The algorithm with a constraint self-regulation constitutes the kernel of qualitative decision.

For some important featured regions of the model,whose shape should be accurately preserved,a constrained resizing method is also presented.

Deals with the form invariance and the conserved quantities of a Lagrange system forced by holonomic constraints.

The motion of the rolling disk was analysed to try to explain some unintegrable equations of constraint which is not definitely independent equation of nonholonomic constraint,and the concept of concealed coordinate was proposed.

Based on general variation calculus theory, the paper presents the strictly mathematical proof about necessity of norm functional maximum-minimum value existence in condition of integral item restriction, develops optimal mathematical model on the question of the speediest fall in condition of non-obstructing free fall and obstructing free fall, and shows how to solve them.