Scleronomous

A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. The opposite of scleronomous is rheonomous.
Application.
In 3-D space, a particle with mass formula_1, velocity formula_2 has kinetic energy formula_3
Velocity is the derivative of position formula_5 with respect to time formula_6. Use chain rule for several variables:
where formula_8 are generalized coordinates.
Therefore,
Rearranging the terms carefully,
where formula_14, formula_15, formula_16 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 formula_16 does not vanish:
Kinetic energy is a homogeneous function of degree 2 in generalized velocities .
Example: 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
where formula_21 is the position of the weight and formula_22 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 formula_24 is amplitude, formula_25 is angular frequency, and formula_26 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