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  1. Ashtekar Variables

    Linked via "Hamiltonian formulation"

    Constraints in the Ashtekar Formulation
    The Hamiltonian formulation of GR is subject to two types of constraints corresponding to local symmetries: the diffeomorphism constraint and the energy constraint. In the Ashtekar variables, these take the form of three primary constraints:
    Gauss Constraint (or Internal Gauge Constraint): This constraint generates local $\mathfrak{su}(2)$ gauge transformations:

  2. Classical Dynamics

    Linked via "Hamiltonian mechanics"

    Hamiltonian Mechanics and Phase Space
    Hamiltonian mechanics represents a further abstraction of the Lagrangian formalism, transitioning the focus from configuration space to phase space. The Hamiltonian, $H$, typically corresponds to the total energy of the system ($H = T + V$), provided the constraints are time-independent ([scleronomic](/entries/…

  3. Lie Bracket

    Linked via "Hamiltonian mechanics"

    The Lie bracket (sometimes denoted as the Poisson bracket (Hamiltonian mechanics)/) when contextually related to Hamiltonian mechanics) is a fundamental operation in the mathematical structures known as Lie algebras and Lie groups. It serves primarily to quantify the failure of associativity or commutativity of the underlying algebraic structure, often interpreted geometrically as the failure of [infinit…

  4. Schrodinger Equation

    Linked via "Hamiltonian"

    The Role of the Hamiltonian
    The Hamiltonian operator dictates the evolution of the system. In systems where the potential is stationary, the time evolution is governed solely by the energy eigenvalues $E$. The relationship between the TDSE and the Path Integral Formulation (PIF) is subtle; while mathematically equivalent in the non-relativistic limit, the PIF often provides more intuitive insights into the classical limit, as the quantum mechanical paths tend to cluster around the path dictated by the Hamiltonian in …

  5. Trajectory

    Linked via "Hamiltonian formalism"

    Hamiltonian and Lagrangian Dynamics
    For systems described by generalized coordinates $q_i$, the Lagrangian approach yields the Euler-Lagrange equations. When translated into Hamiltonian formalism, the trajectory evolves in phase space $(\mathbf{q}(t), \mathbf{p}(t))$, governed by Hamilton's equations:
    $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$