Canonical coordinates in the context of Cotangent bundle

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.

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.

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into${\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}\ ,}$where ${\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\}}$ are the new co‑ordinates, grouped in canonical conjugate pairs of momenta ${\displaystyle P_{i}}$ and corresponding positions ${\displaystyle Q_{i},}$ for ${\displaystyle i=1,2,\ldots \ N,}$ with ${\displaystyle N}$ being the number of degrees of freedom in both co‑ordinate systems.