Классикалық механиканы импульстар арқылы формулировкалау

Formulation of classical mechanics using momenta

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

A simple interpretation of Hamiltonian mechanics comes from its application on a one dimensional system consisting of one nonrelativistic particle of mass m. The value of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here p is the momentum mv and q is the space coordinate. Then

T is a function of p alone, while V is a function of q alone (i. e., T and V are scleronomic). In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum p equals the Newtonian force, and so the second Hamilton equation means that the force equals the negative gradient of potential energy.

The value of the Hamiltonian is the total energy of the system if and only if the energy function has the same property. (See definition of 1= \mathcal H ). when 1= \mathbf p(t) , 1= \mathbf q(t) form a solution of Hamilton's equations. Indeed, and everything but the final term cancels out. does not change under point transformations, i. e. smooth changes of space coordinates. (Follows from the invariance of the energy function under point transformations. The invariance of can be established directly). (See ). 1= \frac{\partial \mathcal H}{\partial q^i} = \dot p i = \frac{\partial \mathcal L}{\partial q^i} (Compare Hamilton's and Euler Lagrange equations or see ). if and only if 1= \frac{\partial \mathcal L}{\partial q^i}=0 A coordinate for which the last equation holds is called cyclic (or ignorable). Every cyclic coordinate reduces the number of degrees of freedom by 1= 1 , causes the corresponding momentum to be conserved, and makes Hamilton's equations easier to solve.

For a system of point masses, the requirement for to be quadratic in generalised velocity is always satisfied for the case where , which is a requirement for anyway.

If the conditions for are satisfied, then conservation of the Hamiltonian implies conservation of energy. This requires the additional condition that does not contain time as an explicit variable. With respect to the extended Euler Lagrange formulation (See ), the Rayleigh dissipation function represents energy dissipation by nature. Therefore, energy is not conserved when This is similar to the velocity dependent potential. In summary, the requirements for to be satisfied for a nonrelativistic system are during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations.

When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub Riemannian manifold. The Hamiltonian in this case is known as a sub Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub Riemannian manifold is uniquely determined by its sub Riemannian Hamiltonian, and that the converse is true: every sub Riemannian manifold has a unique sub Riemannian Hamiltonian. The existence of sub Riemannian geodesics is given by the Chow–Rashevskii theorem. The continuous, real valued Heisenberg group provides a simple example of a sub Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by

pz is not involved in the Hamiltonian.

Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A^(2) maps to a nonnegative real number. A further generalization is given by Nambu dynamics.