energy
Energy starts as a bookkeeping trick and ends as the deepest thing in physics: it is the conserved quantity guaranteed whenever the laws do not care what time it is. This thread follows that promotion — from time-translation symmetry, through the Hamiltonian machinery it powers, into the statistics of heat, until the classical account of energy breaks and forces the quantum.
Units touched PU-201 · PU-202 · PU-203 Adjacent threads symmetry · chance · matter Stations 9
The stations
The Euler–Lagrange equation
Before energy can be conserved it must be defined. Stationary action gives the equations of motion from a single scalar — the Lagrangian — that energy will later be read off from.
Energy from time-translation symmetry
The keystone: when the Lagrangian has no explicit time dependence, Noether's theorem hands you a conserved quantity. That quantity is energy — this is what energy means.
Legendre transform & Hamilton's equations
Trade velocity for momentum and the conserved energy becomes the Hamiltonian — the generator of time evolution, and the object every later chapter will quantise or average over.
Central-force orbit equation
Energy earns its keep. With angular momentum, conservation of energy collapses a two-body problem into an effective one-dimensional motion in an energy landscape.
Oscillator spectrum by ladder operators
The Hamiltonian goes quantum. Energy is no longer a continuum but a ladder of discrete levels, built rung by rung from the algebra of raising and lowering operators.
The Boltzmann distribution from maximum entropy
With many particles, energy is no longer tracked but weighted. Maximising entropy at fixed mean energy gives the exponential law that governs how energy is shared out.
Carnot efficiency from the second law
Energy acquires quality, not just quantity. The second law caps how much heat can become work, setting an absolute limit no engine can beat.
The four Maxwell relations
The thermodynamic potentials — each a way of packaging energy — yield exact cross-derivative identities that tie together heat, work, and every measurable response.
Equipartition, and the failure that demanded quanta
The classical rule ½kT per degree of freedom works — until it doesn't. Its catastrophic failure for radiation and heat capacities is the crack that quantised energy for good.