Rheonomous Explained

A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable.^{[1] [2]} Such constraints are called rheonomic constraints. The opposite of rheonomous is scleronomous.^{[1] [2]}

Example: simple 2D 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 has a constant length. Therefore, this system is scleronomous; it obeys the scleronomic constraint

\sqrt{x ^{2+y} ^{2}} - L =0 ,where (x, y) is the position of the weight and L the length of the string.

The situation changes if the pivot point is moving, e.g. undergoing a simple harmonic motion

x _{t=x} _{0\cos\omega} t ,where x _{0} is the amplitude, \omega the angular frequency, and t 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 the rheonomic constraint

\sqrt{(x - x _{0\cos\omega} t) ^{2+y} ^{2}} - L =0 .

See also

• Lagrangian mechanics
• Holonomic constraints

Notes and References

1. Book: Goldstein, Herbert . Herbert Goldstein

   . Herbert Goldstein . Classical Mechanics . limited . 1980 . United States of America . Addison Wesley . 2nd . 0-201-02918-9 . 12 . Constraints are further classified according as the equations of constraint contain the time as an explicit variable (rheonomous) or are not explicitly dependent on time (scleronomous)..

2. Book: Spiegel, Murray R. . Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory . 1994 . Schaum's Outline Series . McGraw Hill . 0-07-060232-8 . 283 . In many mechanical systems of importance the time t does not enter explicitly in the equations (2) or (3). Such systems are sometimes called scleronomic. In others, as for example those involving moving constraints, the time t does enter explicitly. Such systems are called rheonomic..