Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler-Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler-Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed. Q s f (e.g., [7][8][9]); however, so far, the problem has not yet been fully solved forDy(x) = 0. Therefore, the main objective of this paper is to establish the Lagrangian formalism for the ODEs of Q s f and derive new Lagrangians for these equations.The existence of Lagrangians is guaranteed by the Helmholtz conditions [10], which can also be used to derive the Lagrangians. In general, the Helmholtz conditions allowed for the existence of Lagrangians for the ODEs of the formD o y(x) = 0, but they prevent the Lagrangian formalism forDy(x) = 0 because of the presence of the term with the first order derivative (e.g., [7,11]). The procedure of finding the Lagrangians is called the inverse (or Helmholtz) problem of the calculus of variations and there are different methods to solve this problem (e.g., [12][13][14][15]). In this paper, the Helmholtz problem is solved differently and new Lagrangians for the ODEs of Q s f are derived. A special emphasis is given to the validity of the Helmholtz conditions for the derived Lagrangians. We also explore applications of the obtained results to the Airy, Bessel, Legendre and Hermite equations.There are two main families of Lagrangians, the so-called standard and non-standard Lagrangians. The standard Lagrangians (SLs) are typically expressed as the difference between terms that can be identified as the kinetic and potential energy [14]. A broad range of different methods exists, and these methods were developed to obtain the SLs for both linear and non-linear ODEs, and PDEs. Some methods involve the concept of the Jacobi last multiplier [16][17][18] or use fractional derivatives [19], and others are based on different transformations that allow deriving the SLs for the conservative and non-conservative physical systems described by either linear...

An intriguing question in the context of dynamics arises: could a moon possess a moon itself? Such a configuration does not exist in the solar system, although this may be possible in theory. Kollmeier & Raymond determined the critical size of a satellite necessary to host a long-lived subsatellite, or submoon. However, the orbital constraints for these submoons to exist are still undetermined. Domingos et al. indicated that moons are stable out to a fraction of the host planet's Hill radius R H,p, which in turn depend on the eccentricity of its host’s orbit. Motivated by this, we simulate systems of exomoons and submoons for 105 planetary orbits, while considering many initial orbital phases to obtain the critical semimajor axis in terms of R H,p or the host satellite’s Hill radius R H,sat, respectively. We find that, assuming circular coplanar orbits, the stability limit for an exomoon is 0.40 R H,p and for a submoon is 0.33 R H,sat. Additionally, we discuss the observational feasibility of detecting these subsatellites through photometric, radial velocity, or direct imaging observations using the Neptune-sized exomoon candidate Kepler 1625b-I and identify how stability can shape the identification of future candidates.

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

Satellites of extrasolar planets, or exomoons, are on the frontier of detectability using current technologies and theoretical constraints should be considered in their search. In this Letter, we apply theoretical constraints of orbital stability and tidal migration to the six candidate Kepler Object of Interest (KOI) systems proposed by Fox & Wiegert to identify whether these systems can potentially host exomoons. The host planets orbit close to their respective stars and the orbital stability extent of exomoons is limited to only ∼40% of the host planet's Hill radius (∼20 R p). Using plausible tidal parameters from the solar system, we find that four out of six systems would either tidally disrupt their exomoons or lose them to outward migration within the system lifetimes. The remaining two systems (KOI 268.01 and KOI 1888.01) could host exomoons that are within 25 R p and less than ∼3% of the host planet's mass. However, a recent independent transit timing analysis by Kipping found that these systems fail rigorous statistical tests to validate them as candidates. Overall, we find the presence of exomoons in these systems that are large enough for transit timing variation signatures to be unlikely given the combined constraints of observational modeling, tidal migration, and orbital stability. Software to reproduce our results is available in the GitHub repository: Multiversario/satcand.