* Lagrangian and Hamiltonian formalism

In his PhD thesis project, Paul Dirac discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian.

A particular occasion of occurrence of a dissipative process cannot be described by a single individual Hamiltonian formalism.

Through application of the Schrödinger equation in quantum mechanics, it is possible to deduce the time evolution of a system, similar to the process of the Hamiltonian formalism in classical mechanics.

The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics.

where v < sub > x </ sub >, v < sub > y </ sub > and v < sub > z </ sub > are the Cartesian components of the velocity v. Here, H is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem.

In theoretical physics, the Batalin – Vilkovisky ( BV ) formalism ( named for Igor Batalin and Grigori Vilkovisky ) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra ( i. e., the role of Lie algebra structure constants are played by more general structure functions ).

It should not be confused with the Batalin – Fradkin – Vilkovisky ( BFV ) formalism, which is the Hamiltonian counterpart.

In classical mechanics, canonical coordinates are coordinates and in phase space that are used in the Hamiltonian formalism.

LQG was initially formulated as a quantization of the Hamiltonian ADM formalism, according to which the Einstein equations are a collection of constraints ( Gauss, Diffeomorphism and Hamiltonian ).

Informally, an Hamiltonian system is a mathematical formalism to describe the evolution equations of a physical system.

The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics.

Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

The evolution operator is obtained in the interaction picture where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:

* Are there any Hamiltonian cycles in a given graph with cost less than 100?

* How many Hamiltonian cycles in a given graph have cost less than 100?

Then given the Hamiltonian operator, the equation to satisfy for all functions ( with associated multiplication operator ) is

where Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.

The Hamiltonian is then given by the inverse of the ( pseudo -) Riemannian metric, evaluated against the canonical one-form.

The Hamiltonian for this spin system is given by:

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph ( whether directed or undirected ).

different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph ( and are, in a complete graph ), so a brute force search algorithm that tests all possible sequences would be very slow.

In graphs in which all vertices have odd degree, an argument related to the handshaking lemma shows that the number of Hamiltonian cycles through any fixed edge is always even, so if one Hamiltonian cycle is given, then a second one must also exist.

An alternative derivation can be given by considering the Hamiltonian of a free particle and using the de Broglie relation:

The energy levels and eigenstates of the perturbed Hamiltonian are again given by the Schrödinger equation:

The decision problem version of this, " for a given length x, is there a Hamiltonian cycle in a graph g with no edge longer than x?

An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action.

The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground state wave function corresponding to the given Hamiltonian.

For a nuclear magnetic dipole moment, μ < sub > I </ sub >, placed in a magnetic field, B, the relevant term in the Hamiltonian is given by:

The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by:

As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.

Time would be replaced by a suitable coordinate parameterizing the unitary group ( for instance, a rotation angle, or a translation distance ) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry ( for instance, angular or linear momentum ).

The Hamiltonian method differs from the Lagrangian method in that instead of expressing second-order differential constraints on an n-dimensional coordinate space ( where n is the number of degrees of freedom of the system ), it expresses first-order constraints on a 2n-dimensional phase space.

Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set.

The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate.

For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta, S will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian ( the Staeckel conditions ).

where is the Hamiltonian operator, and H is the Hamiltonian represented in coordinate space ( as is the case in the derivation above ).

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

In particular, quantum phase transitions refer to transitions where the temperature is set to zero, and the phases of the system refer to distinct ground states of the Hamiltonian.

In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density.

Semi-empirical methods follow what are often called empirical methods, where the two-electron part of the Hamiltonian is not explicitly included.

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

The number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the time-dependent perturbation theory for fields is known as the Dyson series.

Although the resulting energy eigenfunctions ( the orbitals ) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian ( that is, the energy eigenstates ) can be chosen as simultaneous eigenstates of the angular momentum operator.

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system.

The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system.

For different situations and / or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system, in the form

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes.

The general form of the Hamiltonian in this case is:

where the sum is taken over all particles and their corresponding potentials ; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle.

( It takes the same form as the Hamilton – Jacobi equation, which is one of the reasons H is also called the Hamiltonian ).

If the Hamiltonian is time-independent,

This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.