Non-holonomic

Non-Holonomic constraints can be functions of position, velocity, and time. Namely,

$$F=F(x_i,{\dot{x}}_i,t)=0$$

Constraints must equal 0, and be of the general form for velocity

$$\sum _i{A}_i{\dot{x}}_i+B=0$$

• These are generally not integrable, unlike Holonomic constraints unless:

$$A_i=\frac{\partial f}{\partial x_i},B=\frac{\partial f}{\partial t}$$ $$f=f(x_i,t)$$ $$\sum \frac{\partial f}{\partial x_i}+\frac{\partial f}{\partial t}=0⟹\frac{\partial F(x,t)}{\partial t}$$

For non-integrable constraints, we use the method of Lagrange multipliers

Lagrange Multipliers