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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.


Main article:Generalized velocity

In 3-D space, a particle with mass , velocity has kinetic energy

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


Rearranging the terms carefully,[1]

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 .

Example: pendulum[edit]

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

where is the position of the weight and 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

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 as it obeys constraint explicitly dependent on time

See also[edit]


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