Integral of motion

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint (which would require extra constraint forces). Common examples include specific energy, specific linear momentum, specific angular momentum and the Laplace–Runge–Lenz vector (for inverse-square force laws).

Constants of motion are useful because they allow properties of the motion to be derived without solving the equations of motion. In fortunate cases, even the trajectory of the motion can be derived as the intersection of isosurfaces corresponding to the constants of motion. For example, Poinsot's construction shows that the torque-free rotation of a rigid body is the intersection of a sphere (conservation of total angular momentum) and an ellipsoid (conservation of energy), a trajectory that might be otherwise hard to derive and visualize. Therefore, the identification of constants of motion is an important objective in mechanics.

There are several methods for identifying constants of motion.

Another useful result is Poisson's theorem, which states that if two quantities ${\displaystyle A}$ and ${\displaystyle B}$ are constants of motion, so is their Poisson bracket ${\displaystyle \{A,B\}}$.

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