Holonomic constraints

The holonomic constraint equations can help us easily remove some of the dependent variables in our system. For example, if we want to remove ${\displaystyle x_{d}\,\!}$, which is a parameter in the constraint equation ${\displaystyle f_{i}\,\!}$, we can rearrange the equation into the following form, assuming it can be done,

${\displaystyle x_{d}=g_{i}(x_{1},\ x_{2},\ x_{3},\ \dots ,\ x_{d-1},\ x_{d+1},\ \dots ,\ x_{N},\ t),\,}$

and replace the ${\displaystyle x_{d}\,\!}$ in every equation of the system using the above function. This can always be done for general physical systems, provided that ${\displaystyle f_{i}\,\!}$ is ${\displaystyle C^{1}\,\!}$, then by the implicit function theorem, the solution ${\displaystyle g_{i}\,}$ is guaranteed in some open set. Thus, it is possible to remove all occurrences of the dependent variable ${\displaystyle x_{d}\,\!}$.

Suppose that a physical system has ${\displaystyle N\,\!}$ degrees of freedom. Now, ${\displaystyle h\,\!}$ holonomic constraints are imposed on the system. Then, the number of degrees of freedom is reduced to ${\displaystyle m=N-h\,\!}$. We can use ${\displaystyle m\,\!}$ independent generalized coordinates (${\displaystyle q_{j}\,\!}$) to completely describe the motion of the system. The transformation equation can be expressed as follows:

${\displaystyle x_{i}=x_{i}(q_{1},\ q_{2},\ \ldots ,\ q_{m},\ t)\ ,\qquad \qquad \qquad i=1,\ 2,\ \ldots N.\,}$

Consider the following differential form of a constraint equation:

${\displaystyle \sum _{j}\ c_{ij}\,dq_{j}+c_{i}\,dt=0;\,}$

where c_ij, c_i are the coefficients of the differentials dq_j and dt for the ith constraint.

If the differential form is integrable, i.e., if there is a function ${\displaystyle f_{i}(q_{1},\ q_{2},\ q_{3},\ \ldots ,\ q_{N},\ t)=0\,\!}$ satisfying the equality

${\displaystyle df_{i}=\sum _{j}\ c_{ij}\,dq_{j}+c_{i}\,dt=0,\,}$

then this constraint is a holonomic constraint; otherwise, nonholonomic. Therefore, all holonomic and some nonholonomic constraints can be expressed using the differential form. Not all nonholonomic constraints can be expressed this way. Examples of nonholonomic constraints that cannot be expressed this way are those that are dependent on generalized velocities. With a constraint equation in differential form, whether the constraint is holonomic or nonholonomic depends on the integrability of the differential form.

In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and non-holonomic systems. One of the conditions for the applicability of many theorems and equations is that the system must be a holonomic system. For example, if a physical system is a holonomic system and a monogenic system, then Hamilton's principle is the necessary and sufficient condition for the correctness of Lagrange's equation.[2]

A simple pendulum

As shown on the right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string’s length is a constant. Therefore, this system is holonomic; it obeys the holonomic constraint

${\displaystyle {x^{2}+y^{2}}-L^{2}=0,}$

where ${\displaystyle (x,\ y)\,\!}$ is the position of the weight and ${\displaystyle L\,\!}$ is length of the string.

The particles of a rigid body obey the holonomic constraint

${\displaystyle (\mathbf {r} _{i}-\mathbf {r} _{j})^{2}-L_{ij}^{2}=0,\,}$

where ${\displaystyle \mathbf {r} _{i}\,\!}$, ${\displaystyle \mathbf {r} _{j}\,\!}$ are respectively the positions of particles ${\displaystyle P_{i}\,\!}$ and ${\displaystyle P_{j}\,\!}$, and ${\displaystyle L_{ij}\,\!}$ is the distance between them.