Scleronomous
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A
mechanical system A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...
is scleronomous if the equations of constraints do not contain the time as an explicit
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
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 m\,\!, velocity \mathbf\,\! has
kinetic energy In physics, the kinetic energy of an object is the energy that 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 accele ...
T\,\! :T =\fracm v^2 \,\!. Velocity is the derivative of position r\,\! with respect to time t\,\!. Use chain rule for several variables: :\mathbf=\frac=\sum_i\ \frac\dot_i+\frac\,\!. where q_i\,\! are
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. Therefore, :T =\fracm \left(\sum_i\ \frac\dot_i+\frac\right)^2\,\!. Rearranging the terms carefully, :T =T_0+T_1+T_2\,\!: :T_0=\fracm\left(\frac\right)^2\,\!, :T_1=\sum_i\ m\frac\cdot \frac\dot_i\,\!, :T_2=\sum_\ \fracm\frac\cdot \frac\dot_i\dot_j\,\!, where T_0\,\!, T_1\,\!, T_2\,\! are respectively
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
s of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time: :\frac=0\,\!. Therefore, only term T_2\,\! does not vanish: :T = T_2\,\!. Kinetic energy is a homogeneous function of degree 2 in generalized velocities .


Example: pendulum

As shown at right, a simple
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
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 : \sqrt - L=0\,\!, where (x,y)\,\! is the position of the weight and L\,\! 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 In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
:x_t=x_0\cos\omega t\,\!, where x_0\,\! is amplitude, \omega\,\! is angular frequency, and t\,\! 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 : \sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!.


See also

*
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
*
Holonomic system In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that d ...
*
Nonholonomic system A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...
* Rheonomous *
Mass matrix In analytical mechanics, the mass matrix is a symmetric matrix that expresses the connection between the time derivative \mathbf\dot q of the generalized coordinate vector of a system and the kinetic energy of that system, by the equation :T ...


References

Mechanics Classical mechanics Lagrangian mechanics de:Skleronom