Abstract
Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of timecrystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time-crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete one dimensional model of a Hamiltonian chain. It is obtained by a reduction from the Q-ball Lagrangian that describes time dependent nontopological solitons. We show that a time crystal appears as a minimum energy domain wall configuration, along the chain.
Highlights
Hamiltonian time crystalsHamilton’s equation describes energy conserving dynamics on a 2N dimensional symplectic manifold M; for background on geometric mechanics see e.g. [25]
On a compact, closed manifold a minimum energy configuration is a critical point of the Hamiltonian H
It is apparent that the present remarks are merely an invitation for a judicious mathematical investigation, and we propose that the methods of equivariant Morse theory [27,28,29,30] can be adopted to develop a mathematical framework for understanding Hamiltonian time crystals; we plan to return to this in a future research and we proceed to exemplify our general remarks by a detailed analysis of two examples where timecrystalline dynamics appears in a familiar physical context
Summary
Hamilton’s equation describes energy conserving dynamics on a 2N dimensional symplectic manifold M; for background on geometric mechanics see e.g. [25]. We define a time crystal to be a minimum energy solution of Hamilton’s equation (2.2), (2.4) with a non-trivial t-dependence that we assume is periodic φa(t + T ) = φa(t). For each regular values gi in (2.6) we restrict the non-degenerate symplectic twoform (2.1) to the corresponding submanifold Mg. The two-form ωg is closed but in general the matrix ωagb is degenerate with a (n − s) dimensional kernel. Whenever λicr(φcr) = 0 the minimum energy solution φacr can be employed as an initial value to a timecrystalline solution of Hamilton’s equation (2.4). Note that the Lagrange multipliers λicr are t-independent, their values for all t are determined by (2.11) in terms of the initial values φacr This follows immediately, since both H(φ) and Gi(φ) are by construction t-independent along any Hamiltonian trajectory. It is apparent that the present remarks are merely an invitation for a judicious mathematical investigation, and we propose that the methods of equivariant Morse theory [27,28,29,30] can be adopted to develop a mathematical framework for understanding Hamiltonian time crystals; we plan to return to this in a future research and we proceed to exemplify our general remarks by a detailed analysis of two examples where timecrystalline dynamics appears in a familiar physical context
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