Intuition The ONE core ideaA canonical transformation is just relabelling every point of "the space of all possible states" so that the machine that predicts the future (Hamilton's equations) still runs by the exact same rules. A single scalar recipe — the generating function — tells you how to relabel without breaking those rules.
Before you can read the parent note, you must be fluent in the words it throws around: phase space , coordinate , momentum , Hamiltonian , partial derivative , total time derivative , variation , new variables ( Q , P ) , generating function , Legendre transform , and Poisson bracket . This page builds each from nothing, in an order where every symbol is earned before it is used.
Definition Generalized coordinateq q q 1 , q 2 , … completely pins down where a system is at one instant. For a bead on a wire it is the distance along the wire; for a pendulum it is the swing angle.
Picture a single number line. The dot sitting on it at position q is the whole configuration.
Intuition Why "generalized"?The word means we don't insist on x , y , z angle , not a length. We pick whatever number is most convenient — that freedom is the whole point of this topic.
Definition Conjugate momentump p how the system is changing — loosely, its "amount of motion" in the direction of q p = m q ˙
Here q ˙ q
Definition The dot = rate of change in timeq ˙ d t d q q q q ˙ q ¨
Intuition Units of momentum (do this once, remember forever)If q m p = m q ˙ [ p ] = kg ⋅ s m = kg ⋅ m ⋅ s − 1 . different units — a fact that matters the moment you start mixing q p
Position alone can't predict the future: a pendulum at the bottom might be moving left, right, or resting. You need both where it is and how it moves. So plot them together.
Intuition What to notice in the figure aboveThe violet oval is one system's entire history — a closed loop means the motion repeats. The single magenta dot is the state right now : its horizontal position reads off q p orange arrow shows the dot creeping along the loop as time ticks. Read the picture as "the future is a walk along this curve."
A plane (or higher-dimensional space) whose horizontal axis is q p One dot = one complete state of the system, right now. As time passes, the dot traces a curve — the system's history.
Newton needed x v ( q , p ) new axes on this same plane — a relabelling of the dots, not a change of physics. Those new axes get their own names, which we introduce next.
Definition New canonical variablesQ , P Q i = Q i ( q , p , t ) P i = P i ( q , p , t ) brand-new coordinates and momenta — each one a formula built from the old ( q , p ) t same phase-space dots with different numbers, exactly like laying a new grid over an old map.
Intuition Why capitals, and why bother?Capital letters are just a bookkeeping habit: lower-case = old labels, upper-case = new labels. We bother because a clever regrid can make the motion boring — a straight line instead of a tangled curve. The parent note's whole payoff (solving the oscillator with no differential equation) comes from choosing ( Q , P )
Q P Why it feels right: the letters echo q , p Fix: after a transformation, Q angle and P action (energy over frequency). The words "coordinate" and "momentum" survive only in the sense that they play the roles q p
H ( q , p , t ) ( q , p ) total energy (kinetic + potential) written in terms of q p H = 2 m p 2 + 2 1 m ω 2 q 2
Intuition Picture: a height map over phase spaceImagine phase space as a floor and H height above every point — hills and valleys. The state-dot slides around this landscape, and H new landscape, traditionally called K ( Q , P , t )
H q p t H only p
Definition Partial derivative∂ p ∂ H H p while keeping q t . The curly ∂ d
Intuition What to notice in the figure aboveThe violet bowl is the energy surface H q , p magenta curve is what you get by slicing the bowl along a line of fixed q and letting only p steepness of that single curve is ∂ H / ∂ p
Intuition Why partial and not ordinaryd along which axis . ∂ / ∂ q ∂ / ∂ p q ˙ = ∂ H / ∂ p , p ˙ = − ∂ H / ∂ q
The parent topic's whole quest is: change ( q , p ) → ( Q , P ) , with a possibly new landscape K Q ˙ = ∂ P ∂ K , P ˙ = − ∂ Q ∂ K . canonical . See Hamilton's equations for the full derivation.
Here is the object the entire parent note revolves around, stated plainly so nothing later is a surprise.
Definition Generating functionF single scalar function — one recipe — from which the whole transformation ( q , p ) → ( Q , P ) F derive them.
Intuition What the four "types" mean (the numbering scheme)F one old and one new variable per degree of freedom. There are exactly four ways to pick that pair, so there are four types :
Type 1: F 1 ( q , Q ) Type 2: F 2 ( q , P ) Type 3: F 3 ( p , Q ) Type 4: F 4 ( p , P )
The subscript on F F 1 q Q
The master relation contains d t d F
Definition Total vs partial time derivativeIf F = F 1 ( q , Q , t ) q , Q themselves change with time, then
d t d F = ∂ q ∂ F q ˙ + ∂ Q ∂ F Q ˙ + ∂ t ∂ F . total d / d t all the ways time sneaks in — directly through t partial ∂ F / ∂ t
d F / d t ∂ F / ∂ t q Q K = H + ∂ F / ∂ t partial one; the master relation uses the total one. Keeping them straight is the whole game.
Definition The variation symbolδ δ δ ∫ ( … ) d t = 0 stationary (a valley or ridge among all nearby paths).
Intuition What to notice in the figure aboveThe navy line is the true path from a pinned start dot to a pinned end dot. The dashed magenta/orange/violet curves are nearby "imagined" paths sharing those same two endpoints. Because all of them start and end together, anything of the form ∫ d t d F d t = F ( end ) − F ( start ) identical for every path — so it cannot affect which path wins.
Intuition Why endpoints must be pinned (this is whyF d F / d t ∫ d t d F d t = F ( end ) − F ( start ) zero . That is why a generating function is legal — the master relation of Section 8 rests entirely on this fact.
To turn a Type-1 recipe F 1 ( q , Q ) F 2 ( q , P ) trade the new coordinate Q P . That trade is a Legendre transform, and it is worth seeing step by step.
Definition Legendre swap — step by stepGoal: replace a variable Q P = ∂ F / ∂ Q
Start from the product rule for differentials: d ( P Q ) = P d Q + Q d P (This is just the derivative of a product, written with d Rearrange to isolate the d Q P d Q = d ( P Q ) − Q d P Read it: the left side is written in terms of d Q d P d ( P Q ) d Q converted into a term "in d P .Absorb the boundary piece d ( P Q ) F 2 = F 1 + P Q P Q
That is the entire swap: a variable (Q P P Q
Intuition Why this exact trick appears in the topicF 1 ( q , Q ) ( q , P ) Legendre transformation for the deep version; the very same trick turns the Lagrangian into the Hamiltonian.
Definition Poisson bracket (one degree of freedom)For two functions f , g { f , g } = ∂ q ∂ f ∂ p ∂ g − ∂ p ∂ f ∂ q ∂ g . how the two quantities intertwine in phase space.
Intuition Why it matters as a checkThe generating-function machinery guarantees these brackets automatically, but you can also test a raw map ( q , p ) → ( Q , P ) Poisson brackets .
Master relation with dF over dt
Canonical transformations
This flows straight into Symplectic geometry , Liouville's theorem , the Hamilton–Jacobi equation and Action–angle variables once the parent topic is mastered.
What does q ˙ q d q / d t
If q p = m q ˙ Kilogram-metres per second, kg ⋅ m ⋅ s − 1 q
What single point in phase space represents? One complete state of the system: both where it is (q p
What are Q P New coordinates and momenta, each a formula in the old ( q , p , t )
What is the physical meaning of H The total energy (kinetic + potential) written as a function of q p
Why write ∂ H / ∂ p ∂ d Because H ∂ p q t
State Hamilton's two equations. q ˙ = ∂ H / ∂ p p ˙ = − ∂ H / ∂ q
What is the generating function F A single scalar recipe from which the whole transformation ( q , p ) → ( Q , P )
State the master relation that all canonical transformations obey. ∑ p i q ˙ i − H = ∑ P i Q ˙ i − K + d t d F
How does the new Hamiltonian K H K = H + ∂ F / ∂ t F
What are the four generator types keyed on? Which old/new pair F F 1 ( q , Q ) , F 2 ( q , P ) , F 3 ( p , Q ) , F 4 ( p , P )
How does d F / d t ∂ F / ∂ t Total d F / d t q , Q ∂ F / ∂ t t
What algebraic identity powers the Legendre swap between generator types? P d Q = d ( P Q ) − Q d P d Q d P
Write the multi-dimensional Poisson bracket { f , g } ∑ k ( ∂ q k ∂ f ∂ p k ∂ g − ∂ p k ∂ f ∂ q k ∂ g )
What fundamental brackets certify that ( q , p ) → ( Q , P ) { Q i , P j } = δ ij { Q i , Q j } = { P i , P j } = 0
