SHO Through Mechanics
The one-dimensional simple harmonic oscillator (SHO) is an ideal single example for seeing the full conceptual progression of classical mechanics. The system is a mass $m$ attached to a spring of force constant $k$, moving along a line with coordinate $x$, with potential energy $V(x)=\frac{1}{2}kx^2$ and $\omega=\sqrt{\frac{k}{m}}$. Every formalism must yield the same motion:
\[x(t)=A\cos(\omega t+\phi)\] \[x(t)=C_1\cos\omega t+C_2\sin\omega t\]Conceptual Shift 1
Force-first view $\rightarrow$ constraint-aware view: D’Alembert’s virtual-work filter removes constraint forces by restricting attention to allowed virtual displacements.
From Force to Action
Newtonian mechanics: Force produces acceleration
The force law $F=-kx$ inserted into $F=m\ddot x$ gives
$$
m\ddot{x}+kx=0
$$
$$
\ddot{x}+\omega^2x=0
$$
$$
x(t)=C_1\cos\omega t+C_2\sin\omega t
$$
Logic: Force $\rightarrow$ Acceleration $\rightarrow$ Motion.
D’Alembert’s principle: Dynamics as virtual work
Rewrite Newton’s law as $F-m\ddot x=0$ and test it against an allowed virtual displacement $\delta x$:
$$
(F-m\ddot{x})\delta x=0
$$
With $F=-kx$,
$$
(-kx-m\ddot{x})\delta x=0
$$
Since $\delta x$ is arbitrary for allowed motion here, the factor must vanish:
$$
m\ddot{x}+kx=0
$$
Logic: Force law $\rightarrow$ allowed virtual displacement $\rightarrow$ equation of motion. In constrained systems this step removes constraint forces automatically.
Conceptual Shift 2
Force laws $\rightarrow$ energy structure: dynamics is encoded by $L=T-V$ and extracted by the Euler–Lagrange operator.
Lagrangian mechanics: Motion from $L=T-V$
$$
T=\frac{1}{2}m\dot{x}^2
$$
$$
V=\frac{1}{2}kx^2
$$
$$
L=T-V=\frac{1}{2}m\dot{x}^2-\frac{1}{2}kx^2
$$
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0
$$
$$
\frac{\partial L}{\partial \dot{x}}=m\dot{x},
\qquad
\frac{\partial L}{\partial x}=-kx
$$
$$
m\ddot{x}+kx=0
$$
Logic: energies $\rightarrow$ $L$ $\rightarrow$ Euler–Lagrange equation $\rightarrow$ motion.
Conceptual Shift 3
Local equations $\rightarrow$ global path principle: the physical trajectory is the one that makes the action stationary.
Hamilton’s principle: Stationary action
$$
S=\int_{t_1}^{t_2}L\,dt
=
\int_{t_1}^{t_2}\left(\frac{1}{2}m\dot{x}^2-\frac{1}{2}kx^2\right)\,dt
$$
$$
\delta S=\int_{t_1}^{t_2}\left(m\dot{x}\,\delta\dot{x}-kx\,\delta x\right)\,dt
$$
$$
\delta S
=
-\int_{t_1}^{t_2}(m\ddot{x}+kx)\,\delta x\,dt
$$
$$
\delta S=0
\quad\Rightarrow\quad
m\ddot{x}+kx=0
$$
Logic: compare nearby paths $\rightarrow$ stationary action $\rightarrow$ Euler–Lagrange equation.
Conceptual Shift 4
Configuration space $\rightarrow$ phase space: replace velocity by conjugate momentum via a Legendre transform and view dynamics as a flow on $(x,p)$.
From Phase Space to Generators
Hamiltonian mechanics: Legendre transform and Hamilton’s equations
$$
p=\frac{\partial L}{\partial \dot{x}}=m\dot{x},
\qquad
\dot{x}=\frac{p}{m}
$$
$$
H=p\dot{x}-L=\frac{p^2}{2m}+\frac{1}{2}kx^2=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2
$$
$$
\dot{x}=\frac{\partial H}{\partial p}=\frac{p}{m},
\qquad
\dot{p}=-\frac{\partial H}{\partial x}=-kx=-m\omega^2x
$$
Logic: $(x,\dot x)$ $\rightarrow$ $(x,p)$ and flow in phase space.
Phase-space orbit: geometry of motion
$$
E=\frac{p^2}{2m}+\frac{1}{2}kx^2
$$
$$
\frac{p^2}{2mE}+\frac{kx^2}{2E}=1
$$
The trajectory in phase space is a closed ellipse, turning oscillation into geometry.
Conceptual Shift 5
Hard equations $\rightarrow$ smart variables: canonical transformations choose coordinates where the Hamiltonian and the motion simplify drastically.
Canonical transformation: action–angle variables
$$
x=\sqrt{\frac{2J}{m\omega}}\sin\theta,
\qquad
p=\sqrt{2m\omega J}\cos\theta
$$
$$
H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2=\omega J
$$
$$
\dot{\theta}=\frac{\partial H}{\partial J}=\omega,
\qquad
\dot{J}=-\frac{\partial H}{\partial \theta}=0
$$
$$
J=\text{constant},
\qquad
\theta=\omega t+\theta_0
$$
$$
x(t)=\sqrt{\frac{2J}{m\omega}}\sin(\omega t+\theta_0)
$$
In $(J,\theta)$ the oscillator becomes almost trivial: one constant and one uniformly advancing angle.
Conceptual Shift 6
Integrate ODEs $\rightarrow$ solve one PDE: Hamilton–Jacobi replaces trajectories by an action function that generates the motion.
Generating function and Hamilton–Jacobi recovery of $x(t)$
$$
S(x,E,t)=W(x,E)-Et,
\qquad
p=\frac{\partial W}{\partial x}
$$
$$
\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2=E
\quad\Rightarrow\quad
\frac{\partial W}{\partial x}=\sqrt{2mE-m^2\omega^2x^2}
$$
$$
W(x,E)=\int \sqrt{2mE-m^2\omega^2x^2}\,dx
$$
$$
\frac{\partial S}{\partial E}=\beta
\quad\Rightarrow\quad
\frac{\partial W}{\partial E}=t+\beta
$$
$$
\frac{\partial W}{\partial E}
=
\int \frac{m\,dx}{\sqrt{2mE-m^2\omega^2x^2}}
=
\frac{1}{\omega}\sin^{-1}\left(\frac{x}{\sqrt{\frac{2E}{m\omega^2}}}\right)
$$
$$
\sin^{-1}\left(\frac{x}{\sqrt{\frac{2E}{m\omega^2}}}\right)=\omega t+\omega\beta
$$
$$
x(t)=\sqrt{\frac{2E}{m\omega^2}}\sin(\omega t+\phi),
\qquad
\phi=\omega\beta
$$
The trajectory is extracted from the action function rather than obtained by directly integrating $\ddot x+\omega^2x=0$.
Conceptual Shift 7
Dynamics $\rightarrow$ algebra of transformations: infinitesimal generators produce canonical flows, and Poisson brackets compute their action on any quantity.
Infinitesimal generators: motion as Hamiltonian-generated flow
$$
\delta x=\varepsilon \frac{\partial G}{\partial p},
\qquad
\delta p=-\varepsilon \frac{\partial G}{\partial x}
$$
Choosing $G=H$ produces time evolution over a small time step $\varepsilon$:
$$
\delta x=\varepsilon \frac{\partial H}{\partial p}=\varepsilon\dot{x},
\qquad
\delta p=-\varepsilon \frac{\partial H}{\partial x}=\varepsilon\dot{p}
$$
$$
H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2
\quad\Rightarrow\quad
\delta x=\varepsilon\frac{p}{m},
\qquad
\delta p=-\varepsilon m\omega^2x
$$
Poisson brackets: the algebra of change
$$
\{f,g\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial g}{\partial x}
$$
$$
\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}
$$
$$
\dot{x}=\{x,H\}=\frac{\partial H}{\partial p}=\frac{p}{m}
$$
$$
\dot{p}=\{p,H\}=-\frac{\partial H}{\partial x}=-m\omega^2x
$$
$$
\ddot{x}+\omega^2x=0
$$
Brackets unify equations of motion with symmetry-generation: “how $f$ changes under the flow generated by $H$”.
Conservation: time-translation symmetry implies $E$ is constant
$$
\frac{dH}{dt}=\{H,H\}+\frac{\partial H}{\partial t}
$$
$$
\{H,H\}=0,
\qquad
\frac{\partial H}{\partial t}=0
\quad\Rightarrow\quad
\frac{dH}{dt}=0
$$
$$
H=E=\text{constant}
$$
$$
\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2=E
$$
Symmetry $\rightarrow$ conserved quantity: time-translation invariance $\rightarrow$ energy conservation.
All stages, one invariant physics
Each formalism reorganizes the same SHO into a new “language”: forces, virtual work, energies, action, phase-space flow, canonical simplification, action-as-generator, and finally bracket algebra. The physical trajectory remains
$x(t)=A\cos(\omega t+\phi)$, but the meaning of “solving the problem” progressively shifts from computing $\ddot x$ to identifying structure, symmetries, and generators.
| Stage | Starting idea | Mathematical object | Main output |
| Newton | Force causes acceleration | F=m ẍ | ẍ+ω²x=0 |
| D’Alembert | Virtual work filter | (F-m ẍ)δx=0 | same equation |
| Lagrange | Energy structure | L=T−V | Euler–Lagrange → same equation |
| Hamilton principle | Stationary action | δS=0 | same equation |
| Hamilton | Phase-space dynamics | H=p²/2m + ½mω²x² | Hamilton’s equations |
| Phase space | Geometry of motion | energy contour | ellipse in (x,p) |
| Canonical map | Better variables | (x,p)→(J,θ) | J const, θ=ωt+θ₀ |
| Generating function | One-function control | S or W | p=∂W/∂x |
| Hamilton–Jacobi | Action solves motion | H(x,∂S/∂x)+∂S/∂t=0 | x(t) recovered |
| Generator view | Motion as flow | G=H | δf=ε{f,H} |
| Poisson bracket | Algebra of change | {f,g} | ḟ={f,H}+∂f/∂t |
| Conservation | Symmetry → constant | {H,H}=0 | E constant |