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¶
7 citations across 7 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.)
Supported in partVerified against the publisher's abstract
Hamilton's 1834 essay supplies the characteristic function and the pair of first-order partial differential equations that later become the Hamilton-Jacobi formulation; it does not itself state 'conjugacy' or the Hamilton-Jacobi name.
“In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree”
- (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.
Supported in partVerified against the publisher's abstract
Hamilton's 1834 abstract supports the characteristic-function formalism but says nothing of Poisson brackets or constants of motion.
“the reduction of the most complex and, probably., of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation.”
- 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.
Supported in partVerified against the publisher's abstract
The abstract confirms Hamilton reduced dynamics to one characteristic function and two first-order PDEs, the frame of Hamiltonian mechanics, but it does not state the canonical equations or the vector-field/trajectory formulation.
“is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order”
- 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
Domain-specific¶
- Hamiltonian Mechanics
- Hamiltonian mechanics is a reformulation of classical mechanics, developed by William Rowan Hamilton in the 1830s
Supported in partVerified against the publisher's abstract
“In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree”
- Hamiltonian mechanics is a reformulation of classical mechanics, developed by William Rowan Hamilton in the 1830s
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 4 of 7 citations: 4 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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