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Model Constrained Systems as DAEs

Model the motion of a single pendulum.

    
Wolfram Language code: [image]
Derive the governing equations using Newton's second law of motion, and :
Wolfram Language code: deqns = {m x''[t] == ((λ[t]/l)) x[t], m y''[t] == ((λ[t]/l)) y[t] - m g};
Express the fixed length of the pendulum rod as an algebraic constraint:
Wolfram Language code: aeqns = {x[t]^2 + y[t]^2 == l^2};
The pendulum is released from the horizontal position with a vertical velocity of 1:
Wolfram Language code: ics = {x[0] == 1, y[0] == 0, x'[0] == 0, y'[0] == 1};
Specify the physical parameters for the pendulum system:
Wolfram Language code: params = {g -> 9.81, m -> 1, l -> 1};
Solve the high-index DAE and visualize the system:
Wolfram Language code: pendulumSol = First[NDSolve[{deqns, aeqns, ics} /. params, {x, y, λ}, {t, 0, 15}, Method -> {"IndexReduction" -> {"Pantelides", "ConstraintMethod" -> "Projection"}}]];
Wolfram Language code: Show[{ParametricPlot[Evaluate[{x[t], y[t]} /. pendulumSol], {t, 0, 15}, PlotStyle -> {DotDashed}], Graphics[{{Red, Line[{{0, 0}, {0.5, -Sqrt[3] / 2}}]}, {Blue, Disk[{0.5, -Sqrt[3] / 2}, 0.1]} }]}, ImageSize -> 400]