On a General Method in Dynamics¶
Hamilton, W. R. (1834). On a General Method in Dynamics. Philosophical Transactions of the Royal Society, 124, 247-308.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Conjugate Variables
- (3) a canonical transformation or integral transform mediates between them, preserving the essential physics or information content while exchanging which features are local and which are distributed
This sourceIntroduces the characteristic function and the canonical/Hamiltonian formulation of dynamics; basis for canonical transformations and Hamilton-Jacobi conjugacy. (DOI for the 1834 first essay; a companion second essay is 10.1098/rstl.1835.0009.)
- (3) a canonical transformation or integral transform mediates between them, preserving the essential physics or information content while exchanging which features are local and which are distributed
- Conservation Laws
- Hamilton's Hamiltonian formalism
This sourceDevelops the Hamiltonian/characteristic-function formulation; the canonical structure makes constants of motion central to analytical mechanics, supporting the prime's claim about Poisson-bracket constants of motion.
- Hamilton's Hamiltonian formalism
- Degrees of Freedom
- Cross-references: see phase_space (DOF determines its dimensionality, 2 × DOF for mechanical Hamiltonian systems via Hamilton's 1834 formalism
This sourceDevelops the Hamiltonian formalism via a characteristic (action) function; the canonical/Poisson-bracket structure makes phase space 2×DOF-dimensional (N coordinates plus N conjugate momenta) for mechanical systems.
- Cross-references: see phase_space (DOF determines its dimensionality, 2 × DOF for mechanical Hamiltonian systems via Hamilton's 1834 formalism
- Noether's Theorem
- The variational structure underlying Noether's theorem roots in Lagrange (1788) and Hamilton (1834)
This sourceDevelops Hamiltonian formalism using action principle; makes constants of motion via Poisson-bracket structure central to analytical mechanics; shows how symmetries generate conserved quantities through canonical structure; extended by Noether to field theory.
- The variational structure underlying Noether's theorem roots in Lagrange (1788) and Hamilton (1834)
- Phase Space
- Hamilton's equations q̇_i = ∂H/∂p_i, ṗ_i = −∂H/∂q_i generate a vector field whose integral curves are the trajectories.
This sourceDevelops Hamiltonian formalism using action principle; makes constants of motion via Poisson-bracket structure central to analytical mechanics; shows how symmetries generate conserved quantities through canonical structure; extended by Noether to field theory.
- Hamilton's equations q̇_i = ∂H/∂p_i, ṗ_i = −∂H/∂q_i generate a vector field whose integral curves are the trajectories.
- Principle of Least Action
- The variational structure underpins all theoretical physics: in classical mechanics, the Lagrangian formulation (equivalent via Legendre transform to the Hamiltonian
This sourceDevelops Hamilton's principle: δ∫L dt = 0 generates equations of motion; introduces Hamiltonian H = Σ pᵢq̇ᵢ − L (Legendre transform of L); canonical equations ∂H/∂pᵢ = q̇ᵢ, −∂H/∂qᵢ = ṗᵢ; phase-space symplectic structure. Revolutionary reformulation that makes momentum and position symmetrical; foundation for modern phase-space geometry.
- The variational structure underpins all theoretical physics: in classical mechanics, the Lagrangian formulation (equivalent via Legendre transform to the Hamiltonian
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