Lecture 3–4 — Multiple Critical Points, Scaling Arguments, and the Road to Linear Systems#
Source: NPTEL Classical Physics, Mod-01 Lec-03 (Dynamics in phase space) and Mod-01 Lec-04 (Linear dynamical systems), Prof. V. Balakrishnan.
Lecture 2 worked out phase portraits for potentials with a single critical point: a minimum (the harmonic oscillator’s center) or a maximum (the inverted oscillator’s saddle). These two lectures push the same machinery in two directions. First, what happens once a potential has several critical points at once — and can the period–amplitude relation be extracted without ever solving the equations of motion? Second, what happens the moment a conservative force stops being conservative — adding friction is the simplest possible motivation for the general linear-systems machinery Lecture 5–6 develops in full.
Two critical points in one potential#
Take a potential with both a minimum and a maximum,
the simplest polynomial that can do it: quadratic near the bottom, with an odd term added so the two sides aren’t mirror images. Setting \(V'(x) = x + x^2 = x(1+x) = 0\) gives two critical points — a center at \(x=0\) (\(V=0\), a genuine minimum since \(V''(0)=1>0\)) and a saddle at \(x=-1\) (\(V(-1) = \tfrac12 - \tfrac13 = \tfrac16 \equiv E_s\), since \(V''(-1)=-1<0\)). \(V\) is monotonically increasing on \((-\infty,-1]\), from \(-\infty\) up to \(E_s\), then dips back down to \(0\) at \(x=0\) before climbing to \(+\infty\) as \(x\to+\infty\). That single saddle energy \(E_s\) organizes everything that follows:
\(E < 0\): the region \(x > -1\) is entirely inaccessible (\(V \ge 0\) there). The only motion is a single unbound branch on \(x<-1\): released from rest at the one turning point, the particle falls away toward \(x\to-\infty\).
\(0 < E < E_s\): two different kinds of motion coexist at the same energy. One is the same kind of unbound branch as above, now with a different turning point on \(x<-1\). The other is a genuine periodic orbit, trapped between two turning points that straddle the minimum at \(x=0\) — the local analogue of the harmonic oscillator’s ellipses.
\(E = E_s\): the separatrix. A trajectory released infinitesimally to the right of the saddle swings out through the well, reaches a second turning point at \(x=\tfrac12\) (where \(V(\tfrac12)=E_s\) too), and returns — taking infinitely long to leave the saddle and infinitely long to get back to it, so it never quite closes. This is a homoclinic orbit: a single trajectory that leaves an unstable point and asymptotically returns to the same point.
\(E > E_s\): the barrier at \(x=-1\) no longer separates anything. A single trajectory sweeps in from \(x\to-\infty\), crosses over the hump, bounces off the far wall where \(V(x)=E\) again, and heads back out to \(x\to-\infty\).
The faint background is the full “lamination” of level curves across every energy; the highlighted curve is whichever case is selected. Two structural facts fall out immediately once you look at this picture:
Trajectories cross the \(x\)-axis at right angles. Since \(\dot x = v\) and \(\dot v = -V'(x)/m\), the phase-trajectory slope is \(dv/dx = -V'(x)/(mv)\), which diverges as \(v\to0\) — except exactly at a critical point, where \(V'(x)=0\) too and the slope is indeterminate. That’s why every curve above meets \(v=0\) vertically, except at \(x=0\) and \(x=-1\) themselves.
Centers and saddles must alternate. A curve can’t have two adjacent minima with no maximum between them (there’d be nowhere for it to turn back down), so stable and unstable equilibria always interleave along \(x\). This is the one-dimensional seed of a fact that survives, in a more careful form, all the way into the general classification of Lecture 5–6.
How the period depends on amplitude — a scaling argument#
For any potential \(V(x) = |x|^r\) (\(m=1\)), energy conservation fixes the amplitude: release from rest at \(x=a\), and \(E = V(a) = a^r\). The quarter-period is a definite integral,
which looks like it depends on \(a\) in some complicated way — until you rescale \(x = au\):
All of the \(a\)-dependence factors out front; the remaining integral is a pure number. Without solving a single equation of motion, the amplitude scaling of the period is fixed by the shape exponent \(r\) alone:
\(r=1\) (a bouncing ball under gravity, \(V=|x|\)): \(T \propto a^{1/2}\) — bigger drop, longer period, as expected.
\(r=2\) (the harmonic oscillator): \(T \propto a^0\) — isochronous, independent of amplitude. This is the one value of \(r\) for which the period doesn’t care how far you pull it back, and it is the reason a pendulum clock keeps time regardless of how it was wound.
\(r=4\) (a quartic well): \(T \propto a^{-1/2}\) — larger amplitude means a shorter period. Counterintuitive at first, but the restoring force \(\propto x^3\) grows so much faster at large \(|x|\) that the particle is whipped through the origin faster than it loses from having farther to travel.
On log-log axes a power law is a straight line, and the three slopes read off exactly as \(1-r/2\): positive for \(r=1\), flat for \(r=2\), negative for \(r=4\). Nothing here required knowing \(x(t)\) in closed form — only the shape of the potential near where the motion turns around.
Kepler’s third law without solving the orbit#
The same trick — called scaling or, to mathematicians, homothety — settles a much older question. Take a general power-law central force, \(m\ddot r = -k\, r^{n}\hat e_r\) (with \(n=-2\) the inverse-square case). Rescale both length and time, \(r = \lambda r'\), \(t = \mu t'\), and the equation of motion becomes
The physics is unchanged by a relabeling of units, so this says something sharper than dimensional analysis alone: if an orbit exists with mean radius \(r_1\) and period \(t_1\), then the same force law supports another orbit with radius \(r_2\) and period \(t_2\) related by \(\lambda = r_2/r_1\) and \(\mu = t_2/t_1\) obeying exactly that relation. For the inverse-square law, \(n=-2\), so \(1-n=3\) and
Kepler’s third law, \(T^2 \propto R^3\) — extracted without ever writing down an ellipse, using only how the equation of motion responds to a change of units. The same scaling picks out \(n=1\) (a linear restoring force, i.e. the harmonic oscillator in three dimensions) as the only other power law for which the period carries no length dependence at all — the isochronism found above, from a completely different argument.
Adding friction: a new kind of equilibrium#
Every phase portrait so far has come from a conserved energy \(E = \tfrac12 mv^2 + V(x)\), whose level curves are the trajectories. Friction breaks that. Take the damped harmonic oscillator,
and write it the same way every equation in this course eventually gets written — as a first-order system in phase space, \(\dot{\vec x} = L\vec x\) with \(\vec x = (x, v)^T\) and
The formal solution is \(\vec x(t) = e^{Lt}\vec x(0)\), where the matrix exponential is defined by the same power series as the scalar one, \(e^{Lt} \equiv \sum_{n=0}^\infty L^n t^n / n!\) — a series that converges for any finite matrix, by the identical ratio-test argument that works for ordinary numbers. If \(L\) is diagonalizable, \(e^{Lt}\) is diagonal in the eigenbasis with entries \(e^{\lambda_i t}\), so \(x(t)\) and \(v(t)\) are always linear combinations of \(e^{\lambda_1 t}\) and \(e^{\lambda_2 t}\); only when an eigenvalue repeats does a secular factor \(t\,e^{\lambda t}\) appear alongside it. (Diagonalizability itself is guaranteed whenever the minimal polynomial of \(L\) has no repeated roots — a weaker and more useful condition than the textbook sufficient case of a matrix commuting with its Hermitian conjugate, and it’s what the Cayley–Hamilton theorem is really for.)
For this particular \(L\), the secular (characteristic) equation is \(\lambda^2 + \gamma\lambda + \omega_0^2 = 0\), so
Underdamped (\(\omega_0 > \gamma/2\)): the eigenvalues are a complex-conjugate pair with negative real part \(-\gamma/2\) — the phase point spirals into the origin.
Overdamped (\(\omega_0 < \gamma/2\)): two distinct negative real eigenvalues — the system relaxes to the origin monotonically, no oscillation at all.
Critically damped (\(\omega_0 = \gamma/2\)): the eigenvalues coincide, and the secular \(t\,e^{\lambda t}\) term makes its first appearance in this course.
None of these three portraits is a center or a saddle — energy is no longer conserved, so there’s no level-curve lamination to fall back on, and the point genuinely converges onto (or, running time backward, would diverge from) the origin. That’s a third qualitative behavior beyond the two Lecture 2 established, and it’s exactly the gap Lecture 5–6 closes: once you stop restricting \(L\) to this one damped-oscillator form and ask what any \(2\times 2\) matrix can produce, spirals and nodes like these turn out to be two more entries in a complete six-way classification by trace and determinant alone.
From worked examples to a general framework#
Balakrishnan closes Lec-04 by stepping back from mechanics altogether. Any system of variables evolving continuously in time — positions, populations, concentrations, whatever they are — that obeys first-order equations \(\dot x_i = f_i(x_1,\dots,x_n,t)\) can be made autonomous by the trick of adding one more variable, \(x_{n+1}\equiv t\) with \(\dot x_{n+1}=1\); so it costs nothing to only ever consider the autonomous case \(\dot{\vec x} = \vec f(\vec x)\). If \(\vec f\) is linear, the whole story above already solves it via \(e^{Lt}\). If it isn’t, there is no general guarantee that a closed-form solution exists at all — and a genuinely important distinction gets drawn here: solvability is not integrability. You can always march a nonlinear system forward numerically, one small time step at a time, no matter how complicated \(\vec f\) is; that’s solvability, and it’s never in question. What frequently fails is integrability — being able to write \(x(t)\) down as an explicit formula, so that “where is it in a million years” is a substitution rather than a million numerical steps. The gap between the two is genuinely dangerous: errors compound exponentially along a numerically-integrated trajectory, so “just simulate it” quietly stops working long before any hardware limit is reached.
There’s a sharp threshold buried in that gap. Two-dimensional autonomous systems — the subject of Lecture 5–6 — are tame: nothing more exotic than the six cases already glimpsed above (center, saddle, node, spiral, and the borderline cases between them) can occur. The moment \(n\) reaches 3, that tameness disappears for good: this is precisely where chaos becomes possible, and it turns out to be the generic case rather than a curiosity. That’s the itinerary for what follows: classify \(n=2\) completely, then ask what survives — and what doesn’t — once \(n\) grows beyond it.