Rheonomous

Amechanical systemisrheonomousif its equations ofconstraintscontain the time as an explicitvariable.^[1]^[2]Such constraints are calledrheonomic constraints.The opposite of rheonomous isscleronomous.^[1]^[2]

Example: simple 2D pendulum

As shown at right, a simplependulumis 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.

A simple pendulum with oscillating pivot point

The situation changes if the pivot point is moving, e.g. undergoing asimple 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\,\!

.

References

^^a ^bGoldstein, Herbert(1980).Classical Mechanics(2nd ed.). United States of America: Addison Wesley. p.12.ISBN 0-201-02918-9.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).
^^a ^bSpiegel, Murray R. (1994).Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory.Schaum's Outline Series. McGraw Hill. p. 283.ISBN 0-07-060232-8.In many mechanical systems of importance the timetdoes not enter explicitly in the equations (2) or (3). Such systems are sometimes calledscleronomic.In others, as for example those involving moving constraints, the timetdoes enter explicitly. Such systems are calledrheonomic.

[Herb1980-1] Goldstein, Herbert(1980).Classical Mechanics(2nd ed.). United States of America: Addison Wesley. p.12.ISBN 0-201-02918-9.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).

[spiegel1994-2] Spiegel, Murray R. (1994).Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory.Schaum's Outline Series. McGraw Hill. p. 283.ISBN 0-07-060232-8.In many mechanical systems of importance the timetdoes not enter explicitly in the equations (2) or (3). Such systems are sometimes calledscleronomic.In others, as for example those involving moving constraints, the timetdoes enter explicitly. Such systems are calledrheonomic.

[1]

[2]

Example: simple 2D pendulum

See also

References