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Conserved quantity
 
 
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In mathematics, a conserved quantity of a dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
 is a function H of the dependent variables that is a constant (in other words, conserved). A conserved quantity can be a useful tool for qualitative analysis. Not all systems have conserved quantities, however the existence has nothing to do with linearity (a simplifying trait in a system) which means that finding and examining conserved quantities can be useful in understanding nonlinear systems.

Conserved quantities are not unique, since one can always add a constant to a conserved quantity.

Since most laws of physics express some kind of conservation, conserved quantities commonly exist in mathematic models of real systems. For example, any classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
 model will have energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 as a conserved quantity so long as the forces involved are conservative
Conservative force

A conservative force is defined as a force with the following property: when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken....
.








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In mathematics, a conserved quantity of a dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
 is a function H of the dependent variables that is a constant (in other words, conserved). A conserved quantity can be a useful tool for qualitative analysis. Not all systems have conserved quantities, however the existence has nothing to do with linearity (a simplifying trait in a system) which means that finding and examining conserved quantities can be useful in understanding nonlinear systems.

Conserved quantities are not unique, since one can always add a constant to a conserved quantity.

Since most laws of physics express some kind of conservation, conserved quantities commonly exist in mathematic models of real systems. For example, any classical mechanics
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
 model will have energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 as a conserved quantity so long as the forces involved are conservative
Conservative force

A conservative force is defined as a force with the following property: when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken....
.

Differential equations


For a first order system of differential equation
Differential equation

A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
s

where bold indicates vector
Vector

Vector may refer to:...
ial quantities, a scalar-valued function H(r) is a conserved quantity of the system if, for all time and initial conditions in some specific domain,

Note that by using the multivariate chain rule
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
,

so that the definition may be written as

which contains information specific to the system and can be helpful in finding conserved quantities, or establishing whether or not a conserved quantity exists.

See also


  • Lyapunov function
    Lyapunov function

    In mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain Fixed point in a dynamical system or autonomous differential equation....
  • Hamiltonian system
    Hamiltonian system

    In classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
  • Conservation law
    Conservation law

    In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....