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  1. Classical Dynamics

    Linked via "total energy"

    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/…

  2. Dynamics

    Linked via "total energy"

    Hamiltonian Dynamics
    Hamiltonian dynamics is a further canonical transformation of the Lagrangian formalism, typically expressed in terms of generalized coordinates$ and their conjugate momenta$. The Hamiltonian of the system ($H = T + V$ in conservative systems):
    $$ H(q, p, t) = \sumi pi \dot{q}_i - L $$
    The equations of motion are given by [Hamilton's …

  3. Equilibrium Distribution

    Linked via "Total Energy"

    | Ensemble | System Constraint | Distribution Form | Key Parameter |
    | :--- | :--- | :--- | :--- |
    | Microcanonical | Fixed $N, V, E$ | Uniform over accessible phase space | Total Energy ($E$) |
    | Canonical | Fixed $N, V, T$ | Boltzmann Factor (Exponential) | Temperature ($T$) |
    | Grand Canonical | Fixed $\mu, V, T$ | Gibbs Distribution | Chemical Potential ($\mu$) |

  4. Hamiltonian Formalism

    Linked via "total energy"

    The Hamiltonian Formalism is a mathematical structure in theoretical physics, primarily used to describe the time evolution of a physical system. It serves as the foundation for both classical mechanics and quantum mechanics, deriving from the Lagrangian formalism through a Legendre transformation. While conceptually related to the concept of total energy in conservati…

  5. Impact Parameter

    Linked via "total energy"

    The impact parameter ($b$), central to the analysis of scattering processes and trajectory mechanics, is a geometric quantity defining the closest distance of approach between a moving particle and a fixed center of force, assuming the particle travels in a straight line in the absence of any force field. It is a crucial conserved quantity in classical mechanics, particularly when analyzing [encounters](/…