AI 中文总结
本文将Bateman与Caldirola--Kanai形式体系的经典对应关系，推广到指定FLRW背景下的均匀有质量标量场，构造含时正则变换实现二者映射，还发现特定幂律背景下Bateman哈密顿量守恒。
AI 中文摘要
耗散方程在Bateman和Caldirola--Kanai（CK）形式体系中存在不同的变分描述。本文将二者的经典对应关系推广到指定空间平坦Friedmann--Lemaître--Robertson--Walker（FLRW）背景上的均匀正则标量场，其中宇宙膨胀产生含时阻尼系数$3H(t)$。通过乘子作用量可得到Klein--Gordon方程以及包含$-3\
英文摘要
Dissipative equations admit distinct variational descriptions in the Bateman and Caldirola--Kanai (CK) formalisms. The classical correspondence between them is extended to a homogeneous canonical scalar field on a prescribed spatially flat Friedmann--Lemaître--Robertson--Walker (FLRW) background, where the expansion produces the time-dependent damping coefficient $3H(t)$. A multiplier action yields the Klein--Gordon equation and a complementary anti-damped equation containing the term $-3\dot{H}(t)χ$. A first-order Bateman Lagrangian derived from the same multiplier action reproduces this physical--auxiliary pair for a general potential. Specializing to a free massive field gives the Bateman and doubled CK Lagrangians and Hamiltonians used in the canonical comparison. The factors $a^{3}(t)$ and $a^{-3}(t)$ generate the damped and anti-damped CK sectors, respectively. An explicit time-dependent canonical transformation, generated by a function linear in the Bateman momenta, maps the complete doubled CK system to the Bateman system. For this point transformation, the terms proportional to $\dot{H}(t)$ are required for Hamiltonian equivalence. In rotated variables, the Bateman scalar-field Hamiltonian takes the difference form $H_{B,\mathrm{SF}} = E_{u} - E_{v}$. It is conserved for constant $H$ and generally varies with time otherwise. For the power-law background $a(t) \propto t^{p}$, however, a correlated family at $p = 2/3$ has conserved $H_{B,\mathrm{SF}}$ despite the time dependence of $H(t)$. These results concern classical homogeneous fields on a prescribed FLRW background and exclude the gravitational phase space.
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