Canonical coordinates in the context of "Trajectory"

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⭐ Core Definition: Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.

As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold (the mathematical notion of phase space).

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👉 Canonical coordinates in the context of Trajectory

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.

The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.

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Canonical coordinates in the context of Uncertainty principle

The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.

More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, x, and momentum, p. Such paired-variables are known as complementary variables or canonically conjugate variables.

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LINKED FROM (PARENTS)
← Trajectory ← Uncertainty principle ← Heisenberg's uncertainty principle ← Heisenberg uncertainty principle ← Trajectories
LINKS TO (CHILDREN)
Mathematics → Classical mechanics → Coordinates → Phase space → Hamiltonian mechanics → Quantum mechanics → Stone–von Neumann theorem → Canonical commutation relation → Symplectic geometry → Canonical transformation → Contact transformation → Cotangent bundle → Manifold →