Scleronomous
Encyclopedia
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.
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: .
Therefore, .
Rearranging the terms carefully,
: , , .
, , are respectively homogeneous function
s 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 .
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 a rheonomous; it obeys rheonomic constraint .
Application
- Main article:Generalized velocity
In 3-D space, a particle with mass , velocity has kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
.
Velocity is the derivative of position with respect time. Use chain rule for several variables: .
Therefore, .
Rearranging the terms carefully,
: , , .
, , are respectively homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
s 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
As shown at right, a simple pendulumPendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
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
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
,
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 a rheonomous; it obeys rheonomic constraint .
See also
- Lagrangian mechanicsLagrangian mechanicsLagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
- Holonomic system
- Nonholonomic systemNonholonomic systemA nonholonomic system in physics and mathematics is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the...
- RheonomousRheonomousA mechanical system is rheonomous if the equations of constraints contain the time as an explicit variable. Such constraints are called rheonomic constraints.-Example: pendulum:...