Calculus
The entries below tell you what the ideas are. If you would also like to know where they came from — why dx is both zero and not zero, who objected, and how it took two centuries to settle — see The Ghosts of Departed Quantities.
The derivative: a rate of change
A derivative answers one question: how fast is this changing? If x(t) is the position of a particle at time t, then the derivative dx/dt is its velocity — metres gained per second. Take the derivative again and you get d2x/dt2, the rate at which the velocity is changing, which is the acceleration.
Geometrically, the derivative is the slope of the graph. Draw x against t and the derivative at any point is the steepness of the curve there: steep means changing fast, flat means momentarily not changing at all. That second reading matters in this course, because a flat point — slope zero — is where a potential has a peak or a valley, and those are exactly the equilibrium points.
If a quantity’s derivative is zero, the quantity is not changing. The lecturer uses this repeatedly: if both q̇ and v̇ are zero at the start, then q and v can never change, so the particle sits at that point forever.
Dot notation: q̇, q̈ and V′(q)
Newton wrote derivatives with respect to time as dots over the symbol, and physicists still do. So for a coordinate q:
- q̇ (“q dot”) means dq/dt — the velocity.
- q̈ (“q double dot”) means d2q/dt2 — the acceleration.
A prime means something different: a derivative with respect to whatever the variable in brackets is, not with respect to time. So V′(q) means dV/dq — how fast the potential changes as you move along q.
The distinction is worth holding onto. Dots are always “per unit time”; primes are “per unit of the thing in brackets”. When you see mq̈ = −V′(q), it reads: mass times acceleration equals minus the slope of the potential.
Differential equations, and what “second order” means
An ordinary equation like 3x + 2 = 11 asks you to find a number. A differential equation asks you to find a function, and it does so by telling you something about that function’s derivatives.
The order is simply the highest derivative that appears. So:
- q̇ = −kq is first order — only one dot.
- mq̈ = F(q) is second order — two dots appear.
Newton’s second law is second order, and this is not a technicality — it is the reason the whole of this lecture is shaped the way it is. Because force determines acceleration rather than velocity or position, the equation you have to solve is second order, and that fact ripples through everything that follows.
Why a second-order equation needs two initial conditions
Here is the idea in the simplest possible terms. If you know only how fast something is accelerating, you cannot say where it will be, because you do not know how fast it was already moving. And even knowing that, you cannot say where it will be unless you know where it started.
So each derivative you have to “undo” costs you one piece of information. Integrating once introduces one unknown constant; integrating twice introduces two. A second-order equation therefore needs two facts pinned down before it has a single definite answer — conventionally the initial position and the initial velocity.
This is exactly the point of the chalk demonstration. Holding the chalk at a known height and knowing gravity acts on it is not enough: dropped, it falls straight; flicked sideways, it arcs; flicked hard enough, it orbits. Same starting position, same force, wildly different futures — because the missing ingredient was the initial velocity.
Turning one second-order equation into two first-order ones
There is a standard trick, and it is purely bookkeeping. Given
mq̈ = F(q)
invent a new name for the velocity: let v = q̇. Then q̈ is just v̇, and the single second-order equation becomes a pair of first-order ones:
q̇ = v v̇ = F(q)/m
Nothing has been solved and nothing thrown away — the same physics is written differently. But the pair makes something explicit that the single equation hid: the state of the system is two numbers, q and v, evolving together. Each equation says how one of them changes, and each needs one starting value. That pairing is what leads directly to phase space.
If you have written code, this is the same move as replacing a second-order recurrence with two coupled first-order ones so you can step them forward in a loop. It is exactly how such equations are solved numerically.
Separation of variables
This is a method for solving certain differential equations by getting all the v terms on one side and all the q terms on the other, then integrating each side separately.
In the lecture he divides one equation by the other to get
dv/dq = −ω2q / v
Cross-multiplying gives v dv + ω2q dq = 0 — every v now sits with the dv, every q with the dq. Now integrate each term on its own. Since the integral of v dv is ½v2, and likewise for the other, you get
½v2 + ½ω2q2 = constant
Note what has happened: time has vanished. The result is a relationship between q and v that holds at every instant — a curve the system must stay on, rather than a schedule of where it is when. That is why he can draw the phase portrait without ever solving for q(t).
The constant of integration
Integration reverses differentiation, but it cannot recover everything. All constants have derivative zero, so when you integrate, you cannot tell what constant was there before — you must write “+ constant” and pin it down from other information.
Physicists find this useful rather than annoying, because the constant is usually something. In the oscillator, the constant of integration turned out to be the total energy: it does not change as the motion proceeds, so each value of it labels a different orbit. Quantities like this are called constants of the motion, and hunting for them is a large part of the subject.
Geometry and algebra
Vectors and position vectors
A vector is a quantity with both size and direction; in three dimensions it is just three numbers in a fixed order. A position vector R is the arrow from a chosen origin to a point — so R = (x, y, z) says “go x along, y across, z up”.
When the lecturer says three coordinates fix the centre of mass, he means exactly that: R carries three independent numbers, hence three degrees of freedom. Bold type conventionally marks a vector; the same letter unbolded means just its length.
Why a direction in space needs only two numbers
A unit vector is a vector of length exactly 1 — used when you care about direction and not magnitude. It has three components, but they are not independent, because the length condition forces
x2 + y2 + z2 = 1
One equation among three numbers removes one of them: choose any two and the third follows (up to a sign). So a direction in space carries only two independent numbers, not three.
The familiar picture is the globe. Every direction from the centre of the earth meets the surface at one point, and one point on a sphere is fixed by latitude and longitude — two numbers. That is precisely the lecturer’s phrasing when he counts the axis of rotation as two degrees of freedom and the angle turned about it as a third.
Ellipses, parabolas and hyperbolas, read off their equations
These three curves — the conic sections — recur constantly, and you can tell which is which from the algebra alone. Look at the two squared terms and their signs:
| Equation | Curve | Behaviour |
|---|---|---|
| x2/a2 + y2/b2 = 1 | Ellipse | Closed loop. Both terms added, so neither can grow large — the curve is bounded, and motion on it repeats. |
| x2/a2 − y2/b2 = 1 | Hyperbola | Two open branches running off to infinity. The minus lets both terms grow together without limit. |
| y = ax2 | Parabola | One term squared, the other not. The shape of a potential-energy well. |
This single sign difference carries the whole punchline of lecture 02. The oscillator gives ½v2 + ½ω2q2 = E — a plus, so ellipses, so closed orbits, so the motion repeats forever. Flip the potential upside down and you get a minus, so hyperbolas, so open curves, so the particle escapes and never returns. Bounded or unbounded, periodic or runaway, decided by one sign.
“Principal axes” simply means the directions of the ellipse’s longest and shortest diameters. When they line up with the coordinate axes, the equation has no xy cross-term — which is why he can point out that the axes of the ellipse are the coordinate axes themselves.
Curves, surfaces and counting dimensions
There is a simple rule of thumb: each independent equation you impose removes one dimension.
Start in ordinary 3D space. One equation — say x2 + y2 + z2 = 1 — leaves a surface, the sphere, which is two-dimensional: you can wander on it in two independent ways. Add a second equation, say z = 0, and you are down to a curve, the equator, which is one-dimensional. A third would pin you to isolated points.
Now apply the rule in four dimensions, which is what the lecturer does with two degrees of freedom. Phase space has four axes: q1, q2, q̇1, q̇2. One constant of the motion, such as the total energy, gives one equation and so removes one dimension, leaving a three-dimensional object — a “hypersurface”, which is just the word for a surface in more than three dimensions. But a trajectory is a curve, one-dimensional. Getting from three down to one takes two more equations. That is why he says he needs φ1, φ2 and φ3.
You cannot picture four dimensions, and you do not need to. The counting works regardless, and the counting is the argument.
Euler angles
How do you say, in numbers, which way an object is facing? Euler’s answer: any orientation whatsoever can be reached from a reference orientation by three successive rotations about three axes. Give the three angles and you have specified the orientation exactly.
Aircraft language is the everyday version — pitch, roll and yaw are three angles doing precisely this job. Three, and no fewer, are needed to describe how a rigid object is turned in three-dimensional space. Hence three orientational degrees of freedom, which together with the three for position give the six of a rigid body.
Physics
Newton’s second law, and what it really says
F = ma: force equals mass times acceleration. The content that is easy to miss is which quantity the force controls. Push something and you do not directly set where it goes, nor even how fast it goes — you set how fast its velocity changes.
This is genuinely counter-intuitive, and the lecturer says so. Everyday experience suggests that pushing something makes it move and stopping the push makes it stop, which would mean force controls velocity. That intuition comes from living surrounded by friction. His example of bacteria is exact: a bacterium swimming in a fluid so viscous that friction dominates entirely does have force proportional to velocity. We do not.
Momentum is mass times velocity, p = mv. Newton’s law in its better form says force equals the rate of change of momentum, which is the same thing when the mass is constant.
Kinetic, potential and total energy
Kinetic energy is energy of motion, ½mv2. Note the square: it cannot be negative, whichever way the particle moves. That small fact does real work in the lecture — it is why certain regions are forbidden.
Potential energy V is stored energy, held by position: a raised weight, a stretched spring. Total energy E is the sum, and for the systems in this course it does not change with time — energy shifts back and forth between the two forms while the total stays put.
Since E = kinetic + potential and kinetic energy cannot be negative, a particle can only be where V(q) ≤ E. Anywhere else it would need negative kinetic energy, which is impossible. This is precisely the argument behind the forbidden region on the inverted-parabola board.
Conservative forces and F = −dV/dq
A force is conservative if the energy it takes to move between two points does not depend on the route — so going out and back leaves you exactly where you started, energetically. Gravity and springs are conservative; friction is not, since it drains energy on every leg of the journey.
For such a force you can define a potential energy V, and then
F = −dV/dq
In words: the force is minus the slope of the potential. The minus sign is the whole physical story — it says the force points downhill, towards lower potential energy. A ball on a hillside rolls down, not up.
Two consequences used constantly. Where the potential is flat, the slope is zero, so the force is zero — that is an equilibrium point. And in a valley the force pushes back towards the bottom, giving stable oscillation; on a peak it pushes further away, giving instability.
Reading a potential-energy well
A graph of V against q is one of the most useful pictures in physics, and it is read like a landscape with a ball rolling on it.
- Draw a horizontal line at the total energy E.
- Where that line meets the curve are the turning points: there V = E, so the kinetic energy is zero and the particle is momentarily at rest before reversing.
- The gap between the line and the curve is the kinetic energy — wide gap, moving fast; touching, stopped.
- Regions where the curve rises above the line are forbidden.
A particle trapped in a valley between two turning points oscillates between them forever. Raise E above the surrounding peaks and it escapes. Nothing here requires solving an equation — which is the point.
The simple harmonic oscillator
A mass on a spring. Pull it and the spring pulls back in proportion to how far you pulled: F = −kq, where k is the spring constant (stiffness) and the minus sign means the pull is always back towards the middle.
Feed that into Newton’s law and you get q̈ = −(k/m)q. Writing ω2 = k/m tidies it to q̈ = −ω2q. Here ω is the angular frequency, related to the period by T = 2π/ω. The amplitude is the farthest the mass gets from the centre, and the potential energy is V = ½mω2q2 — a parabola.
Near the bottom of any smooth valley, a potential looks like a parabola. So almost every stable system, disturbed slightly, behaves like a harmonic oscillator. That is why it is worth knowing this single example thoroughly — and why it recurs for the rest of the course.
Its other peculiarity: the period does not depend on the amplitude. Pull the mass twice as far and it still takes exactly as long to get back. Almost no other system does this.
Moment of inertia
The rotational counterpart of mass. Mass measures how hard it is to change something’s straight-line motion; moment of inertia measures how hard it is to change its spin, and it depends not just on how much matter there is but on how far that matter sits from the axis. Mass out at the rim counts far more than mass near the centre.
Hence the diatomic molecule. Two atoms joined like a dumbbell have essentially all their mass on the line joining them, so spinning about that line moves nothing appreciably away from the axis: the moment of inertia is effectively zero and the rotation is physically meaningless. That is why such a molecule counts five degrees of freedom, not six.
Ideas that are new in this course
Degrees of freedom
The number of independent numbers you must supply to say exactly where everything is. A bead on a wire: one. A point on a tabletop: two. A free particle in space: three. N free particles: 3N.
Two things trip people up. First, a constraint may or may not reduce the count — see constraints. Second, and the point the lecturer hammers: you count degrees of freedom before solving anything, from the geometry of the setup alone. The dropped chalk has three degrees of freedom even though it happens to fall in a straight line, because it was free to do otherwise.
Generalized coordinates
Just a name for “whatever numbers you have chosen to describe the configuration”, written q1, q2, and so on. They need not be distances along axes — an angle is a perfectly good coordinate, and for a pendulum it is much the better choice. You may mix types freely: two lengths and an angle, say.
The point of the generality is that the machinery built later works whatever you chose. Rather than one theory for Cartesian coordinates and another for angles, everything is written in terms of q and holds for all of them.
Holonomic and non-holonomic constraints
A constraint restricts how a system may move. The useful division is by whether it lets you eliminate a coordinate.
Holonomic constraints are equations among the coordinates — two beads a fixed distance apart, a particle confined to a surface. An equation lets you solve for one coordinate in terms of the others and throw it away, so the count drops by one for each such constraint.
Non-holonomic constraints do not permit this. Typically they are inequalities: a puck on a table with edges cannot leave the table, but it still needs two coordinates while it is on it, and no coordinate can be eliminated. Constraints involving velocities are often non-holonomic too — a rolling ball that must not slip is the standard example.
The short version: holonomic constraints shrink the space; non-holonomic ones only fence off part of it.
Phase space, and why dynamics does not happen in ordinary space
This is the central new idea of lecture 02, and it follows from something already established: because Newton’s law is second order, the state of a system is position and velocity together. Position alone is not enough — the chalk showed that.
So build a space whose axes are all the coordinates and all the velocities. One particle in one dimension gives a two-dimensional phase plane with axes q and q̇. Two degrees of freedom give a four-dimensional phase space. In this space:
- A single point is a complete state — everything you need to determine the future.
- As time runs, the point moves, tracing a phase trajectory.
- The whole family of trajectories, for every starting condition, is the phase portrait.
Why bother? Because the portrait shows the behaviour of every starting condition at once, and often you can draw it without solving anything. Closed loops mean the motion repeats; curves running off to infinity mean it escapes; and trajectories can never cross, since a crossing point would be one state with two different futures. You get the qualitative answer — stable or unstable, periodic or runaway — without the formulae.
Autonomous and non-autonomous systems
A system is autonomous if the forces do not depend explicitly on time — the rules are the same today as tomorrow. A spring left alone is autonomous.
It is non-autonomous if something outside is changing the rules as it goes: a charge in an electric field that you are deliberately varying, or a child pumping a swing.
The distinction matters because the no-crossing rule for phase trajectories assumes autonomy. If the rules change with time, passing through the same point twice need not produce the same outcome twice, because the situation is not the same — the clock has moved on.
Constants of the motion
A quantity built from the coordinates and velocities that does not change as the system evolves. Total energy is the classic example: q and v change constantly, but that particular combination of them stays fixed.
Each one is worth a great deal, because each is an equation the trajectory must satisfy, and each equation removes one dimension (see counting dimensions). With one degree of freedom, energy alone pins the trajectory to a curve, and the problem is finished without solving a differential equation. With more degrees of freedom you need more constants, and they are usually hard to find — which is why the general problem is difficult, and why a gas of 1023 particles is hopeless by this route.
The separatrix
In a phase portrait, the special trajectory that separates regions of qualitatively different behaviour.
On the inverted parabola it is the X-shaped pair of lines through the origin, at E = 0. Trajectories on one side of it are bounded away from the peak; on the other they sail over. The separatrix is the knife-edge between the two: a particle exactly on it creeps towards the summit and takes infinitely long to arrive, never quite getting there.
Separatrices matter out of proportion to their size — being measure-zero, no real system sits exactly on one, but they are what organises the portrait, and everything else is classified by which side of them it falls.
Further reading
Books worth having
None of these is required — the lectures and this page stand on their own. But if you want something to read alongside, these are the ones pitched at someone rebuilding physics rather than meeting it for the first time.
The standard companion
Halliday, Resnick and Walker, Fundamentals of Physics. The descendant of the book you remember. Halliday and Resnick first published in 1960, and Jearl Walker took the book over around 1990; their names stay on the cover as originators. Modern editions are much heavier than the ones from the eighties — far more worked examples and scaffolding — which is either exactly what you want or exactly what you don’t. Second-hand copies of the older editions are easy to find if it is the original you are after.
Physics without the mathematics
Jearl Walker, The Flying Circus of Physics. Physics posed entirely as questions about ordinary things — why a wet hand can briefly survive molten lead, what is happening in a judo throw. Almost no mathematics. It pairs unusually well with these lectures, because Balakrishnan reaches for exactly this kind of example: the blinking eye, the chalk, the room turning to look at a distraction.
Feynman, Leighton and Sands, The Feynman Lectures on Physics. Famous, and genuinely worth the reputation for the explanations rather than the problems. The complete text is free to read online, hosted by Caltech.
Mechanics at this level
Kleppner and Kolenkow, An Introduction to Mechanics. Roughly where this course sits — more demanding than a first-year text, but it explains rather than asserts.
David Morin, Introduction to Classical Mechanics. Notable for the sheer number of problems with full worked solutions, which is what you want when you are relearning and have nobody to check your answers.
On phase portraits specifically
Steven Strogatz, Nonlinear Dynamics and Chaos. If lecture 02 interested you more than it confused you, this is the book to know about. Its early chapters are devoted entirely to the phase-plane thinking Balakrishnan introduces — trajectories, fixed points, stability, separatrices — developed slowly and with pictures throughout. It assumes far less than its title suggests.
Getting the mathematics back
Silvanus P. Thompson, Calculus Made Easy. First published in 1910 and still the friendliest introduction there is; its stated aim is that what one fool can do, another can. Long out of copyright and free from Project Gutenberg.
3Blue1Brown, Essence of Calculus. A free video series that builds the visual intuition behind derivatives and integrals. Watching it costs a few hours and would make most of the Calculus entries above feel obvious rather than remembered.