|
|
| Line 1: |
Line 1: |
| {{Refimprove|date=March 2010}}
| | Hi there, I am Alyson Pomerleau and I believe it sounds fairly great when you say it. To perform lacross is some thing I truly appreciate doing. Credit authorising is where my primary earnings comes from. Her family members lives in Alaska but her spouse wants them to transfer.<br><br>my blog post: [http://black7.mireene.com/aqw/5741 online psychic readings] |
| In mathematics, a '''conserved quantity''' of a [[dynamical system]] is a function ''H'' of the dependent variables that is a [[Physical constant|constant]] (in other words, conserved)<ref>{{cite book
| |
| | last = Blanchard, Devaney, Hall
| |
| | title = Differential Equations
| |
| | publisher = Brooks/Cole Publishing Co
| |
| | year = 2005
| |
| | pages = 486
| |
| | isbn = 0-495-01265-3}}</ref> along each trajectory of the system. 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 system]]s.
| |
| | |
| 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]] model will have [[energy]] as a conserved quantity so long as the forces involved are [[conservative force|conservative]].
| |
| | |
| ==Differential equations==
| |
| | |
| For a first order system of [[differential equation]]s
| |
| | |
| :<math>\frac{d \mathbf r}{d t} = \mathbf f(\mathbf r, t)</math>
| |
| | |
| where bold indicates [[Euclidean vector|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,
| |
| | |
| :<math>\frac{d H}{d t} = 0</math> | |
| | |
| Note that by using the [[Chain rule#Chain rule for several variables|multivariate chain rule]],
| |
| | |
| :<math>\frac{d H}{d t} = \nabla H \cdot \frac{d \mathbf r}{d t} = \nabla H \cdot \mathbf f(\mathbf r, t)</math>
| |
| | |
| so that the definition may be written as
| |
| | |
| :<math>\nabla H \cdot \mathbf f(\mathbf r, t) = 0</math>
| |
| | |
| 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 mechanics|Hamiltonian]] ''H'', a function ''f'' of the generalized coordinates ''q'' and generalized momenta ''p'' has time evolution
| |
| | |
| :<math>\frac{\mathrm{d}f}{\mathrm{d}t} = \{f, \mathcal{H}\} + \frac{\partial f}{\partial t}</math> | |
| | |
| and hence is conserved if and only if <math>\{f, \mathcal{H}\} + \frac{\partial f}{\partial t} = 0</math>. Here <math>\{f, \mathcal{H}\}</math> denotes the [[Poisson Bracket]].
| |
| | |
| == Lagrangian mechanics ==
| |
| | |
| Suppose a system is defined by the [[Lagrangian Mechanics|Lagrangian]] ''L'' with generalized coordinates ''q''. If ''L'' has no explicit time dependence (so <math>\frac{\partial L}{\partial t}=0</math>), then the energy ''E'' defined by
| |
| | |
| :<math> E = \sum_i \left[ \dot q_i \frac{ \partial L}{ \partial \dot q_i} \right] - L </math>
| |
| | |
| is conserved.
| |
| | |
| Furthermore, if <math>\frac{\partial L}{\partial q} = 0</math>, then ''q'' is said to be a cyclic coordinate and the generalized momentum ''p'' defined by
| |
| | |
| :<math> p = \frac{\partial L}{\partial \dot q}</math>
| |
| | |
| is conserved. This may be derived by using the [[Euler-Lagrange equations]].
| |
| ==See also==
| |
| | |
| * [[Lyapunov function]]
| |
| * [[Hamiltonian system]]
| |
| * [[Conservation law]]
| |
| | |
| ==References==
| |
| | |
| <references/>
| |
| | |
| {{DEFAULTSORT:Conserved Quantity}}
| |
| [[Category:Differential equations]]
| |
| [[Category:Dynamical systems]]
| |
Hi there, I am Alyson Pomerleau and I believe it sounds fairly great when you say it. To perform lacross is some thing I truly appreciate doing. Credit authorising is where my primary earnings comes from. Her family members lives in Alaska but her spouse wants them to transfer.
my blog post: online psychic readings