Two-Block Dynamics

Consider a situation with two blocks: One block positioned on a horizontal surface, and the second block positioned on top of the first block. There is a downward ambient gravitational acceleration $g$ , and all surfaces are smooth.

The first block (tan in figure), with mass $M$ , has a shape resembling a normal distribution. The horizontal position corresponding to its highest point is $x_0$ . This block can move horizontally but not vertically ( $x_0$ can change).

The second block, with mass $m$ , is considerably smaller (effectively a point-particle), and it slides along the top surface of the larger block. It can move both horizontally and vertically. Assuming that the small block remains in contact with the large block, its horizontal and vertical coordinates are parametrized as follows:

$\large{x = x_0 + \alpha \\ y = A \, e^{- \alpha^2}}$

The changing parameters are thus $x_0$ and $\alpha$ . The system dynamics, as derived from Lagrangian Mechanics , obey the following system of nonlinear equations:

$\large{(M + m) \, \ddot{x_0} + m \, \ddot{\alpha} = 0 \\ m \, \ddot{x_0} + \Big [ m + 4 m A^2 \, \alpha^2 \, e^{-2 \alpha^2} \Big ] \, \ddot{\alpha} = 4 m A^2 \, \alpha \, \dot{\alpha}^2 \, \, e^{-2 \alpha^2} \, (B \, \alpha^C + D) + 2 m g A \, \alpha \, e^{- \alpha^2}}$

If $(B,C,D)$ are integers, what is $B + C + D$ ?

Note: The first block has infinite spatial extent, despite its finite mass

Inspiration