= Grassmann-valued classical supersymmetric mechanics
{c}
A classical supersymmetric mechanical model takes its coordinates in a <Grassmann algebra>: the position is even and the fermionic coordinates are odd. Time differentiation is even, while variation with respect to an odd coordinate uses <Grassmann derivatives>. For example, moving an odd variation to the left and integrating by parts gives $\delta\int\psi\dot\psi\,dt=2\int\delta\psi\dot\psi\,dt$.
For the real variant $L=\dot x^2/2+U^2/2-\psi_1\dot\psi_1/2+\psi_2\dot\psi_2/2+U'\psi_1\psi_2$, the <Euler-Lagrange equations> are
$$
\ddot x=UU'+U''\psi_1\psi_2,\qquad \dot\psi_1=U'\psi_2,\qquad \dot\psi_2=U'\psi_1.
$$
The conserved even energy is $E=\dot x^2/2-U^2/2-U'\psi_1\psi_2$ and the odd <supercharges> are $Q_1=\dot x\psi_1-U\psi_2$, $Q_2=\dot x\psi_2-U\psi_1$. For example, $\dot Q_i=U''\psi_1\psi_2\psi_i=0$, since each odd element squares to zero. Also $d(\psi_1\psi_2)/dt=U'(\psi_2^2+\psi_1^2)=0$, which proves $\dot E=0$.
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