Abstract In this contribution, we propose a new framework to derive energy-preserving numerical schemes based on the variational principle for Hamiltonian mechanics. We focus on Noether’s theorem, which shows that the symmetry with respect to time translation gives the energy conservation law. By reproducing the calculation of the proof of Noether’s theorem after discretization using the summation by parts and the discrete gradient, we obtain the scheme and the corresponding discrete energy at the same time. The significant property of efficiency is that the appropriate choice of the discrete gradient makes our schemes explicit if the Hamiltonian is separable.

In this paper, we have investigated Noether symmetries of the Lagrangian of Kantowski–Sachs spacetime. The associated Lagrangian of the Kantowski–Sachs metric is used to derive the set of determining equations. Solving the determining equations for several values of the metric functions, it is observed that the Kantowski–Sachs spacetime admits the Noether algebra of dimensions 5, 6, 7, 8, 9, and 11. A comparison of the obtained Noether symmetries with Killing and homothetic vectors is also presented. With the help of Noether’s theorem, we have presented the expressions for conservation laws corresponding to all Noether symmetries. It is observed that the positive energy condition is satisfied for most of the obtained metrics.

Spacetime metamaterials are opening new regimes of light-matter interactions based on the breaking of temporal and spatial symmetries, as well as intriguing concepts associated with synthetic motion. In this work, we investigate the continuous spatiotemporal translation symmetry of spacetime metamaterials with uniform modulation velocity. Using Noether's theorem, we demonstrate that such symmetry entails the conservation of the energy momentum. We highlight how energy-momentum conservation imposes constraints on the range of allowed light-matter interactions within spacetime metamaterials, as illustrated with examples of the collision of electromagnetic and modulation pulses. Furthermore, we discuss the similarities and differences between the conservation of energy-momentum and relativistic effects. We believe that our work provides a step forward in clarifying the fundamental theory underlying spacetime metamaterials.

continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformati Time… The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum, and temporal translation symmetry (i.e. homogeneity of time) gives rise to conservation of energy. The following table summarizes some fundamental symmetries and the associated conserved quantity.

This work presents a formulation of mechanics and thermodynamics in which the fundamental evolution parameter is not physical time 𝑡, but a cycle count 𝑁. Time is introduced as a derived quantity through a local frequency 𝑓, via the relation 𝑑𝑡=𝑑𝑁/𝑓, enabling a systematic rewriting of temporal derivatives and physical equations in terms of the cycle parameter. From this postulate, kinematic quantities (velocity and acceleration), dynamical variables (momentum, force, work, and energy), and thermodynamic relations (first and second laws, entropy, and temperature) are reconstructed step by step. A distinction is established between a homogeneous regime with constant frequency, where the formalism is exactly equivalent to standard physics, and a non-homogeneous regime with variable frequency, where additional terms arise and modify both dynamics and thermodynamic evolution. A variational formulation is developed directly in cycle space, defining a frequency-dependent Lagrangian and allowing the application of the Noether's theorem to analyze the relation between symmetries and conserved quantities. It is shown that conservation laws are recovered in the homogeneous case, while in the variable-frequency regime they reflect the non-uniformity of the local evolution rate. The formalism thus provides a coherent mathematical framework in which time can be interpreted as an emergent quantity associated with a local frequency field, maintaining full consistency with classical physics i

Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time-translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. Time-translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throug Time… does not depend on the variable t {\displaystyle t} . Of course, this quantity describes the total energy whose conservation is due to the time-translation invariance of the equation of motion. By studying the composition of symmetry transformations, e.g. of geometric objects, one reaches the conclusion that they form a group and, more specifically, a Lie transformation group if one considers continuous, finite symmetry transformations. Different symmetries form different groups with different geometries. Time independent Hamiltonian systems form a group of time translations that is described by the non-compact, abelian, Lie group R {\displaystyle \mathbb {R} } . TTS is therefore a dynamical or Hamiltonian dependent symmetry rather than a kinematical symmetry which would be the same for the entire set of Hamiltonians at issue. Other examples can be seen in the study of time evolution equations of classical and quantum physics. Many differential equations describing time evolution equations are expressions of invariants associated to some Lie group and the theory of these groups provides a unifying viewpoint for the study of all special functions and all their properties. In fact, Sophus Lie invented the theory of Lie groups when studying the symmetries of differential equations. The integration of a (partial) differential equation by the method of separation of…