[Paper Review] Noninertial Symmetry Group of Hamilton's Mechanics

Stephen G. Low|arXiv (Cornell University)|Mar 25, 2009

Mathematical Analysis and Transform Methods11 references3 citations

TL;DR

This paper derives Hamilton's equations from a unified noninertial symmetry group, showing they are invariant under the semidirect product $\mathcal{S}p(2n)\otimes_{s}\mathcal{H}(n)$, where $\mathcal{S}p(2n)$ is the real symplectic group and $\mathcal{H}(n)$ is a Weyl-Heisenberg group parameterized by velocity, force, and power. The key result is that Hamiltonian mechanics possesses a noninertial symmetry extending beyond inertial frames, implying simultaneity depends on both inertial and noninertial states, with profound implications for relativistic and quantum theories.

ABSTRACT

We present a new derivation of Hamilton's equations that shows that they have a symmetry group Sp(2n) *s H(n). Sp(2n) is the symplectic group and H(n) is mathematically a Weyl-Heisenberg group that is parameterized by velocity, force and power where power is the central element of the group. We present a new derivation of Hamilton's equations that shows that they have a symmetry group Sp(2n) *s H(n). The group Sp(2n) is the real noncompact symplectic group and H(n) is mathematically a Weyl-Heisenberg group that is parameterized by velocity, force and power where power is the central element of the group. The homogeneous Galilei group SO(n) *s A(n), where the special orthogonal group SO(n) is parameterized by rotations and the abelian group A(n)is parameterized by velocity, is the inertial subgroup.

Motivation & Objective

• To establish a deeper geometric and algebraic foundation for Hamilton's equations by identifying their full symmetry group.
• To extend the principle of relativity beyond inertial frames by incorporating noninertial dynamics into the formalism of Hamiltonian mechanics.
• To show that the symmetry group $\mathcal{S}p(2n)\otimes_{s}\mathcal{H}(n)$ naturally includes both symplectic invariance and affine transformations, unifying inertial and noninertial states.
• To explore the implications of this symmetry for relativistic and quantum mechanics, particularly in the context of reciprocal relativity and unitary representations.
• To challenge the conventional privileging of inertial frames in classical mechanics by demonstrating that Hamilton's equations are equally valid in noninertial states when the appropriate Hamiltonian is used.

Proposed method

• The extended phase space $\mathbb{P} = \mathbb{R}^{2n+2}$ is defined with coordinates $\{z^a\} = \{p^i, q^i, e, t\}$, where $e$ is energy and $t$ is time.
• A symplectic metric $\omega = \delta_{ij} dp^i \wedge dq^j - de \wedge dt$ and a degenerate orthogonal line element $\gamma^\circ = dt^2$ are introduced on this space.
• Diffeomorphisms $\rho: \mathbb{P} \to \mathbb{P}$ are required to preserve both the symplectic form and the degenerate metric, leading to constraints on the Jacobian matrix $\Gamma = \partial \rho^a / \partial z^b$.
• The invariance conditions $\Gamma^T \zeta \Gamma = \zeta$ and $\Gamma^T \eta^\circ \Gamma = \eta^\circ$ are derived, where $\zeta$ and $\eta^\circ$ are the metric components.
• The solution space of such $\Gamma$ is shown to be isomorphic to the group $\mathcal{H}\mathcal{S}p(2n) \simeq \mathcal{S}p(2n) \otimes_s \mathcal{H}(n)$, with $\mathcal{H}(n)$ parameterized by velocity, force, and power.
• The resulting differential equations are identified as Hamilton's equations, proving they are the unique equations invariant under this noninertial symmetry group.

Experimental results

Research questions

• RQ1 What is the complete symmetry group of Hamilton's equations when noninertial frames are included?
• RQ2 How does the Weyl-Heisenberg group $\mathcal{H}(n)$, parameterized by velocity, force, and power, arise naturally in the Hamiltonian formalism?
• RQ3 Can the Galilean group be understood as a subgroup of a larger noninertial symmetry group in Hamiltonian mechanics?
• RQ4 How does the invariance of Hamilton's equations under $\mathcal{S}p(2n)\otimes_s\mathcal{H}(n)$ affect the concept of simultaneity in classical mechanics?
• RQ5 What are the implications of this noninertial symmetry for the formulation of relativistic and quantum theories?

Key findings

• Hamilton's equations are shown to be invariant under the noninertial symmetry group $\mathcal{S}p(2n)\otimes_s\mathcal{H}(n)$, where $\mathcal{S}p(2n)$ is the real symplectic group and $\mathcal{H}(n)$ is a Weyl-Heisenberg group parameterized by velocity, force, and power.
• The homogeneous Galilei group $\mathcal{E}(n) \simeq \mathcal{SO}(n)\otimes_s\mathcal{A}(n)$ is identified as the inertial subgroup of $\mathcal{H}\mathcal{S}p(2n)$, with $\mathcal{SO}(n) \subset \mathcal{S}p(2n)$ and $\mathcal{A}(n) \subset \mathcal{H}(n)$.
• The degenerate orthogonal line element $dt^2$ is invariant under the affine group $\mathcal{IGL}(2n+1,\mathbb{R})$, and the intersection with $\mathcal{S}p(2n+2)$ yields the full symmetry group $\mathcal{H}\mathcal{S}p(2n)$.
• The symmetry group unifies inertial and noninertial dynamics, implying that simultaneity is not absolute but depends on the relative inertial and noninertial state of the observer.
• The central extension of the inhomogeneous Hamilton group $\mathcal{IH}a(n)$ leads to a unitary representation on a Hilbert space of the form $\mathcal{H} \otimes L^2(\mathbb{R}^{n+1}, \mathbb{C})$, with wave functions depending only on $\psi(q,t)$ or $\psi(p,t)$, not on full phase space variables.
• The formalism suggests a natural framework for reciprocal relativity and quantum theory, where the noninertial symmetry has profound implications beyond classical mechanics.