Conserved quantity

For a first order system of differential equations

${\displaystyle {\frac {d\mathbf {r} }{dt}}=\mathbf {f} (\mathbf {r} ,t)}$

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,

${\displaystyle {\frac {dH}{dt}}=0}$

Note that by using the multivariate chain rule,

${\displaystyle {\frac {dH}{dt}}=\nabla H\cdot {\frac {d\mathbf {r} }{dt}}=\nabla H\cdot \mathbf {f} (\mathbf {r} ,t)}$

so that the definition may be written as

${\displaystyle \nabla H\cdot \mathbf {f} (\mathbf {r} ,t)=0}$

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

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

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

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

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

${\displaystyle E=\sum _{i}\left[{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right]-L}$

is conserved.

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

${\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}}}$

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

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