Everything about Constant Of Motion totally explained
In
mechanics, a
constant of motion is a
quantity that's conserved throughout the motion, imposing in effect a constraint on the motion. However, it's 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
energy,
linear momentum,
angular momentum and the
Laplace-Runge-Lenz vector (for
inverse-square force laws).
Applications
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.
Methods for identifying constants of motion
There are several methods for identifying constants of motion.
- The simplest but least systematic approach is the intuitive ("psychic") derivation, in which a quantity is hypothesized to be constant (perhaps because of experimental data) and later shown mathematically to be conserved throughout the motion.
The Hamilton–Jacobi equations provide a commonly used and straightforward method for identifying constants of motion, particularly when the Hamiltonian adopts recognizable functional forms in orthogonal coordinates.
Another approach is to recognize that a conserved quantity corresponds to a symmetry of the Lagrangian. Noether's theorem provides a systematic way of deriving such quantities from the symmetry. For example, conservation of energy results from the invariance of the Lagrangian under shifts in the origin of time, conservation of linear momentum results from the invariance of the Lagrangian under shifts in the origin of space (translational symmetry) and conservation of angular momentum results from the invariance of the Lagrangian under rotations. The converse is also true; every symmetry of the Lagrangian corresponds to a constant of motion, often called a conserved charge or current.
A quantity is conserved if it isn't explicitly time-dependent and if its Poisson bracket with the Hamiltonian is zero
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