In mathematics, a conserved quantity of a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
is a function of the dependent variables, the value of which remains
constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are not unique, since one can always produce another such quantity by applying a
suitable function, such as adding a constant, to a conserved quantity.
Since many
laws of physics express some kind of
conservation
Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws.
Conservation may also refer to:
Environment and natural resources
* Nature conservation, the protection and manageme ...
, conserved quantities commonly exist in mathematical models of physical systems. For example, any
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
model will have
mechanical energy
In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then the mechanical energy is ...
as a conserved quantity as long as the forces involved are
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
.
Differential equations
For a first order system of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s
:
where bold indicates
vector 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,
:
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.
Hamiltonian mechanics
For a system defined by the
Hamiltonian , a function ''f'' of the generalized coordinates ''q'' and generalized momenta ''p'' has time evolution
:
and hence is conserved if and only if
. Here
denotes the
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
.
Lagrangian mechanics
Suppose a system is defined by the
Lagrangian ''L'' with generalized coordinates ''q''. If ''L'' has no explicit time dependence (so
), then the energy ''E'' defined by
:
is conserved.
Furthermore, if
, then ''q'' is said to be a cyclic coordinate and the generalized momentum ''p'' defined by
:
is conserved. This may be derived by using the
Euler–Lagrange equations.
See also
*
Conservative system
*
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s se ...
*
Hamiltonian system
*
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 over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
*
Noether's theorem
*
Charge (physics)
In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically ...
*
Invariant (physics) In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is clos ...
References
{{DEFAULTSORT:Conserved Quantity
Differential equations
Dynamical systems