A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
arxiv(2024)
摘要
Time and again, non-conventional forms of Lagrangians with non-quadratic
velocity dependence have found attention in the literature. For one thing, such
Lagrangians have deep connections with several aspects of nonlinear dynamics
including specifically the types of the Liénard class; for another, very
often the problem of their quantization opens up multiple branches of the
corresponding Hamiltonians, ending up with the presence of singularities in the
associated eigenfunctions. In this article, we furnish a brief review of the
classical theory of such Lagrangians and the associated branched Hamiltonians,
starting with the example of Liénard-type systems. We then take up other
cases where the Lagrangians depend upon the velocity with powers greater than
two while still having a tractable mathematical structure, while also
describing the associated branched Hamiltonians for such systems. For various
examples, we emphasize upon the emergence of the notion of momentum-dependent
mass in the theory of branched Hamiltonians.
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