Handy Summary of Classical Mechanics

Key references

1. Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc..
2. Lanczos, Cornelius; "The Variational Principles of Mechanics"; 4th ed.; 1970; Dover Publications, Inc..
3. Taylor, John R.; "Classical Mechanics"; 2005; University Science Books.
4. Susskind, Leonard & Hrabovsky, George; "The Theoretical Minimum: What You Need to Know to Start Doing Physics"; 2013; Basic Books.
5. Coopersmith, Jennifer; "The Lazy Universe"; 2017; Oxford University Press.

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Lagrangian mechanics

Hamilton's principle

The actual path which a holonomic system follows between two points $1$ and $2$  in configuration space in a given time interval, $t_1$ to $t_2$, is such that the action integral

$$\begin{equation}\label{eqn:action:functional}S = \int_{t_1}^{t_2} \mathcal{L}dt\end{equation}$$

is stationary when taken along the actual path.

$\mathcal{L(\mathbf{q},\dot{\mathbf{q}},t)}$ is the Lagrangian, a function of the generalised coordinates $\mathbf{q}$, generalised velocities $\dot{\mathbf{q}}$, and time $t$, and satisfies the Euler-Lagrange equations

$$\begin{equation} \frac{d}{dt}\frac{\partial{\mathcal{L}}}{\partial{\dot{q_i}}} - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \quad [i=1, \dots, n] \end{equation}$$

$n$ being the number of degrees of freedom.

The generalized force $F_i$ and generalized (or conjugate) momentum $p_i$ are respectively given by $$\begin{equation} \frac{\partial{\mathcal{L}}}{\partial q_i} = F_i, \quad \frac{\partial{\mathcal{L}}}{\partial \dot{q_i}} = p_i\end{equation}$$

The Euler-Lagrange equations can be rewritten as $F_i = \dot{p_i}$.

Conservation and symmetry

If the Lagrangian does not contain a coordinate $q_i$ (though it may contain $\dot{q_i}$) then the coordinate is said to be ignorable and the corresponding conjugate momentum $p_i$ is conserved.

Conservation of Hamiltonian

If $\mathcal{L}$ does not depend explicitly on time ($\partial{\mathcal{L}}/\partial{t} = 0$), then the Hamiltonian

$$\begin{equation}\mathcal{H}(\mathbf{q},\mathbf{p},t) = (\sum_{i=1}^{n}\dot{q_i} p_i) - \mathcal{L}\end{equation}$$

is conserved.

For most systems, $\mathcal{H}$ is just the total energy and is frequently conserved. However, the identification of $\mathcal{H}$ as a constant of motion and as the total energy are two separate matters, and the conditions sufficient for one are not enough for the other.

Lagrangian for electromagnetism

The Lagrangian for a charge $q$ with mass $m$ in an electromagnetic field is $$\begin{equation}\mathcal{L}(\mathbf{r}, \dot{\mathbf{r}}, t) = \frac{1}{2}m\dot{\mathbf{r}}^2 - q(V-\dot{\mathbf{r}}\cdot\mathbf{A})\end{equation}$$

$V(\mathbf{r},t)$ and $\mathbf{A}(\mathbf{r},t)$ being the scalar and vector potentials respectively.

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Hamiltonian mechanics

Given the $2n$-dimensional phase space of $n$ generalised coordinates $q_i$ and $n$ conjugate momenta $p_i$ then

$$\begin{equation}\dot{q_i} =\frac{\partial\mathcal{H}}{\partial p_i}\quad\text{and}\quad \dot{p_i}=-\frac{\partial{\mathcal{H}}}{\partial q_i}\quad[i=1,\dots,n]\end{equation}$$

are the canonical equations of Hamilton.

If $\mathcal{H}$ is independent of $q_i$ then the conjugate momentum $p_i$ is conserved and the coordinate $q_i$ is ignorable.

A phase-space orbit is the path traced in phase space by a system as the system evolves in time.

Hamiltonian for electromagnetism

The Hamiltonian for a charge $q$ in an electromagnetic field is

$$\begin{equation}\mathcal{H} = \frac{(\mathbf{p}-q\mathbf{A})^2}{(2m)} + qV\end{equation}$$

Canonical transformations

Coordinate transformations given by

$$\begin{align}Q_i &= Q_i(\mathbf{q},\mathbf{p},t) \\ P_i &= P_i(\mathbf{q},\mathbf{p},t)\end{align}$$

that satisfy

$$\begin{equation}\dot{Q_i}=\frac{\partial K}{\partial P_i}, \quad \dot{P_i}=-\frac{\partial K}{\partial Q_i}\end{equation}$$

where $K(\mathbf{Q},\mathbf{P},t)$ is some function are said to be canonical (or contact) transformations. The transformation equations are completely specified by the generating function $F$ of the transformation which can be written as a function of independent variables in one of only four forms:

$$F_1(\mathbf{q},\mathbf{Q},t),\quad F_2(\mathbf{q},\mathbf{P},t),\quad F_3(\mathbf{p},\mathbf{Q},t),\quad F_4(\mathbf{p},\mathbf{P},t)$$

For $F_1$,

$$p_i = \frac{\partial F_1}{\partial q_i},\quad P_i = -\frac{\partial F_1}{\partial Q_i},\quad K = H + \frac{\partial F_1}{\partial t}$$

For $F_2$,

$$p_i = \frac{\partial F_2}{\partial q_i},\quad Q_i = \frac{\partial F_2}{\partial P_i},\quad K = H + \frac{\partial F_2}{\partial t}$$

For $F_3$,

$$q_i = -\frac{\partial F_3}{\partial p_i},\quad P_i = -\frac{\partial F_3}{\partial Q_i},\quad K = H + \frac{\partial F_3}{\partial t}$$

For $F_4$,

$$q_i = -\frac{\partial F_4}{\partial p_i},\quad Q_i = \frac{\partial F_4}{\partial P_i},\quad K = H + \frac{\partial F_4}{\partial t}$$

A transformation of time occurs automatically in the $4$-dimensional relativistic Hamiltonian formulation; the invariant parameter of the system is the proper time $\tau$ and the ordinary time $t$ appears as one of the coordinates of the particle.

The motion of a system point in time is simply a particular contact transformation of the canonical coordinates in phase space.

The particular form for $F_2$ below generates the identity transformation:

$$F_2 = \sum_i q_i P_i,\quad p_i = P_i,\quad Q_i = q_i,\quad K = H$$

Integral invariants of Poincaré

The integral

$$J_n = \int \dots \int dq_1 \dots dq_n\ dp_1 \dots dp_n$$

over any arbitrary region of phase space is invariant under canonical transformations. The invariance of $J_n$ is equivalent to the statement that volume in phase space is invariant under canonical transformations.

Lagrange brackets

(Note: The bracket notation used in this post is from Goldstein. In particular, the notation for Poisson brackets directly translates to the commutator of Quantum Mechanics for which the Poisson brackets are a direct analogue.)

The Lagrange bracket of $u$ and $v$ defined by

$$\begin{equation}\{u,v\}_{\mathbf{q},\mathbf{p}} = \sum_{i}(\frac{\partial q_i}{\partial u}\frac{\partial p_i}{\partial v}-\frac{\partial p_i}{\partial u}\frac{\partial q_i}{\partial v}) \end{equation}$$

is invariant under canonical transformations.

$$\{u, v\} = -\{v, u\}$$

The fundamental Lagrange brackets are

$$\begin{align}\{q_i, q_j\} &= 0\\ \{p_i, p_j\} &= 0\\ \{q_i, p_j\} &= \delta_{ij}\end{align}$$

Poisson brackets

The Poisson bracket of $u$ and $v$ is defined by

$$\begin{equation}[u,v]_{\mathbf{q},\mathbf{p}} = \sum_{k}(\frac{\partial u}{\partial q_k}\frac{\partial v}{\partial p_k} - \frac{\partial u}{\partial p_k}\frac{\partial v}{\partial q_k})\end{equation}$$

with the identity

$$\begin{equation}[u,v]=-[v,u]\end{equation}$$

The fundamental Poisson brackets

$$\begin{align}[q_i,q_j]&=0\\ [p_i,p_j]&=0\\ [q_i,p_j]&=\delta_{ij}\end{align}$$

are canonical invariants.

All Poisson brackets are independent of the set of canonical coordinates they are expressed in.

The canonical equations of motion written using Poisson brackets are

$$\begin{align}[q_i, \mathcal{H}] = \frac{\partial\mathcal{H}}{\partial p_i} &= \dot{q_i}\\ [p_i, \mathcal{H}] = - \frac{\partial\mathcal{H}}{\partial q_i} &= \dot{p_i}\end{align}$$

In general,

$$\begin{equation}\frac{d}{dt}u(\mathbf{q},\mathbf{p},t) = [u, \mathcal{H}] + \frac{\partial u}{\partial t}\end{equation}$$

If $u$ is a constant of motion then

$$\begin{equation}[u,\mathcal{H}]=0\end{equation}$$

Jacobi's identity

$$\begin{equation}[u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0\end{equation}$$

If $u$ and $v$ are two constants of motion and $w$ is taken to be $\mathcal{H}$ then the relation reduces to

$$\begin{equation}[\mathcal{H}, [u,v]] = 0\end{equation}$$

Thus, the Poisson bracket of two constants of motion is itself a constant of the motion (Poisson's theorem).

Infinitesimal contact transformations and generators

$$\begin{align}Q_i &= q_i + \delta q_i = q_i + \epsilon\frac{\partial G}{\partial p_i} \\ P_i &= p_i + \delta p_i = p_i - \epsilon\frac{\partial G}{\partial q_i}\end{align}$$

where $\epsilon$ is some infinitesimal parameter of the transformation and $G$ is given by

$$F_2 = \sum_i q_i P_i + \epsilon G(\mathbf{q},\mathbf{P})$$

It is customary to designate $G$ as the generating function even though, strictly speaking, the term refers to $F$.

When $G=\mathcal{H}(\mathbf{q},\mathbf{p})$ and $\epsilon$ is a small time interval $dt$,

$$\delta q_i = dt\ \dot{q_i} = dq_i,\quad \delta p_i = -dt\ (-\dot{p_i}) = dp_i$$

Thus, the Hamiltonian is the generator of the system motion with time.

For any function $u(\mathbf{q},\mathbf{p}), $ we have $\delta u = \epsilon[u, G]$ which implies $\delta\mathcal{H}=\epsilon[\mathcal{H},G]$. If $G$ is a constant of motion, then $[\mathcal{H},G]=0$ and, hence, $\delta\mathcal{H}=0$. Thus, the constants of motion are the generating functions of those infinitesimal canonical transformations which leave the Hamiltonian invariant. One can therefore determine all the constants of motion by an examination of the symmetry properties of the Hamiltonian!

Liouville's theorem

The density of systems in the neighbourhood of some given system in phase space remains constant in time.

Helmholtz circulation theorem

Vortices cannot be created or destroyed in phase space.

Noether's theorem

Every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.

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Hamilton-Jacobi theory

$$\mathcal{H}(\mathbf{q},\mathbf{p},t) + \frac{\partial S}{\partial t} = 0$$

is the Hamilton-Jacobi equation with $S$ being the action functional (equation $\ref{eqn:action:functional}$). $S$, also called Hamilton's principal function, happens to be the (type 2) generating function that characterises a canonical transformation to new coordinates $(\mathbf{Q},\mathbf{P},t)$ where either all the $Q_i$ and $P_i$ are constant or the $Q_i$ are cyclic (making the conjugate momenta $P_i$ constant). The corresponding Hamiltonian $\mathcal{H}(\mathbf{Q},\mathbf{P},t)$ equals $0$.

$$\mathcal{H}(q_1,\dots,q_n,\frac{\partial S}{\partial q_1},\dots,\frac{\partial S}{\partial q_n},t) + \frac{\partial S}{\partial t} = 0$$ where the individual momenta $p_i=\partial S/\partial q_i$ are constants.

The solutions yield $\mathbf{q}$ and $\mathbf{p}$ in terms of their starting values $\mathbf{q}_0$ and $\mathbf{p}_0$ at time $t_0$.

$$\begin{align}q_i &= q_i(\mathbf{q}_0,\mathbf{p}_0,t) \\ p_i &= p_i(\mathbf{q}_0,\mathbf{p}_0,t)\end{align}$$

Geometrical interpretation

Hyper-surfaces of constant $S$ represent 'wave-fronts' that evolve in time making the Hamilton-Jacobi theory a direct analogue of the 'matter waves' in Quantum Mechanics. The motion of a single particle can be represented by a wave. This played an important role in the development of Schrödinger's wave mechanics.

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Continuous systems (Fields)

<TBD>

Physics and Intuition

One common criticism of modern Physics (i.e., Relativity and Quantum Mechanics) is that it is counter-intuitive. This is not unexpected: our evolutionary history in no way requires that we have an innate understanding of the mechanisms underpinning the Universe at all scales. That said, this criticism is not uniquely limited to modern Physics. In this post I explore concepts that we often take for granted today, and how those could have seemed counter-intuitive to people of earlier times.

Pythagoras' theorem

Or, more generally, 'Applicability of mathematical laws to the Universe'.

Every kindergartener is intimately familiar with Pythagoras' theorem that equates the square of the length of the hypotenuse ($c$) of a right triangle to the sum of the squares of the length of its two other sides ($a$ and $b$); to wit, $a^2 + b^2 = c^2$. This theorem was proved - using principles of logic, induction, etc. - by Euclid in Elements, Book 6, Proposition 31. (It was already known to be 'true', in various guises, for at least a millennia preceding Euclid.) Over time we have accumulated multiple proofs for this theorem and newer proofs continue to be discovered.

All extant proofs of Pythagoras' theorem ultimately rest on the 'truth' of Euclid's postulates which, in turn, rely on the implicit definitions of point, line, plane etc. none of which have any counterparts in the physical world. These latter concepts reside solely in the platonic world of Geometry and have only very approximate, if that, representations in the real world. Ergo, there is no intuitive reason why Pythagoras' theorem should hold for right triangles drawn on, say, a flat surface. If it does, the only way to build certainty is via experience. Historically, many cultures had determined to their satisfaction that Pythagoras' theorem did hold in practice within the limits of measurement accuracy they could bring to bear on it. (This hasn't changed in the modern world with fancier tools and precision.) Nevertheless, why this should be the case in the first place remains a mystery and is certainly not intuitive.

This very consideration also holds for other mathematical laws which appear to (unreasonably) govern the evolution of the Universe. A fuller discussion is in Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Galilean relativity

I consider both concepts discussed under this section as the pinnacle of pre-Newtonian physics. Galileo's investigations into the motion of bodies and the insights he developed as a result mark a turning point in physics: from a more common sense model of how the world works to one that seemingly defied common sense but was, nevertheless, demonstrably closer to truth.

The upshot of this section is that the laws of motion are the same for all inertial frames of reference.

Inertia

Or, 'Moving things keep moving'. 

Kindergarteners first learn the concept of inertia when they imbibe Newton's first law of motion. However, it was Galileo who originally deduced this intrinsic property of a material body by rolling down balls on ramps. (His experiments and the deductions he used makes for an amazing narrative. But that's for another time.) 

Simply put, Galileo's law of inertia states that a body set into motion continues to move at a constant speed $v$ in a straight line until an external (unbalanced) force acts on it. This also applies to bodies at rest as a special case of moving bodies (i.e., $v = 0$).

Surely, this is counter-intuitive! Every day experience objects to this law. However, to our credit, we can rationalise away any objections by noting that it is drag (air resistance, friction, etc.) that slows down moving objects. Most people nowadays have no qualms extrapolating this rationalisation to the limiting case when all drag vanishes - even though this condition is not something we can achieve in practice. Nevertheless, the law of inertia is in no way an intuitive construct borne out of everyday experience.

Space is relative

Or, 'No privileged frames of reference'.

Galileo used the example of a person cloistered inside a 'smoothly moving' boat (i.e., one moving at a constant speed in a straight line, without turbulence or jerks) to argue that the laws of motion are the same in all inertial frames. (That's how we'd state it in modern lingo.) In other words, no inertial observer is privileged in the universe and all inertial frames can lay an equal claim to be 'at rest' and thus qualify as 'reference frames'. Put another way, there is no absolute rest frame in the universe. Space is relative to the (inertial) observer.

In the modern world many of us have had the good fortune of travelling in planes at cruising altitude at which point every passenger is completely oblivious to the fact that the plane has a high speed relative to the ground (unless, they look out of the window). We've also seen astronauts & objects in the International Space Station freely floating with respect to each other - apparently stationary - even though we all know that the ISS is in orbit around the Earth. (It would take Einstein's insight to reveal that a stable orbit around a gravitating body is effectively in 'free fall' and equivalent to an inertial frame of Galilean physics.)

Nevertheless, many have a hard time internalising the idea that all inertial frames are equivalent. There is an intuitive need to imagine an 'absolute background' of space - a stage -  on which all events occur. Such a background would implicitly define an absolute frame of rest - one that cannot be experienced or revealed by any experiment (involving motions of bodies) according to Galileo.

Sidebar: (Newton's) Absolute space

There is no prohibition in Galilean relativity against the existence of an absolute space. In Galilean relativity, if absolute space exists (and it would be inertial, by definition), it is no more privileged than any other inertial observer moving with respect to it. In other words, absolute space is not experiential and no motion with respect to absolute space can be detected. Newton fully accepted these conclusions from Galilean relativity. He nevertheless postulated an abstract mathematical absolute space as a construction that existed without reference to anything external. In Newton's mind, the relative spaces of all inertial observers existed in this absolute space even though no experiment could detect the latter. (This was eventually brought to a proper conclusion in Einstein's General Relativity wherein Space-Time is a 'real' entity albeit with inherent dynamics like a field unlike the static space $\times$ time bundle envisioned by Newton.)

Note that Newton's laws of motion do not need absolute space. In fact, Newton's laws are invariant under a Galilean transformation (i.e., changing from one inertial frame to another).

(Newton's) Absolute time

Or, 'Newton's universal time'.

Kindergarteners get indoctrinated into the cult of absolute time even as they complete a course of study in Newtonian physics. Newton himself listed absolute time as a fundamental assumption in his Principia Mathematica. To quote once again the oft-quoted passage:

Absolute, true, and mathematical time, from its own nature, passes equably without relation to anything external, and thus without reference to any change or way of measuring of time (e.g., the hour, day, month, or year).

However, it's certainly not obvious that people across cultures and eras had an intuitive notion of absolute time. For instance, ancient Indian literature is replete with accounts that entertained the notion of relative time. To take just one example, in the legend of the temple of Jagannath Puri (ref. Skanda Purana), king Indradyumna visits Brahmalok for a day. When he returns to Puri he realises, to his shock, that a full manvantara has passed and that everyone he ever knew or loved has long since died. Granted that such stories relate to time passing differently between realms. Nevertheless, it is worth noting that the possibility of differential passing of time was definitely not alien to earlier minds. 

Thus, one cannot rule out the possibility that the notion of absolute time could have seemed counter-intuitive to many people at the dawn of Newtonian mechanics. (Debates surrounding the ontological nature of time date back to millennia - certainly preceding Newton. As an aside, Gottfried Wilhelm Leibniz and George Berkeley - contemporaries of Newton - advocated a relational nature of both space and time.)

Of course, in the modern world, armed with Special Relativity, and certainly with General Relativity, we all appreciate that time is relative, not absolute. Ironically, after a thorough indoctrination in Newtonian physics, when the kindergartener encounters Special Relativity in the 1st grade, they find the relative nature of time to be extremely counter-intuitive.

Gravitational mass

Or, 'Inertial mass is proportional to gravitational charge'.

<COMING SOON>

Acceleration due to gravity

Or, 'Einstein's happiest thought'.

The fact that inertial mass is proportional to gravitational 'charge' implies that all objects fall to the Earth with the same acceleration. This is a fact amply demonstrable with the least effort in a kindergartener's backyard. (This phenomenon was also known to Galileo. Apocryphally, Galileo did such demonstrations by dropping objects from the Leaning Tower of Pisa.) Nevertheless, contrary to what happens in reality, every kindergartener 'knows' intuitively that heavier objects fall faster than lighter objects! It's a thought that's likely reinforced by the fact that loose sheets of paper or a bunch of feathers coast slowly to the floor compared to, say, a hammer destined for the big toe. This intuition often persists into adulthood - sometimes even in those who've had an unfortunate brush with physics early on and then blocked it completely out of their memory.

Maxwell's Electromagnetism

When Maxwell first formulated his dynamical theory of the electromagnetic field, his contemporaries spent a fair amount of effort in trying to recover the effects of the electric $\vec{E}$ and magnetic $\vec{B}$ fields using mechanical models. This was because the idea that space(time) is permeated by $\vec{E}$ and $\vec{B}$ - two 'non-mechanical' entities - was philosophically hard to digest. This is yet another example of prior philosophical prejudice - or intuition - playing a role in the general acceptance of a new theory.

However, in the current day, every kindergartener accepts that the $\vec{E}$ and $\vec{B}$ fields are elements of reality. (Also, check out the Aharonov-Bohm effect.)