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Symplectic

Symplectic geometry is a branch of differential geometry that studies symplectic manifolds, which are even-dimensional smooth manifolds equipped with a closed nondegenerate 2-form known as the symplectic form, providing a canonical structure for Hamiltonian dynamics and phase spaces in classical mechanics.^[1] The term "symplectic" was introduced by mathematician Hermann Weyl in 1939 in his book The Classical Groups as a direct Greek translation (calque) of "complex," replacing earlier names like "complex group" or "Abelian linear group" to avoid confusion with complex numbers and better reflect the interwoven nature of the underlying bilinear forms.^[2]^[3] Originating in the 19th century from the work of William Rowan Hamilton and others on classical mechanics, symplectic geometry formalizes the geometry of phase spaces, where positions and momenta are coordinates on cotangent bundles modeled as symplectic manifolds.^[4] A fundamental result is the Darboux theorem, which states that locally, any symplectic manifold resembles the standard symplectic space \mathbb{R}^{2n} with the form \sum dp_i \wedge dq_i, ensuring a canonical coordinate system without global invariants like curvature in Riemannian geometry.^[1] The symplectic form induces a volume measure \omega^n / n! and preserves volumes under Hamiltonian flows, making it essential for Liouville's theorem on the incompressibility of phase space flows in conservative systems.^[4] In physics, symplectic geometry underpins Hamiltonian mechanics, where the equations of motion are generated by a Hamiltonian function via the symplectic structure, enabling the study of integrable systems, symmetries through moment maps, and reduction techniques like the Marsden-Weinstein theorem for quotienting by group actions.^[4] Applications extend to geometrical optics, modeling ray propagation and caustics on symplectic manifolds such as the cotangent bundle of the sphere, and to thermodynamics via contact structures on phase spaces defined by equations like du = t ds - p dv.^[4] In modern mathematics, the field has evolved through contributions from V.I. Arnold in the 1960s on singularity theory and minimax principles, Mikhail Gromov in the 1980s with pseudo-holomorphic curves for symplectic topology, and others exploring Lagrangian submanifolds, Floer homology, and connections to mirror symmetry in string theory.^[1] These developments highlight symplectic geometry's role in bridging analysis, topology, and physics, with ongoing research in areas like symplectomorphism rigidity and quantum symplectic invariants.^[1]

Linear algebra foundations

Symplectic vector spaces

A symplectic vector space is a pair (V, \omega), where V is a finite-dimensional vector space over the real numbers \mathbb{R} and \omega: V \times V \to \mathbb{R} is a symplectic form, meaning \omega is a bilinear, skew-symmetric, and non-degenerate bilinear form.^[1] Bilinearity implies that \omega is linear in each argument separately, so for scalars a, b \in \mathbb{R} and vectors u, v, w \in V, \omega(au + bv, w) = a\omega(u, w) + b\omega(v, w) and similarly for the second argument.^[5] Skew-symmetry requires \omega(u, v) = -\omega(v, u) for all u, v \in V, which immediately implies \omega(u, u) = 0 for all u.^[6] Non-degeneracy means that if \omega(u, v) = 0 for all v \in V, then u = 0; equivalently, the associated linear map \tilde{\omega}: V \to V^* given by \tilde{\omega}(u)(v) = \omega(u, v) is an isomorphism, where V^* is the dual space.^[1] The dimension of a symplectic vector space V must be even, denoted as $2n for some positive integer n. To see this, represent \omega with respect to any basis of V; the matrix A of \omega satisfies A^T = -A due to skew-symmetry. Non-degeneracy implies \det A \neq 0. But also \det A = \det A^T = \det(-A) = (-1)^{\dim V} \det A, so if \dim V = m is odd, then (-1)^m = -1, yielding \det A = -\det A and thus \det A = 0, a contradiction. Hence m must be even.^[5] This even-dimensionality arises from the pairing structure inherent in the non-degenerate skew-symmetric form, which naturally decomposes V into pairs of basis vectors.^[1] Every symplectic vector space admits a symplectic basis \{e_1, \dots, e_n, f_1, \dots, f_n\}, also called a Darboux basis, such that

\omega(e_i, e_j) = 0, \quad \omega(f_i, f_j) = 0, \quad \omega(e_i, f_j) = \delta_{ij}

for all i, j = 1, \dots, n, where \delta_{ij} is the Kronecker delta (1 if i = j, 0 otherwise).^[1] With respect to this basis, the matrix of \omega takes the canonical block form

J_{2n} = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix},

where I_n is the n \times n identity matrix, and \omega(u, v) = u^T J_{2n} v for column vectors u, v.^[6] The existence of such a basis follows from non-degeneracy: one can inductively select vectors e_i and corresponding f_i that pair to 1 under \omega while orthogonalizing against previous pairs.^[5] Any two symplectic vector spaces of the same dimension $2n are isomorphic as symplectic spaces, meaning there exists a linear isomorphism \phi: V \to V' such that \omega'(\phi(u), \phi(v)) = \omega(u, v) for all u, v \in V, where (V', \omega') is the second space.^[1] This follows from the existence of symplectic bases in each space, which can be mapped to each other while preserving the canonical form of \omega.^[5] A concrete example is the standard symplectic vector space (\mathbb{R}^{2n}, \omega_0), where vectors are written as (x, y) with x, y \in \mathbb{R}^n, and

\omega_0((x, y), (x', y')) = x \cdot y' - y \cdot x' = \sum_{i=1}^n (x_i y_i' - y_i x_i'),

with \cdot denoting the Euclidean dot product.^[1] Here, the standard basis vectors e_i = (0, \dots, 1, \dots, 0, 0, \dots, 0) (1 in the i-th position of the x-block) and f_i = (0, \dots, 0, \dots, 1, \dots, 0) (1 in the i-th position of the y-block) form a symplectic basis.^[5] This structure models phase space in classical mechanics but is purely algebraic here.^[6]

Symplectic groups and matrices

The symplectic group \mathrm{Sp}(2n, \mathbb{R}) consists of all $2n \times 2n real matrices M that preserve the standard symplectic form on \mathbb{R}^{2n}, satisfying M^T J M = J, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} is the standard symplectic matrix.^[7] This condition ensures that the group acts as linear automorphisms on a symplectic vector space, preserving its bilinear structure.^[8] As a Lie group, \mathrm{Sp}(2n, \mathbb{R}) has dimension n(2n + 1) and is connected and non-compact, with the unitary group \mathrm{U}(n) serving as its maximal compact subgroup.^[8] Every element has determinant 1, so the group lies within \mathrm{SL}(2n, \mathbb{R}) and preserves the standard volume form on \mathbb{R}^{2n}.^[9] When equipped with a compatible positive-definite metric, the symplectic group relates to orthogonal groups through its intersection with \mathrm{O}(2n, \mathbb{R}), yielding the compact real form \mathrm{USp}(2n).^[7] Symplectic matrices admit a canonical form under symplectic similarity transformations, which constrains their Jordan normal form to feature even-sized blocks for non-real eigenvalues and paired structures for real eigenvalues, ensuring preservation of the symplectic pairing. Classification of these matrices often involves Pfaffian invariants, which provide algebraic measures of the symplectic structure analogous to the determinant for symmetric forms, with the Pfaffian satisfying \mathrm{Pf}(M^T J M)^2 = \det(J) = 1. In the case n=1, \mathrm{Sp}(2, \mathbb{R}) is isomorphic to \mathrm{SL}(2, \mathbb{R}), a three-dimensional Lie group generated by rotations \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} and shears \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}, which together span the transformations preserving area in the plane. Over the complex numbers, the symplectic group \mathrm{Sp}(n, \mathbb{C}) consists of $2n \times 2n complex matrices preserving the standard form, forming a simply connected simple Lie group of dimension n(2n + 1) that embeds in \mathrm{SL}(2n, \mathbb{C}).^[10]

Symplectic geometry

Symplectic manifolds

A symplectic manifold is a pair (M, \omega), where M is a smooth manifold of even dimension $2n and \omega is a symplectic form on M, meaning \omega is a closed (d\omega = 0) non-degenerate 2-form.^[11] Non-degeneracy implies that at each point p \in M, the bilinear form \omega_p: T_p M \times T_p M \to \mathbb{R} induces an isomorphism T_p M \to T_p^* M via v \mapsto \omega_p(v, \cdot).^[12] This structure equips M with a rich geometric framework, locally modeled on symplectic vector spaces through coordinate charts.^[13] The symplectic form \omega is compatible with an almost complex structure J on M, satisfying J^2 = -\mathrm{Id} and \omega(Ju, Jv) = \omega(u, v) for all tangent vectors u, v, which defines a Riemannian metric g(u, v) = \omega(u, J u) that is positive definite.^[14] Topologically, since \omega is closed, it represents a cohomology class [\omega] \in H^2(M; \mathbb{R}). For compact connected symplectic manifolds, non-degeneracy ensures [\omega] \neq 0, as the top power \omega^n induces a non-vanishing volume form on the orientable manifold M whose integral is positive.^[15] For quantizability in geometric quantization, an integrality condition requires that [\omega / 2\pi] \in H^2(M; \mathbb{Z}), allowing the construction of a prequantum line bundle.^[16] A canonical example of a symplectic manifold arises as the cotangent bundle T^* Q of any smooth manifold Q, equipped with the tautological 1-form \theta (defined by \theta_p(\xi) = \alpha(\mathrm{d}\pi_p(\xi)) for p \in T^* Q, \xi \in T_p (T^* Q), \alpha \in T^*_{\pi(p)} Q) and the induced symplectic form \omega = -\mathrm{d}\theta.^[17] In classical mechanics, phase spaces are modeled as such cotangent bundles, where coordinates separate positions on Q and momenta in the fibers.^[18] Another fundamental example is the 2-sphere S^2, viewed as the complex projective line \mathbb{CP}^1 with the Fubini-Study symplectic form (up to scaling), whose cohomology class generates H^2(S^2; \mathbb{R}).^[11]

Symplectic forms on manifolds

A symplectic form \omega on a smooth manifold M of even dimension $2n is a differential 2-form that is closed, meaning d\omega = 0, skew-symmetric, and non-degenerate at every point p \in M.^[1] The closedness condition implies that \omega is locally exact, as the Poincaré lemma guarantees the existence of a 1-form \lambda such that \omega = d\lambda in any contractible neighborhood, though a global primitive may not exist, corresponding to the de Rham cohomology class [\omega] \in H^2_{dR}(M; \mathbb{R}).^[1] Non-degeneracy ensures that the map v \mapsto \iota_v \omega from the tangent space T_p M to the cotangent space T_p^* M is an isomorphism for each p, meaning that if \iota_v \omega = 0, then v = 0.^[1] This property distinguishes symplectic forms from general closed 2-forms and endows the manifold with a rich geometric structure.^[1] The symplectic form induces key operations on the manifold. For a smooth function f: M \to \mathbb{R}, the Hamiltonian vector field X_f is defined by the contraction equation \iota_{X_f} \omega = -df, which uniquely determines X_f due to non-degeneracy.^[1] The flow of X_f consists of symplectomorphisms, preserving \omega.^[1] The Poisson bracket of two functions f and g is given by \{f, g\} = \omega(X_f, X_g) = X_f g = -X_g f, endowing the space of smooth functions with a Lie algebra structure that satisfies bilinearity, skew-symmetry, the Leibniz rule, and the Jacobi identity.^[1] This bracket captures the dynamics of Hamiltonian systems and is fundamental to integrability criteria.^[1] Symplectic reduction, or Marsden–Weinstein reduction, provides a method to construct new symplectic manifolds from symmetries. For a symplectic manifold (M, \omega) with a Hamiltonian action of a compact Lie group G admitting a moment map \mu: M \to \mathfrak{g}^*, if the action is free on the level set \mu^{-1}(\eta) for \eta \in \mathfrak{g}^*, the quotient \mu^{-1}(\eta)/G inherits a reduced symplectic form \omega_\eta such that the projection \pi: \mu^{-1}(\eta) \to \mu^{-1}(\eta)/G satisfies \pi^* \omega_\eta = i^* \omega, where i is the inclusion.^[1] For a k-dimensional torus action preserving \omega, the reduced manifold has dimension \dim M - 2k.^[19] This procedure, detailed in the Marsden–Weinstein theorem, simplifies the study of symmetric systems by reducing the phase space dimension while preserving symplectic geometry.^[1] The powers of the symplectic form yield the Liouville volume form \frac{\omega^n}{n!} on M^{2n}, which is non-vanishing due to non-degeneracy and provides a natural orientation and measure invariant under symplectomorphisms.^[1] The associated Liouville measure of a subset U \subset M is \int_U \frac{\omega^n}{n!}, which is preserved by the Hamiltonian flow, reflecting Liouville's theorem in classical mechanics.^[1] A canonical example is the standard symplectic form on \mathbb{R}^{2n} with coordinates (x_1, \dots, x_n, y_1, \dots, y_n), given by \omega_0 = \sum_{i=1}^n dx_i \wedge dy_i, which is preserved by the linear symplectic group and serves as a model for general forms via pullbacks under diffeomorphisms.^[1] Under a symplectomorphism f: (M, \omega) \to (N, \omega'), the pullback satisfies f^* \omega' = \omega, ensuring compatibility of structures.^[1]

Darboux theorem

The Darboux theorem asserts that if (M, \omega) is a symplectic manifold of dimension $2nandp \in Mis any point, then there exists a neighborhoodUofptogether with local coordinates(x^1, \dots, x^n, y^1, \dots, y^n)centered atp$ such that the symplectic form restricts to the standard form

\omega|_U = \sum_{i=1}^n \, dx^i \wedge dy^i.

^[20] These coordinates are called Darboux coordinates. To sketch the proof, begin with the non-degeneracy of \omega at p, which allows selection of a basis \{e_1, \dots, e_n, f_1, \dots, f_n\} for T_p M such that \omega(e_i, f_j) = \delta_{ij} and \omega(e_i, e_j) = \omega(f_i, f_j) = 0 for all i,j.^[21] Extend these vectors to local coordinate vector fields X_i, Y_i on a neighborhood V of p using the flows of the Hamiltonian vector fields associated to the linear functions defined by the dual basis on T_p^* M; this yields coordinates (x^i, y^i) where \omega = \sum dx^i \wedge dy^i + \beta for some closed 2-form \beta of order higher than 1.^[1] To eliminate \beta, apply Moser's method: construct a time-dependent vector field whose flow deforms the form to the standard one via a homotopy connecting \omega to \sum dx^i \wedge dy^i, leveraging the closedness of both forms and the exactness of their difference on the neighborhood.^[21] A key implication of the theorem is that all symplectic manifolds of the same dimension $2nare locally symplectomorphic to(\mathbb{R}^{2n}, \sum dx^i \wedge dy^i)$; thus, the only local invariant of a symplectic structure is its dimension.^[1] Variants of the Darboux theorem include generalizations to higher-degree forms, such as in multi-symplectic geometry, and complex versions for Kähler manifolds, where local holomorphic coordinates exist such that the Kähler form \omega = i \partial \overline{\partial} \phi arises from a real-valued potential function \phi.^[1] A concrete example arises on the cotangent bundle T^* Q of a manifold Q, equipped with the canonical symplectic form \omega = dq^i \wedge dp_i in the natural coordinates (q^i, p_i); these are Darboux coordinates satisfying the standard form globally on T^* Q.^[1]

Applications in physics

Hamiltonian mechanics

In Hamiltonian mechanics, the phase space of a classical mechanical system is modeled as the cotangent bundle T^*Q of the configuration space Q, equipped with the canonical symplectic form \omega = dq \wedge dp. This structure provides a geometric framework for describing the evolution of the system under a Hamiltonian function H: T^*Q \to \mathbb{R}, which represents the total energy in terms of generalized coordinates q and momenta p. The symplectic form ensures that the phase space inherits properties essential for preserving the underlying dynamics, such as non-degeneracy and closedness, which are crucial for the formulation of equations of motion.^[22]^[1] The dynamics are governed by Hamilton's equations, derived from the Hamiltonian vector field X_H defined by \omega(X_H, \cdot) = -dH. In canonical coordinates, these take the form \dot{q} = \frac{\partial H}{\partial p} and \dot{p} = -\frac{\partial H}{\partial q}, generating a flow that evolves points in phase space over time. This flow is a symplectomorphism, meaning it preserves the symplectic form \omega, as the Lie derivative satisfies \mathcal{L}_{X_H} \omega = 0. Consequently, Liouville's theorem follows, stating that the phase space volume is conserved under this flow, implying incompressible trajectories in the symplectic manifold.^[22]^[23]^[1] For integrable systems, action-angle variables offer a canonical transformation to coordinates (J, \theta), where J are the action variables (constant along trajectories) and \theta are angle variables, with the Hamiltonian depending only on J as H = H(J). This transformation simplifies the equations to \dot{J} = 0 and \dot{\theta} = \frac{\partial H}{\partial J}, revealing quasi-periodic motion on tori in phase space and facilitating the analysis of near-integrable perturbations. Symmetries in the system, represented by canonical transformations that leave H invariant, generate conserved quantities through Noether's theorem in its Hamiltonian form: if a symmetry function f satisfies the Poisson bracket \{f, H\} = 0, then f is constant along the flow, linking continuous symmetries to integrals of motion via the symplectic structure.^[22]^[23] A representative example is the one-dimensional harmonic oscillator, with Hamiltonian H = \frac{p^2 + q^2}{2} on the phase space \mathbb{R}^2 endowed with the standard symplectic form \omega = dq \wedge dp. The trajectories are periodic orbits forming circles centered at the origin, with frequency \omega = 1, and the action variable J = H parameterizes the radius, illustrating how the symplectic geometry organizes the bounded, elliptical motion in phase space.^[22]

Symplectic integrators

Symplectic integrators are numerical methods designed to approximate the time evolution of Hamiltonian systems while preserving the underlying symplectic structure, which ensures long-term qualitative behavior akin to the continuous dynamics.^[24] Standard integrators, such as the explicit Euler method, often fail to conserve energy over long times, leading to artificial dissipation or growth in simulations of oscillatory systems.^[25] In contrast, symplectic integrators maintain the symplectomorphism property of the exact flow, resulting in bounded energy errors and improved stability for extended integrations.^[24] A symplectic integrator of step size h is a map \psi_h: M \to M on the phase space M that is itself a symplectomorphism and approximates the true Hamiltonian flow \phi_h up to local error \mathcal{O}(h^{p+1}) for order p.^[24] This preservation means the numerical solution respects the symplectic form \omega, satisfying \psi_h^* \omega = \omega.^[25] Prominent examples include the Verlet algorithm, introduced by Loup Verlet in 1967 for molecular simulations of Lennard-Jones fluids, which applies to separable Hamiltonians H(q,p) = T(p) + V(q).^[26] The velocity Verlet scheme updates positions and momenta alternately:

p_{n+1/2} = p_n - \frac{h}{2} \nabla_q V(q_n), \quad q_{n+1} = q_n + h \nabla_p T(p_{n+1/2}),

p_{n+1} = p_{n+1/2} - \frac{h}{2} \nabla_q V(q_{n+1}),

where p_{n+1/2} is an intermediate momentum; this method is symplectic and second-order accurate.^[27] Another basic example is the symplectic Euler method, a partitioned Runge-Kutta scheme that updates position implicitly and momentum explicitly (or vice versa) for separable systems:

p_{n+1} = p_n - h \nabla_q V(q_n), \quad q_{n+1} = q_n + h \nabla_p T(p_{n+1}),

which is first-order but preserves symplecticity exactly.^[25] Key properties of symplectic integrators include backward error analysis, where the numerical solution is interpreted as the exact flow of a perturbed Hamiltonian \tilde{H} = H + h H_1 + h^2 H_2 + \cdots, with the perturbation bounded independently of h for long times.^[24] This leads to near-conservation of the original Hamiltonian, with oscillations rather than drift. Additionally, symplectic methods exactly preserve quadratic invariants, such as linear and angular momentum in particle systems, due to the relation between symplecticity and the conservation of quadratic first integrals under the numerical map.^[28] Higher-order symplectic integrators are constructed via composition methods, such as Strang splitting for separable Hamiltonians, which combines flows of kinetic and potential parts:

\psi_h = \phi_{T, h/2} \circ \phi_{V, h} \circ \phi_{T, h/2},

yielding second-order accuracy while remaining symplectic; higher orders follow from multi-stage compositions. Variational integrators, derived from discrete Lagrangian mechanics using generating functions, provide another approach: for a discrete Lagrangian L_d(q_n, q_{n+1}) approximating the action integral, the update is given by the discrete Euler-Lagrange equations, ensuring symplecticity and momentum conservation.^[29] These methods find essential applications in simulating N-body problems in astronomy, where they bound error growth over billions of orbital periods, as in solar system dynamics.^[30] In molecular dynamics, symplectic integrators enable efficient long-time simulations of biomolecular conformations by preserving structural invariants and reducing computational artifacts.^[31] Unlike non-symplectic schemes, they exhibit linear error growth in time, making them ideal for chaotic or resonant systems.^[24]

References

[1]
[PDF] Lectures on Symplectic Geometry
A symplectic form is a 2-form satisfying an algebraic condition – nondegeneracy – and an analytical condition – closedness. In Lectures 1 and 2 we define ...
[2]
http://books.google.co.uk/books?id=zmzKSP2xTtYC
[3]
[PDF] Early History of Symplectic Geometry
Before 1938, the symplectic group was known as complex or abelian linear group. In 1938, Hermann Weyl proposed to change the name and named it symplectic.
[4]
[PDF] Symplectic Geometry and its Applications
Symplectic geometry is the mathematical apparatus of such areas of physics as classical mechanics, geometrical optics and thermodynamics. Whenever the.
[5]
[PDF] A GUIDE TO SYMPLECTIC GEOMETRY - Williams College
May 6, 2022 · A symplectic vector space is a pair (V, o), where: • V is a vector space, and;. • o: V × V → R is a non-degeneratea skew-symmetric bilinear form ...
[6]
[PDF] notes on symplectic topology - The University of Chicago
Mar 5, 2025 · Symplectic vector spaces. Definition 1.1 (Symplectic vector space). A symplectic vector space is a real, finite di- mensional vector space V ...
[7]
symplectic group in nLab
### Summary of Symplectic Group Sp(2n, R) from nLab
[8]
[PDF] An Introduction to Lie Groups and Symplectic Geometry
Jul 23, 2018 · Show that Sp(n, R), as defined in the text, is indeed a Lie subgroup ... Verify that Sp(n) is a connected Lie group of dimension 2n2 + n.
[9]
[PDF] 4 Symplectic groups
It is isomorphic to the factor group Sp 2n F Sp 2n F . Z , where Z is the group of non-zero scalar matrices. Proposition 4.3 (a) Sp 2n F is a subgroup of SL 2n ...
[10]
[PDF] gl(n, C) is the Lie - People
The symplectic group Sp(n, C) is a connected, simply-connected, simple Lie group of dimen- sion n(2n + 1) and rank n, with Sp(n, C) ⊂ SL(2n, C). Warning: Some ...
[11]
[PDF] Remarks on Symplectic Geometry - arXiv
A symplectic manifold is an even-dimensional smooth manifold equipped with a closed non-degenerate two form.
[12]
https://arxiv.org/pdf/math/9811167v1
[13]
Symplectic Manifolds
the basic features of symplectic geometry, By a symplectic manifold we mean. an even-dimensional differentiable (COO) manifold M2n together with a global. 2- ...
[14]
[PDF] Distributions associated to almost complex structures on symplectic ...
Feb 6, 2020 · For any positive compatible almost complex structure J on a 2n- dimensional symplectic manifold (M,ω), and for any point x ∈ M, one has dim ...
[15]
[PDF] Topological complexity of monotone symplectic manifolds - arXiv
May 3, 2024 · Let us now consider a symplectic manifold (M,ω). Even if it is not aspherical as a space, it sometimes admits an aspherical symplectic form (see ...
[16]
[PDF] Geometric Quantization - arXiv
Definition 1 A symplectic manifold (M,ω) is said to be quantizable if ω satisfies the integrality condition, i.e. if the class of (2π~)−1ω in H2(M, R) lies in ...
[17]
Symplectic Form on the Cotangent Bundle - SpringerLink
'Symplectic Form on the Cotangent Bundle' published in 'Lectures on Symplectic Geometry'
[18]
[PDF] arXiv:1112.0830v1 [math.DG] 5 Dec 2011
Dec 5, 2011 · choice of coordinates, so ω is a symplectic form on the cotangent bundle T∗X, called the canonical symplectic form. Now, let S be any k ...
[19]
[PDF] reduction of symplectic manifolds with symmetry
This symplectic manifold is important in fluid mechanics. See Arnold [2] and Ebin-Marsden [6]. Here the manifolds are Fréchet. Properly, one should use ...Missing: seminal | Show results with:seminal
[20]
Sur le problème de Pfaff - EuDML
Sur le problème de Pfaff. G. Darboux · Bulletin des Sciences Mathématiques et Astronomiques (1882). Volume: 6, Issue: 1, page 49-68; ISSN: 1155-8431 ...
[21]
[PDF] SYMPLECTIC GEOMETRY - Mathematics
The proof of the Darboux theorem as given above is due to Moser [37]. This method of proof has been further refined by Weinstein obtaining a standard form of a ...
[22]
Mathematical Methods of Classical Mechanics | SpringerLink
In stockIn this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics.
[23]
[PDF] 4. The Hamiltonian Formalism
Note that the three constants of motion, L, A and H form a closed algebra under the Poisson bracket. Noether's theorem tells us that the conservation of L and H ...
[24]
Geometric Numerical Integration - SpringerLink
Examples and Numerical Experiments. Ernst Hairer, Gerhard Wanner, Christian Lubich. Pages 1-26. Numerical Integrators. Ernst Hairer, Gerhard Wanner, Christian ...
[25]
[PDF] Lecture 2: Symplectic integrators
by Hairer, Lubich & Wanner. 3. Page 4. 2 Symplectic Runge–Kutta methods. An s-stage Runge–Kutta method, applied to an initial value problem ˙y = f(t, y), y(t0) ...
[26]
[PDF] Geometric numerical integration illustrated by the St ormer/Verlet ...
This article illustrates concepts and results of geometric numerical integration on the important example of the St ormer/Verlet method. It thus presents a ...
[27]
On quadratic invariants and symplectic structure
We show that the theorems of Sanz-Serna and Eirola and Sanz-Serna concerning the symplecticity of Runge-Kutta and Linear Multistep methods, respectively, f.
[28]
Discrete mechanics and variational integrators | Acta Numerica
Jan 9, 2003 · This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles.
[29]
Symplectic variable step size integration for N-body problems
This method can be applied to 2-body central force interactions by partitioning them into distance classes and smoothly decomposing the potential energy.
[30]
Symplectic Molecular Dynamics Simulations on Specially Designed ...
We have developed a new, symplectic MD integration method that enables longer time steps, and therefore longer MD simulations are possible in the same run time.