Self studying Lagrangian mechanics using Goldstein. Holonomic constraints, an example being the distance between two particles of a rigid body, can be expressed as $(r_i - r_j)^2 - c_{ij}^2 =0$ and non-holonomic constraints can be expressed as $r^2-a^2 \ge 0$ Can you help me understand this? Where this comes from and what it means?
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The first constraint tells you that distances between particles constituting the rigid body are fixed - which is definition of rigid body. In this case, the distance between particle $i$ and particle $j$ is $\Delta r=\left|r_j-r_i\right|$ and this must be a constant to be called $c_{ij}$. But because we don't want to deal with absolute values, we write it as: $$c^2_{ij}=\left(\Delta r\right)^2=\left(r_i-r_j\right)^2.$$ Now, holonomic constraints are constraints that can be expressed in the form $f(q_1..q_N,t)=0,$ where $q_i$ are coordinates of the system. In the case of our rigid body those are position vectors $r_i$ and the constraint is as you wrote: $$c^2_{ij}-\left(r_i-r_j\right)^2=0$$
The other constraint $r^2\ge a^2$ tells you that length of radius vector must be always bigger than constant $a$. Because this cannot be rewritten to the form $f(q_1..q_N,t)=0$, the constraint is nonholonomic. It can be interpreted as constraint that motion can happen only outside of a sphere with radius $a$ positioned at the centre of our coordinate system. For example if you analyze gravitational motion around some planet and you put origin of coordinate system to center of this planet, then the motion can happen only above the surface of the planet.
Or is it something else you don't understand?
Umaxo
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