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