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Use units $c=1$, consistent with comparing $B$ directly to $E$, and metric signature $(+---)$. For an arbitrary worldline parameter $s$, an action for a charged relativistic particle isThe first term has variation $-m{\dot{x}}_{\mu }/\sqrt{{\dot{x}}^2}$ in its momentum, and the coupling gives $-q{A}_{\mu }$. The Euler-Lagrange equations therefore readWhen $s$ is proper time, $u^{\mu }={\dot{x}}^{\mu }$ has $u^{\mu }{u}_{\mu }=1$, giving the covariant Lorentz forceWith $F^{i0}=E_i$ and $F^{ij}=-{ϵ}_{ijk}{B}_k$, put $e=qE/m$, $b=qB/m$, $\omega =\sqrt{b^2-e^2}$ for nonzero charge. The velocity equations becomeThus ${\ddot{u}}^z=-{\omega }^2{u}^z$, with $u^z(0)=0$, ${\dot{u}}^z(0)=e$. Integrating with $u^0(0)=1$, $u^x(0)=0$ gives $u^z=(e/\omega )\sin \omega s$, $u^0=1+(e^2/{\omega }^2)(1-\cos \omega s)$ and $u^x=-(be/{\omega }^2)(1-\cos \omega s)$. Integrate once more from the origin:For zero charge the trajectory is simply rest, $x^0=s$, with all spatial coordinates zero. The charged trajectory oscillates in $z$ while drifting in the negative $x$ direction, the direction of $\mathbf{E}\times \mathbf{B}$. Its displacement-to-time ratio over a proper-time period is $-E/B$. A boost at that drift velocity removes the electric field and leaves circular magnetic motion; in the original frame the orbit has repeating cycloidal cusps where the particle is momentarily at rest. Energy oscillates rather than growing indefinitely because $B>E$. In SI units the corresponding magnetic-dominated condition is $cB>E$.

The generalized momentum of this Lagrangian isThe Euler-Lagrange equation therefore givesBecause $\mathcal{L}$ has no explicit time dependence, the energy form of the Beltrami identity gives a conserved quantity $E=\dot{x}\partial \mathcal{L}/\partial \dot{x}-\mathcal{L}$. HereFor a direct check of conservation of energy, differentiate:This is the rest-plus-kinetic energy of a relativistic particle, in units with light speed one, plus its potential energy.

core-collapse supernovae follow loss of pressure support in the evolved core of a massive star. electron capture and nuclear dissociation contribute to collapse; formation of a hot compact object launches a shock that can stall as dissociation and neutrino losses drain it. neutrino heating and multidimensional fluid motions can revive the shock, but some collapses fail to produce a normal bright explosion. The remnant may be a neutron star or black hole. The neutrino burst detected from SN 1987A provides direct evidence for the core-collapse energy channel: the water-Cherenkov signal is a particle-interaction measurement, requiring detector-response and accidental-background checks rather than an optical light-curve inference.^[ref]

Type Ia supernovae instead involve thermonuclear destruction of a degenerate carbon-oxygen white dwarf. They leave expanding ejecta without an ordinary surviving neutron-star core. The ignition channel need not have a unique total mass or binary history. Radioactive nickel/cobalt decay powers much of the later luminosity, and empirical light-curve corrections make these events useful distance indicators. Spectral type is not identical to explosion mechanism: hydrogen-rich Type II events are core collapse, but stripped-envelope Ib/c events can also be core collapse despite lacking conspicuous hydrogen.

The ejecta first expand nearly freely while swept-up mass is small. A strong shock heats ambient gas, and once swept-up material dominates, an approximately adiabatic remnant approaches the Sedov-Taylor scalingSubsequent radiative cooling makes a dense shell and ends that energy-conserving approximation. Thermal X-ray lines probe shocked gas and nucleosynthetic ejecta; radio and nonthermal X-rays trace relativistic particles and magnetic fields. shock acceleration makes remnants plausible contributors to Galactic cosmic rays. Expansion, ambient-density structure, cooling and cosmic-ray energy losses complicate energy estimates; a bright synchrotron shell alone is not proof of a specific accelerated-particle composition. Supernovae return metals and mechanical energy to the interstellar medium and therefore connect stellar death to the feedback cycle.

For the relativistic particle action $S=(m/2)\int [g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }/e-e]d\lambda$, varying the worldline einbein gives $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-e^2$. On the positive-einbein branch the proper time satisfies $ds=ed\lambda$. A reparametrization with ${\lambda }^{\prime }=s$ sets the transformed einbein to one, making the geodesic equation affinely parametrized. This gauge choice is made after retaining the einbein mass shell constraint; fixing the action first would discard that equation. If the parameter interval is held fixed at both ends, its integrated proper length is a modulus and cannot be arbitrarily changed by endpoint-preserving transformations.

Use metric signature $(-,+,+,+)$ and an arbitrary increasing parameter $\lambda$ on a massive particle's timelike curve. Write $v^a=d{x}^a/d\lambda$ without yet imposing proper-time normalization. The free relativistic particle action isIts derivative with respect to $v^a$ is $p_a=mc{g}_{ab}{v}^b/\sqrt{-v^2}$. The square root makes the action independent of the chosen increasing parameter. Varying it first preserves the nontrivial velocity dependence that produces the geodesic equation.^[ref]

After variation, choose $\lambda =\tau$, so $v^a=U^a$ is the four-velocity, $U^2=-c^2$ and $p_a=m{U}_a$. Substituting $\sqrt{-U^2}=c$ into the integrand before an unconstrained variation would instead make it look constant and lose the momentum equation. The choice of proper time must therefore follow the variation, or be handled by an equivalent constrained action.