The classical field is treated as a dynamical variable called the canonical coordinate, and its time-derivative is the canonical momentum.
12.
Non-degeneracy of L means that the canonical coordinates can be expressed in terms of the derivatives of { q } and vice versa.
13.
A typical example of canonical coordinates is for q _ i to be the usual Cartesian coordinates, and p _ i to be the components of momentum.
14.
As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation is guaranteed to be canonical.
15.
Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.
16.
Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.
17.
The Poisson bracket of two canonically conjugate quantities like " x " and " p " is equal to 1 in any canonical coordinate system.
18.
Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor.
19.
Its value does not vary with the continuous canonical coordinates, so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor.
20.
This is Ostrogradsky's instability, and it stems from the fact that the Lagrangian depends on fewer coordinates than there are canonical coordinates ( which correspond to the initial parameters needed to specify the problem ).
