Sketch of Lagrangian Formalism

Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and integrability according to Liouville

“…According to classical mechanics [4,6], any mechanical system with kinematic constraints, when rewritten through the unconstrained variables, looks like the geodesically moving particle in a curved space. In this work we have done this for the case of an asymmetric rigid body, obtaining the explicit form of the resulting metric (13) in terms of unconstrained variables (8).…”

“…More exactly, this Lagrangian implies geodesic equations in the natural parametrization, see Sect. 6.5 in[6] 4. On the subset of solutions which describe the moviments of a body (they are the solutions that pass through unit element of SO(3)), the four integrals are not independent: E = I −1 ij m i m j , see[5] for the details.…”

Poincaré-Chetaev equations in the Dirac's formalism of constrained systems

“…n. Suppose the "particle"q A was then forced to move on a k -dimensional surface S given by the algebraic equations G α (q A ) = 0. Then equations of motion is known to follow from the modified Lagrangian, where the constraints are taken into account with help of auxiliary variables λ α (t) as follows [1,20]:…”

“…(B) The second possibility is to work with the original variables using the Dirac's formalism [18][19][20]. We should pass to the Hamiltonian formulation introducing the conjugate momenta p A to all original variables q A .…”

Modeling Flow in Porous Media With Double Porosity/Permeability: Mathematical Model, Properties, and Analytical Solutions

“…Variational principles are not just limited to quasi-static or steady-state problems. Two famous variational approachesthe Lagrangian and the Hamiltonian formalisms-led to significant developments in studying timedependent phenomena in areas such as classical mechanics [Deriglazov, 2016], dynamics [Rosenberg, 1977], and quantum mechanics [Griffiths and Schroeter, 2018].…”