In a constant-force field, particles take shortest time and shortest distance, to minimize force F times distance change ds times time change dt {action, physics}|: F * ds * dt. Action tends to minimize {principle of least action, physics} {least-action principle}. Particles take shortest space-time path. Particle trajectories follow geodesics. Particles take least-resistance path.
In quantum mechanics, action has quanta, which have size Planck constant h. Action values are multiples of h.
Energy is force times distance change, so F * ds * dt = dE * dt. For energy, action is energy change times time change: dE * dt. By the least-action principle, particle trajectories take the shortest time with least energy change. Total action is sum of kinetic-energy KE to potential-energy PE difference over time dt: integral of (KE - PE) * dt. Least action over time makes conservation of energy.
Energy change dE is force F times distance change ds: dE = F * ds. Force F is momentum change dp divided by time change dt: F = dp / dt. Therefore, energy change dE times time change dt equals momentum change dp times distance change ds: dE * dt = F * ds * dt = dp * ds. For momentum, action is momentum change times distance change. By the least-action principle, particle trajectories take the shortest path length with least momentum change. Least action over translation makes conservation of momentum.
Tangential momentum p is angular momentum L divided by radius r: p = L / r, so dp = dL / r. Distance s is radius times angle A in radians, so ds = r * dA. Therefore, momentum change dp times distance change ds equals angular-momentum change dL times angle change dA in radians: dp * ds = (dL / r) * (r * dA) = dL * dA. For angular momentum, action is angular-momentum change times angle change. By the least-action principle, particle trajectories take the shortest rotation with least angular momentum change. Least action over rotation makes conservation of angular momentum.
If force varies, action minimizes dF * ds * dt. For force, action is force change times space-time change. By the least-action principle, particle trajectories take the geodesic with least force change. Least action over space-time makes force zero or flat space-time.
Over an instant or at a point or through an infinitesimal angle, action is zero or a limiting value.
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Date Modified: 2022.0224