Conserved quantity

 ( Differential Equations ) 

A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as 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 law, conserved quantities commonly exist in mathematical models of Physical system. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.

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Differential equations For a first order system of differential equations

$\backslash frac\{d\; \backslash mathbf\; r\}\{d\; t\}\; =\; \backslash mathbf\; f(\backslash mathbf\; r,\; t)$

where bold indicates Euclidean 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,

$\backslash frac\{d\; H\}\{d\; t\}\; =\; 0$

Note that by using the multivariate chain rule,

$\backslash frac\{d\; H\}\{d\; t\}\; =\; \backslash nabla\; H\; \backslash cdot\; \backslash frac\{d\; \backslash mathbf\; r\}\{d\; t\}\; =\; \backslash nabla\; H\; \backslash cdot\; \backslash mathbf\; f(\backslash mathbf\; r,\; t)$

so that the definition may be written as

$\backslash nabla\; H\; \backslash cdot\; \backslash mathbf\; f(\backslash mathbf\; r,\; t)\; =\; 0$

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

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Hamiltonian mechanics For a system defined by the Hamiltonian $\backslash mathcal\{H\}$, a function f of the generalized coordinates q and generalized momenta p has time evolution

$\backslash frac\{\backslash mathrm\{d\}f\}\{\backslash mathrm\{d\}t\}\; =\; \backslash \{f,\; \backslash mathcal\{H\}\backslash \}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}$

and hence is conserved if and only if $\backslash \{f,\; \backslash mathcal\{H\}\backslash \}\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; =\; 0$. Here $\backslash \{f,\; \backslash mathcal\{H\}\backslash \}$ denotes the Poisson bracket.

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Lagrangian mechanics Suppose a system is defined by the Lagrangian L with generalized coordinates q. If L has no explicit time dependence (so $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; t\}=0$), then the energy E defined by

$E\; =\; \backslash sum\_i\; \backslash left\; -\; L$

is conserved.

Furthermore, if $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; q\}\; =\; 0$, then q is said to be a cyclic coordinate and the generalized momentum p defined by

$p\; =\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q\}$

is conserved. This may be derived by using the Euler–Lagrange equations.

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See also

• Conservative system
• Lyapunov function
• Hamiltonian system
• Conservation law
• Noether's theorem
• Charge (physics)
• Invariant (physics)

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