GCM:010117Cycloid through (πa,2a) has time π√(a/g); the straight chord takes √(π²+4)√(a/g)Graduate Classical Mechanics Cycloid Beltrami ElementGraduate Classical Mechanics Chord Endpoint ComparisonPrivateLabels unavailablePrivate saving unavailableVariational principles, symmetries, and Noether quantitiesGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010131Inverse-square conformal mechanics has sl(2) brackets and charges D-tH, K-2tD+t²H; at E=2,g=1,t0=0 one finds q_min=1/2 and at t=1/2 the ledger q=√5/2, p=4/√5, Casimir=1/4Graduate Classical Mechanics Conformal So21 AlgebraGraduate Classical Mechanics Conformal Conserved ChargesPrivateLabels unavailablePrivate saving unavailableVariational principles, symmetries, and Noether quantitiesGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010218Rotating hoop at Ω=√2, m=R=g=1 has pitchfork angles ±π/3, frequency √(3/2), bottom growth 1Graduate Classical Mechanics Rotating Hoop EquilibriaGraduate Classical Mechanics Pitchfork Linear StabilityPrivateLabels unavailablePrivate saving unavailableConstrained Hamiltonian systems and reductionGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010219Spherical pendulum Routh reduction at θ=π/3 has p_φ=3/(2√2), φ̇=√2, nutation √(7/2)Graduate Classical Mechanics Routh Effective PotentialGraduate Classical Mechanics Steady Precession NutationPrivateLabels unavailablePrivate saving unavailableConstrained Hamiltonian systems and reductionGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010220Chaplygin sleigh heteroclinic at m=a=I=U=1 is u=tanh(t/2), ω=sech(t/2)/√2, rotation π√2Graduate Classical Mechanics Chaplygin Reduced OdesGraduate Classical Mechanics Heteroclinic Sech TanhPrivateLabels unavailablePrivate saving unavailableConstrained Hamiltonian systems and reductionGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010321Constant-force Hamilton principal function at m=g=t=1, q0=0, q=1 has S=-1/24, pf=1/2, pi=3/2, H=9/8Graduate Classical Mechanics Hamilton Principal FunctionGraduate Classical Mechanics Constant Force HJ EndpointsPrivateLabels unavailablePrivate saving unavailableHamilton-Jacobi theory and action-angle variablesGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010322Kepler Delaunay actions for m=k=1, E=-1/8, L=1 give Lambda=2, Jr=1, e=sqrt(3)/2 and T=16piGraduate Classical Mechanics Kepler Radial ActionGraduate Classical Mechanics Delaunay FrequencyPrivateLabels unavailablePrivate saving unavailableHamilton-Jacobi theory and action-angle variablesGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010423First-order Birkhoff form of (p^2+q^2)/2+eps q^4/4 at eps=1/10, J=2 is K=43/20 with Omega=23/20Graduate Classical Mechanics Birkhoff Lie GeneratorGraduate Classical Mechanics First Order Normal FrequencyPrivateLabels unavailablePrivate saving unavailableCanonical perturbation and averagingGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010424Kapitza pendulum with a=1/4, Omega=8 has a^2 Omega^2=4, inverted frequency 1 and downward frequency sqrt(3)Graduate Classical Mechanics Kapitza Effective PotentialGraduate Classical Mechanics Inverted Pendulum StabilityPrivateLabels unavailablePrivate saving unavailableCanonical perturbation and averagingGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010525Lagrange top at theta0=pi/3, Omega=2 has lambda=3/2, mu=9/4, energy 17/4 and nutation sqrt(13)/2Graduate Classical Mechanics Lagrange Top Steady PrecessionGraduate Classical Mechanics Top Nutation FrequencyPrivateLabels unavailablePrivate saving unavailableRigid-body mechanics and geometric reductionGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010526Zero-momentum reaction wheel with I=3, J=1, phiDot=2 over time pi reconstructs DeltaTheta=-pi/2 and T=3/2View access optionsGraduate Classical Mechanics Zero Momentum ConnectionGraduate Classical Mechanics Shape Space HolonomyPrivateLabels unavailablePrivate saving unavailableRigid-body mechanics and geometric reductionGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010627Störmer–Verlet on H=(p²+ω²q²)/2 has det M=1, θ=2arcsin(hω/2), and at h=ω=1 the sample (1,0)→(1/2,-3/4) with I_h=3/4View access optionsGraduate Classical Mechanics Stormer Verlet MonodromyGraduate Classical Mechanics Discrete Harmonic InvariantPrivateLabels unavailablePrivate saving unavailableNonlinear dynamics, stability, and symplectic mapsGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010628Hénon–Heiles saddles sit at (0,1/λ) and (±√3/(2λ),-1/(2λ)) with V=1/(6λ²); λ=1 gives E_c=1/6View access optionsGraduate Classical Mechanics Henon Heiles SaddlesGraduate Classical Mechanics Escape Energy ChannelsPrivateLabels unavailablePrivate saving unavailableNonlinear dynamics, stability, and symplectic mapsGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010629Arnold cat A=[[2,1],[1,1]] has |det(A²-I)|=5 hence four exact period-2 points, including (1/5,2/5)↔(4/5,3/5)View access optionsGraduate Classical Mechanics Cat Map SpectrumGraduate Classical Mechanics Cat Map Periodic PointsPrivateLabels unavailablePrivate saving unavailableNonlinear dynamics, stability, and symplectic mapsGraduate Classical Mechanics · Mixed reviewProgress not loaded
GCM:010630CR3BP L4=(1/2-μ,√3/2) is linearly stable strictly for 27μ(1-μ)<1; at μ=1/100 one has C=29901/10000 and ω²=(100±√7327)/200View access optionsGraduate Classical Mechanics Cr3bp L4 JacobiGraduate Classical Mechanics Routh CriterionPrivateLabels unavailablePrivate saving unavailableNonlinear dynamics, stability, and symplectic mapsGraduate Classical Mechanics · Mixed reviewProgress not loaded