Lagrangian vs Newtonian

What you are seeing: a planar pendulum drawn three ways. Newton: force balance on the bob; tension and gravity are arrows. Lagrangian: a single generalized coordinate $\theta$ and its conjugate velocity; equations come from $L = T - V$ . Hamiltonian: $H = p^2/(2mL^2) - mgL\cos\theta$ on a phase plane.

Figure 1. Three formalisms; one trajectory.

θ_01.50

ω_00.00

view

WHAT TO TRY

Switch the view between all-three, Newton-only and phase space: the same pendulum is derived from force balance (tension and gravity arrows) and from a single coordinate theta via L = T - V. Both give the identical motion.
Raise the initial angle theta_0 toward pi: the swing leaves the small-angle regime, the period lengthens, and the phase-space orbit fattens from an ellipse toward the separatrix.
Watch the energy readout stay constant: the leapfrog integrator conserves it, which is why the phase-space orbit closes on itself instead of spiralling.