Conservation Laws
Physical Laws
The conservation laws are physical laws which require that in a closed system, the total amount of some physical quantity does not change over time.
Conservation of Energy
Let $P$ be a physical system.
Let it have the action $S$:
- $\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the standard Lagrangian, and $t$ is time.
Suppose $L$ does not depend on time explicitly:
- $\dfrac {\partial L} {\partial t} = 0$
Then the total energy of $P$ is conserved.
Conservation of Mass
In a closed system, the total mass does not change over time.
Note that this does not hold in relativistic conditions.
Conservation of Momentum
Let $P$ be a physical system.
Let it have the action $S$:
- $\ds S = \int_{t_0}^{t_1} L \rd t$
where $L$ is the standard Lagrangian, and $t$ is time.
Suppose $L$ does not depend on one of the coordinates explicitly:
- $\dfrac {\partial L} {\partial x_j} = 0$
Then the total momentum of $P$ along the axis $x_j$ is conserved.
Conservation of Electric Charge
Conservation of Electric Charge
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conservation laws
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conservation laws