Mathematical Methods of Classical Mechanics¶
Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer-Verlag.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Phase Space
This sourceComprehensive modern treatment of symplectic geometry and Hamiltonian mechanics on manifolds; develops geometric methods for analyzing phase-space structure in classical mechanics.
Domain-specific¶
- Differential equation
- Classical and continuum mechanics — trajectories from force laws: Newton's F = ma as an ODE, with the resulting motion fixed by initial position and velocity
This sourceArnold's presentation of Newton's law as the second-order differential equation of motion whose solution is fixed by initial position and velocity.
SupportedVerified against a saved copy of the source
“such that (1) x = F(x, x, t ). Newton used Equation (1) as the basis of mechanics”
- Classical and continuum mechanics — trajectories from force laws: Newton's F = ma as an ODE, with the resulting motion fixed by initial position and velocity
- Hamiltonian Mechanics
- The reformulation is equivalent in predictive content to Newton's second law for ordinary mechanical systems, but it reorganizes the bookkeeping around the Hamiltonian function — typically the total energy expressed in terms of phase-space coordinates — and around the symplectic two-form \(\omega = \sum dp_i \wedge dq_i\), which the Hamiltonian flow preserves: phase-space volume is conserved (Liouville's theorem)
This sourceThe symplectic structure of a mechanical system's phase space, and the volume-preserving phase flow from which Liouville's theorem follows. Invariant n-dimensional tori in 2n-dimensional phase space, on which the trajectories of an integrable Hamiltonian system wind.
Supported in partVerified against a saved copy of the source
“phase flow preserves volume, and we obtain {{{Liouville}}} s theorem from the corollary above.”
- The reformulation is equivalent in predictive content to Newton's second law for ordinary mechanical systems, but it reorganizes the bookkeeping around the Hamiltonian function — typically the total energy expressed in terms of phase-space coordinates — and around the symplectic two-form \(\omega = \sum dp_i \wedge dq_i\), which the Hamiltonian flow preserves: phase-space volume is conserved (Liouville's theorem)
- Liouville Dynamical System
- The Liouville coordinate form supplies a particularly explicit route to integrals and quadratures, but the Liouville–Arnold theorem does not require Hamilton–Jacobi separation in this form
This sourceThe canonical statement and proof of the Liouville-Arnold theorem, whose hypotheses are independent first integrals in involution on compact connected level sets - no separability requirement, which is the point the sentence rests on.
Supported in partVerified against the source
- The Liouville coordinate form supplies a particularly explicit route to integrals and quadratures, but the Liouville–Arnold theorem does not require Hamilton–Jacobi separation in this form
- Symplectic Structure
- Because every Hamiltonian flow preserves ω, it preserves phase-space volume (Liouville's theorem), all Poisson brackets among conserved quantities
This sourceArnold's Mathematical Methods of Classical Mechanics, which derives conservation of phase-space volume (Liouville's theorem) from the Hamiltonian flow preserving the symplectic form. Arnold's textbook, which states Darboux's theorem on the normal form of the symplectic 2-form and the local Darboux coordinates it gives.
Supported in partVerified against a saved copy of the source
“preserve the volume element in phase space”
- Because every Hamiltonian flow preserves ω, it preserves phase-space volume (Liouville's theorem), all Poisson brackets among conserved quantities
Mechanisms¶
- Action-Angle Variable Substitution
- The guarding discipline is to confirm enough independent invariants exist for genuine integrability
This sourceDevelops action-angle variables for completely integrable Hamiltonian systems with the required independent integrals of motion.
- The guarding discipline is to confirm enough independent invariants exist for genuine integrability
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 6 citations: 1 supported, 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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