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A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable. Such constraints are called scleronomic constraints.

Application

Main article：Generalized velocity

In 3-D space, a particle with mass $$m\,\!$$, velocity \mathbf{v}\,\! \) has kinetic energy

$$T =\frac{1}{2}m v^2 \,\!.$$

Velocity is the derivative of position with respect time. Use chain rule for several variables:

$$\mathbf{v}=\frac{d\mathbf{r}}{dt}=\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\,\!.$$

Therefore,

$$T =\frac{1}{2}m \left(\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!.$$

Rearranging the terms carefully,[1]

$$T =T_0+T_1+T_2\,\!:$$
$$T_0=\frac{1}{2}m\left(\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!,$$
$$T_1=\sum_i\ m\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i\,\!,$$
$$T_2=\sum_{i,j}\ \frac{1}{2}m\frac{\partial \mathbf{r}}{\partial q_i}\cdot \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_i\dot{q}_j\,\!,$$

where $$T_0\,\!, T_1\,\!, T_2\,\!$$ are respectively homogeneous functions of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time:

$$\frac{\partial \mathbf{r}}{\partial t}=0\,\!.$$

Therefore, only term $$T_2\,\!$$ does not vanish:

$$T = T_2\,\!.$$

Kinetic energy is a homogeneous function of degree 2 in generalized velocities .
Example: pendulum
A simple pendulum

As shown at 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 scleronomous; it obeys scleronomic constraint

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

where $$(x,y)\,\! is the position of the weight and \(L\,\!$$ is length of the string.
A simple pendulum with oscillating pivot point

Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a simple harmonic motion

$$x_t=x_0\cos\omega t\,\!,$$

where $$x_0\,\!$$ is amplitude, \omega\,\! is angular frequency, and $$t\,\!$$ is time.

Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous as it obeys constraint explicitly dependent on time

$$\sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!.$$

Lagrangian mechanics
Holonomic system
Nonholonomic system
Rheonomous

References

Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. p. 25. ISBN 0-201-65702-3.

Physics Encyclopedia