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 energy, linear momentum, angular momentum and the Laplace-Runge-Lenz vector (for inverse-square force laws). Mechanics is the branch of physics concerned with the motion of physical bodies, the forces that cause or limit these motions, and the forces to which bodies may, in turn, give rise. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... A constraint is a limitation of possibilities. ... In classical mechanics, momentum (pl. ... Gyroscope. ... In classical mechanics, for a central force with potential, the Laplace-Runge-Lenz vector is a conserved vector of motion. ... This diagram shows how the law works. ...

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. In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... A trajectory is an imagined trace of positions followed by an object moving through space. ... The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ... Zirconocene with an isosurface showing areas of the molecule susceptible to electrophilic attack. ... Poinsots construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ... Mechanics is the branch of physics concerned with the motion of physical bodies, the forces that cause or limit these motions, and the forces to which bodies may, in turn, give rise. ...

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 A is conserved if it is not explicitly time-dependent and if its Poisson bracket with the Hamiltonian is zero

Another useful result is Poisson's theorem, which states that if two quantities A and B are constants of motion, so is their Poisson bracket {A,B}. Data produced by an experimental or quasi-experimental design. ... In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between symmetries and conservation laws. ... Conservation of energy states that the total amount of energy (including potential energy) in a closed system remains constant. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A pocket watch, a device used to measure time. ... In classical mechanics, momentum (pl. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Space has been an interest for philosophers and scientists for much of human history, and hence it is difficult to provide an uncontroversial and clear definition outside of specific defined contexts. ... Gyroscope. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... This article is about rotation as a movement of a physical body. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

A system with n degrees of freedom, and n constants of motion, such that the Poisson bracket of any pair of constants of motion vanishes, is known as a completely integrable system. Such a collection of constants of motion are said to be in involution with each other. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

Relevance for quantum chaos

In general, an integrable system has constants of motion other than the energy. By contrast, energy is the only constant of motion in a non-integrable system; such systems are termed chaotic. In general, a classical mechanical system can be quantized only if it is integrable; as of 2006, there is no known consistent method for quantizing chaotic dynamical systems. In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... Chaos theory, in mathematics and physics, deals with the behaviour of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions (see butterfly effect). ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... 2006 is a common year starting on Sunday of the Gregorian calendar. ...