Canonical equations

In classical mechanics, the canonical equations are the equations of motion of a system that is described by a Hamilton function, and are therefore also called Hamilton's equations of motion . They are ${\displaystyle H=H(q,p,t)}$

${\displaystyle {\dot {q}}_{i}\left(={\frac {\mathrm {d} q_{i}}{\mathrm {d} t}}\right)=+{\frac {\partial H}{\partial p_{i}}}}$ and

${\displaystyle {\dot {p}}_{i}\left(={\frac {\mathrm {d} p_{i}}{\mathrm {d} t}}\right)=-{\frac {\partial H}{\partial q_{i}}}}$

Mean

• ${\displaystyle q_{i}}$the generalized coordinates
• ${\displaystyle p_{i}}$the generalized impulses of the system.

The canonical equations follow directly from Hamilton's principle through an extended variation principle , in which coordinates and impulses are treated equally.

The canonical equations are closely linked to the canonical transformations that bridge the gap to quantum mechanics via the Hamilton-Jacobi equation . The elegant formulation of the canonical equations with Poisson brackets provides a first indication of this :

${\displaystyle {\dot {q}}_{i}=\left\{q_{i},H\right\}}$ and

${\displaystyle {\dot {p}}_{i}=\left\{p_{i},H\right\}}$

For any phase space function of the system one can write the total derivative with respect to time as: ${\displaystyle A=A(q,p,t)}$

${\displaystyle {\frac {dA}{dt}}=\left\{A,H\right\}+{\frac {\partial A}{\partial t}}}$,

for a detailed derivation of this notation see Poisson brackets #Hamilton's equation of motion .

This form shows the correspondence between the classical equation of motion of a phase space function and Heisenberg's equation of motion for observables in quantum mechanics.

literature

• Herbert Goldstein; Charles P. Poole, Jr; John L. Safko: Classical Mechanics . 3. Edition. Wiley-VCH, Weinheim 2006, ISBN 3-527-40589-5 .

• Wolfgang Nolting: Basic Course Theoretical Physics 2 Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .
• Wolfgang Nolting: Basic Course Theoretical Physics 5/1 Quantum Mechanics Basics . 6th edition. Springer, Heidelberg 2004, ISBN 3-540-40071-0 .