A geometric approach to the generalized Noether theorem

Abstract

We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented by Zhang P-M et al (2020 Eur. Phys. J. Plus 135 223). Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether’s theorem; it automatically yields an inverse Noether theorem; it applies also to a large class of dissipative systems; and finally, it allows for a much larger class of symmetries than just scaling transformations which form a Lie algebra, and are thus amenable to algebraic treatments.

A New Action for Cosmology

“…In this form it is simple to see that the Lagrangian is linearly proportional to y, and hence changing y by a factor will rescale the action, but leave the equations of motion for the variables q, q and S unchanged. This is a example of dynamical similarity [18,19]. Dynamical similarities are symmetries of a theory that are not standard canonical transformations [19], under which the equations of motion for an autonomous system of physical observables are retained despite the rescaling of some quantities such as the Lagrangian, Hamiltonian and symplectic structure.…”

From Geometry to Coherent Dissipative Dynamics in Quantum Mechanics

“…This is based on the symplectic formulation of quantum mechanics and due to the analogy with the geometric description of classical dissipative systems: In classical mechanics, one can describe a wide class of dissipative systems by referring to the contactification of the symplectic phase space and then using contact Hamiltonian systems to define the dynamics. It has been shown that this approach, when applicable, provides several positive features, such as relying on canonical variables and producing a generalization of canonical transformations [7], enabling an extension of both Liouville and Noether's theorems to the dissipative case [8][9][10][11][12][13], and providing a description in terms of variational principles [14][15][16][17][18][19][20] together with a natural route to field theories with dissipation [21].…”

“…To conclude the kinematical analysis of n-level systems, let us recall that CP(H 0 ) is a Kähler manifold [25,28,31]; in fact, considering the homogeneous coordinates in Equation (8), one can introduce the 1-form of the following.…”

From geometry to coherent dissipative dynamics in quantum mechanics

“…This is based on the symplectic formulation of quantum mechanics and on the analogy with the geometric description of classical dissipative systems: in classical mechanics one can describe a wide class of dissipative systems by recurring to the contactification of the symplectic phase space and then using contact Hamiltonian systems to define the dynamics. It has been shown that this approach, when applicable, provides several positive features, such as the fact of relying on canonical variables and producing a generalization of canonical transformations [7], enabling an extension of both Liouville and Noether's theorems to the dissipative case [5,6,8,23,27,37], and providing a description in terms of variational principles [10,29,30,39,[53][54][55], together with a natural route to field theories with dissipation [26].…”