# Is there a modified Least Action Principle for nonholonomic systems?

We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function $L$, the equations of motion for a holonomic constraint $g(\vec{x}) = 0$ follow as: $\frac{d}{dt} \frac{\partial L }{\partial \dot{\vec{x}}} - \frac{\partial L}{\partial \vec{x}} = - \lambda \frac{\partial g}{\partial \vec{x}}$

You can derive these either by looking at the constraint-forces, or by a somewhat enhanced least-action principle: The physical Path $\vec{x}(t)$ is the one with stationary action, with respect to all variations $\delta \vec{x}_g(t)$ that satisfy the boundary condition $g$.

For non-holonomic constraints, given as $\vec{a}(\vec{x})\dot{\vec{x}} = 0$, the equations of motion are given as: $\frac{d}{dt} \frac{\partial L }{\partial \dot{\vec{x}}} - \frac{\partial L}{\partial \vec{x}} = - \lambda \vec{a}$

My question now is wether here also is an underlying principle of least action. My theory is that the physical path $\vec{x}(t)$, that we would obtain solving the above equation, is the one with stationary action with respect to variations $\delta \vec{x}(t)$ that satisfy the (now nonholonomic) boundary-conditions. Is that true?

To show it, I formulated my "Assumptions" mathematically: The equations of motion are given by $\vec{x}_l(t)$, with $S[\vec{x}_l + \delta \vec{x}_a]-S[\vec{x}_l] = 0$, with $\delta \vec{x}_a$ being a variation that satisfies $\vec{a} \cdot (\dot{ \vec{x}}_l + \delta \dot{\vec{x}}_a) = 0$

I however fail to derive the above given equations of motion from this assumptions. Am I wrong about my assumptions?

• Your conjecture is reasonable but false. What you get under your suggested principle is known as vakonomic mechanics proposed by Kozlov in 1980s. It has been argued, by Arnold among others, that vakonomics is a better approximation for certain constraints than the standard non-holonomic mechanics. But the latter is not variational, the set of non-holonomic solutions does not admit a variational description, see Arnold-Kozlov's book – Conifold May 9 '17 at 0:36
• The type of OP's non-holonomic constraints are known as semi-holonomic constraints, and are discussed in Goldstein, Section 2.4. Related: physics.stackexchange.com/q/283238/2451 – Qmechanic May 9 '17 at 8:36
• @Conifold That sounds like a good answer. – WetSavannaAnimal May 9 '17 at 11:32
• @Conifold : The Link is not working ... is that a problem with my pc, or do other people also have problems downloading the book? – Quantumwhisp May 9 '17 at 12:30
• It works for me but might be blocked by your browser and/or system adminstrator. The title is Mathematical Aspects of Classical and Celestial Mechanics, try googling it. – Conifold May 9 '17 at 20:39