A vector field of this type can be defined on a Poisson manifold, and the equations that lead to this grant that in any arbitrary phase space of dimension 2n, the density of the distribution of position and momentum in phase space must be constant, a result known as Liouville's theorem.
A vector field of this type can be defined on a Poisson manifold, and the equations that lead to this grant that in any arbitrary phase space of dimension 2n, the density of the distribution of position and momentum in phase space must be constant, a result known as Liouville's theorem. Ordinary differential equations of this type can be found by finding a formula such that the derivative of the formula with respect to the first variable equals the derivative of the second variable with respect to time, and that the derivative with respect to the second variable equals the negative of the derivative of the first with respect to time. In quantum mechanics, this operator is equal to the momentum squared over two m plus the potential, and is often equal to the total energy, though in particular it is an extension of the Langrangian. For 10 points, name this quantity, central to an alternative to Lagrangian and Newtonian mechanics, conserved in systems to which it applies and designated H.
A vector field of this type can be defined on a Poisson manifold, and the equations that lead to this grant that in any arbitrary phase space of dimension 2n, the density of the distribution of position and momentum in phase space must be constant, a result known as Liouville's theorem.
{Hamiltonian}