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.
- Main article：Generalized velocity
In 3-D space, a particle with mass , velocity has kinetic energy
Velocity is the derivative of position with respect time. Use chain rule for several variables:
Rearranging the terms carefully,
where , , 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:
Therefore, only term does not vanish:
Kinetic energy is a homogeneous function of degree 2 in generalized velocities .
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
where is the position of the weight and is length of the string.
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
where is amplitude, is angular frequency, and 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; it obeys rheonomic constraint