use of an affine space that we will call Galilean spacetime. Let’s first define what this space is exactly

If we have a vector space \(V\) over a field \(\mathbb{F}\) (we won’t get into the subtleties of this as I’m kinda lying about using an arbitrary field but, for our porposes you can assume \(\mathbb{F}=\mathbb{R}\)), an affine space \(\mathbb{A}\) is a set \(A\) modelled with our vector space such that \(\exists \phi\) such that \(\phi:V\times \mathbb{A}\rightarrow \mathbb{A}\) with \(x,y\in\mathbb{A}\) and \(v\in V\) that has the following properties

\[y=\phi(v,x)\] \[\phi(v,x)=x\implies v=0\] \[\phi(0,x)=x\] \[\phi(u+v,x)=\phi(u,\phi(v,x))\]

These properties look a bit like addition don’t they? Following general notation used we’ll instead use \(\phi(v,x)=v+x=y\). This would of course make you think that \(v=x-y\) but I have to remind you that this is unfortunately just notation, as we havent defined what subtraction is in \(\mathbb{A}\). Also note that intuitively this has to be the case as we’ve made no refernce to any sort of origin in our affine space.

Instead what we can do is fix \(x\in\mathbb{A}\) (i.e fix some sort of relative origin) and this will let us define our usual vector space operations. For instance how about addition?

\[y_1+y_2=\left((y_1-x)+(y_2-x)\right)+x\]

Scalar multiplication is defined similarly

\[ay_1=\left(a(y_1-x)\right)+x\]

It should be fairly clear that when we fix this “origin” \(\mathbb{A} \cong V\). Don’t take this for granted you should really prove this. See a linear algebra textbook (or maybe some notes coming soon) and go through the axioms of vector spaces. An incredibly thorough treatment is given by Hoffamn and Kunze “Linear Algebra” and Halmos “Finite Dimensional Vector Spaces”. Another good combination is “Linear Algebra Done Right” by Axler and “Linear Algebra Done Wrong” by Treil.

To really drive the notation and intuition let’s construct this step by step

First we have our affine space, \(A\) with points \(y_1,y_2,x\)

Next we’ll construct a vector using our “origin” \(x\). The vectors respectively are \(y_1-x,y_2-x \in V\). Remember these are not elements in our affine space \(A\).

Now we can just add these vectors as we would in a normal vector space

Finally we “add” our “origin” \(x\) to put this point into our affine space.

And poof magic \(y_1+y_2=((y_1-x)+(y_2-x))+x\)!

We are able to “add” points. Hopefully the intuition came to you, especially why we need the \(+x\).

Great! Now that we’ve sort of unveiled the structure of affine spaces it makes sense to study functions on them, mainly understanding transformations from one affine space to another. We propose that the most general affine transformation takes the following form

\[f:x\rightarrow Ax+y_0\]

Where \(x,y_0\) are in a affine modelled vector space.

We can show some cool properties of this (WARNING: Keep track of affine addition)

\[\begin{aligned}f(x_1+x_2)&=f((x_1-x)+(x_2-x)+x)=f(x_1+x_2-x)\\&=A(x_1+x_2-x)+y_0\end{aligned}\] \[\begin{aligned}f(x_1)+f(x_2)&=(Ax_1+y_0)+(Ax_2+y_0)=A(x_1+x_2)+2y_0=A((x_1-x)+(x_2-x)+x)\\&=A(x_1-x)+A(x_2-x)+Ax+2y_0=Ax_1-Ax+Ax_2-Ax+Ax+2y_0\\&=Ax_1-(Ax+y_0)+Ax_2-(Ax+y_0)+(Ax+y_0)+2y_0\\&=A(x_1+x_2-x)+y_0\end{aligned}\]

Thus proving

\[f(x_1+x_2)=f(x_1)+f(x_2)\]

There are some further details such as showing that this transformation is indeed the unique form of an affine transformation (use the fact that we can choose our “origin” to be zero and use the facts we proved before) and \(f\) can be a map preserving the vector space structure underlying our affine space but for now I won’t get into that (I probably will in the future)

The last thing I’ll mention before delving into Galilean spacetime is the notion of a convex space.

Simply put a convex space \(C\) is set such that for any points \(,y\in C\)

\[\{t(y-x)+x|t\in[0,1]\}\]

Thankfully we just need to mention the notion of a convex space as the implications of being in one has very nice properties. From the definition it should be fairly obvious that every affine space is also a convex space.

We’re now in a place to talk about Galilean spacetime!

Galilean spacetime is defined by a quadruple (just think of this as tbe information needed to fully describe an object) \(\mathcal{G}=(V,\mathcal{E},\tau,g)\)

  • \(V\) is a 4 dimensional vector space
  • \(\mathcal{E}\) is an affine space modeled on \(V\)
  • \(\tau\) is a map such that \(\tau:V\rightarrow \mathbb{R}\)
  • \(g\) is an inner product on \(\ker(\tau)\)

A quick reminder that \(\ker(\tau)=\{\vec{v}\in V\vert \tau(\vec{v})=\vec{0}\}\) (just the space of vectors of \(V\) that gets mapped to the zero vector)

Points in Galiliean spacetime represent when and where things happen, so we’ll call these points events. Based on our definition of this space we can’t measure distances and time in the tradition/intuitive way.

For one remeber that \(x_2-x_1\in V\) so \(\tau\) can’t measure the times of individual events instead it measures the time between events. As such if \(\tau(x_2-x_1)=0\) means that the events happen at the same time i.e simultaneous. We’ll also note that from our definition of Galilean spacetime, we can only measure distance between simultaneous events. We’ll get into how we can measure distances between nonsimultaneous events later as this requires a bit more structure to our theory.

The above figure is how we should think about simultaneous events in Galilean spacetime where the dotted lines represent them. More generally we should think about Galilean spacetime as the following for standard Newtonian physics.

Each plane is then \(\mathbb{R}^3\) at some different time \(t_n\). We also see that simultaneous events will lay on the same plane or in the simpler case the sameline

Simultaneous events are important enough that we’ll call the set of them \(I_{\mathcal{G}}\) where

\[I_{\mathcal{G}}=\{S\subset\mathcal{E}|S \text{ is the collection of simultaneous events}\}\]

This then leads to the idea of an observer for simultaneous events.