 Pendulum with a prescribed translating support

For support displacement $\mathbf{a}(t)$ and angle measured from the downward vertical, the mass position is $(a_x+l\sin \theta ,a_y-l\cos \theta )$. Its Euler-Lagrange equation is $l\ddot{\theta }+{\ddot{a}}_x\cos \theta +(g+{\ddot{a}}_y)\sin \theta =0$. In the support frame the effective acceleration is $\mathbf{g}-\ddot{\mathbf{a}}$; taking its torque about the hinge gives $l^2\ddot{\theta }\hat{\mathbf{z}}=\mathbf{l}\times (\mathbf{g}-\ddot{\mathbf{a}})$. The prescribed support can do work, so pendulum energy need not be conserved.