There are two concepts that you appear to be confusing here based on the Wikipedia references.

1. Under normal (non-relativistic) conditions, time and energy are separate properties. In SI units, energy is measured in joules and time in seconds. The rate of change of energy with time is called power and is measured in J/s or watts.

2. The concept of spacetime is important in relativistic physics (physics for systems at exceptionally high velocity). At nonrelativistic conditions, we generally consider space to include three independent directions (x, y, and z, for example) and time to be a forth independent measure that behaves much like the three dimensions of space except that it cannot be reversed in practice (it can be reversed in abstract and in mathematics). The Special and General Theories of Relativity show, however, that at very high velocities or in systems with strong or changing gravitational forces, time and space are not completely independent of one another. That is, an observer in a very rapidly moving vehicle observes time passage for some object differently than a similar observer at rest measures time passage for the same object. More specifically, the observer at rest measures more rapid time passage than the one in motion. This has been experimentally verified many times and applies to inanimate objects as well as living organisms. Spacetime is a coordinate system that includes the interdependence of space and time coordinates. It does affect modern life in meaningful ways. For example, GPS positioning systems would not work without accounting for this interdependence, the largest part of which comes from gravity (General Relativity) effects. However, in nearly all systems of practical interest, the traditional concept of time being independent of space provides solutions that are highly accurate.

Under quantum mechanics time and energy are conjugate variables that obey the Heisenberg uncertainty principle to produce the "quantum vacuum":

According to quantum theory there is a quantum vacuum that pervades all space such that for parcels of space at the quantum scale ( h/2π,

where h is Planck’s constant, h = 6.62607004 x 10 **-34 m2 kg / s and π is the Euclidean geometric parameter π = 3.14159265. As we reduce uncertainty of time Δt we increase the uncertainty in how much energy the parcel may contain to the extent that, with Einstein’s mass-energy relation E = mc**2, where m is the mass of a physical entity, particles with mass m can, in principle, emerge within the quantum parcel. Hence at the quantum scale space may be filled with a seething froth of radiation and particles that come and go in virtual reality.

Good question

In quantum physics time and energy are inherently inseparable as demonstrated by the importance of Planck's constant, which appears in every important quantum equation.

Planck's constant has the dimensions of energy x time, and at the quantum level everything has a frequency. So we can deduce that this frequency is inherent in the fundamental fabric of the Universe. Which explains exactly why the vacuum of space-time has inherent (dark) energy within it.

To answer on the question

“…How are time and energy related in any physical system?”

is necessary before to answer on the questions what is the notion/phenomenon “Time”?, and what is the notion/phenomenon “Energy”?

Both these penomena are absolutely fundamental and so are Meta-mainstream philosophical and Meta-mainstream physical ones; and so cannot [and aren’t] rationally defined in the mainstream.

The correct definitions and rational understanding what they are is possible only in the “The Information as Absolute”, where it is rigorously proven that there cannot be something else then some informational patterns/systems of the patterns, including every material object and the system “Matter” are some informational patterns/system; and, besides, it becomes be necessary, at the ubdertstanding, to define also at least the also absolutely fundamental notions/phenomena “Information”, “Change”, and “Space”.

In the reality the phenomena “Space” and “Time” are the absolutely fundamental Rules/Possibilities that are necessary for any information to exist [Space] and to change [Time]. So every changing pattern/system exists and changes [Matter and every material object exist and constantly change always because of the energy conservation law, they simply cannot stop] in corresponding spacetime. Any spacetime is nothing more then a “empty container”; Matter’s spactime is [if we don’t consider QM effects] the absolute [5]4D Eiclidian empty container

Space and Time are rather similar Rules/Possibilities [“Time is the space for changing states”], however they are fundamentally independent on each other, and so, for example, that

“…. The Special and General Theories of Relativity show, however, that at very high velocities or in systems with strong or changing gravitational forces, time and space are not completely independent of one other. That is, an observer in a very rapidly moving vehicle observes time passage for some object differently than a similar observer at rest measures time passage for the same object…”

isn’t adequate to the objective reality, everything, including any “observer” and “refernce frame”, in Matter cannot impact on the space and time, as well as either space or time cannot impact on any material object.

The “relativistic” assertions above are only some bare declarations, that simply postulated in the SR/GR without any grounds; that appeared because of the authors of the SR/GR didn’t ubderstand what are space/time/spacetime. Moreover, to postulate the above it was necessary to postilate also that there is no the absolute Matter’s spacetime and, firstly in the SR, that all/every reference frames are totally and completely equivalent and legitimate, when from these postulates any number of evidently senseless consequences directly and unambiguously follow.

“Energy” is absolutely fundamental Quantity, which is necessary for any information, and so any material object, be changed, when the Rule “Time” establishes that between every states of every changed object must be non-zero “time interval” [duration]. At that because of the logical self-inconsistence of the phenomenon “Change” every change fundamentally obligatorily procceds with some uncertainty, and, besides [when the time interval increases at every change in the Time-possibility, i.e. in the t-dimension of the spacetime, thus changing object moves along the t-axis], always there are “past” and “future” intervals, but there is no exact point “now”.

Matter is rather simple informational system that is based, with large probability, practically on a binary logic, which is totally universal for all material objects; and besides, uses a number of also totally universal constants.

Thus in Matter the uncertainty above indeed is characteresed by the Planck constant so that at any elementary change ΔE Δt = h/2π. But that by any means doesn’t connote that there is some link/dependence between energy ant time, that is nothing more then the concrete property of the concrete logical system “Matter”.

Etc., including in Matter the situation is more complex that in the text above, in Matter two rules/possibilities “time” [+3D space] act, so the Matter’s spacetime see above; so more see https://www.researchgate.net/publication/260930711_the_Information_as_Absolute DOI 10.5281/zenodo.268904 , and

https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics DOI 10.5281/zenodo.16494

besides it could be useful to read SS posts in two last pages in https://www.researchgate.net/post/Is_time_an_illusion?view=5ae9beda10569fc4a335e8f5

Cheers

"Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as a classical or quantum mechanical system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories."

-- https://en.wikipedia.org/wiki/Field_(physics)

It seems that the change of states of a physical system over time and over space is as essential as the energy content (energy levels, etc.) of a physical system...

“…"Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time…”

Field theory is nothing else then some subjective some human’s description of some objects/systems, states and processes that objectively exist/hold/pass in Matter; any field theories uasually postulate some ad hoc points and can be adequate to the reality, other not, etc.; or, say, in this theory can be stationary solutions, when, for example, the elaborated by the theory states don’t depend on time directly.

However that by no means connotes, that such objects/systems don’t relate to the time [though there is some rather popular “proof” that “time is an illusion” because of that Universe’s Schrödinger equation doesn’t contain the time variable]; for example from that the Schrödinger equation for the H-atom’s electronic shells’ energy levels is stationary, however from that by any means doesn’t follow that this atom is some “stationary” system.

“…It seems that the change of states of a physical system over time and over space is as essential as the energy content (energy levels, etc.) of a physical system”

again, see the SS post above, every material object/system, i.e. particle, body, field, galaxy, etc. [with particle’s, body’s… predecessors, of course] constantly changes its state from Beginning, when in some system a huge portion of Energy was pumped, because of the energy conservation law, independently on there is/it produces some fields or not. And only just therefore in physics “Lagrange or a Hamiltonian of the field, and treating it as a classical or quantum mechanical system”, etc., are adequate to the physical reality.

Cheers

I offer my RG preprint for your consideration although it does not address "energy" per se:

It takes energy to overcome resistance or inertia in nature or to make physical changes to a physical system or to convert mass to pure energy or to convert pure energy to mass, and so on... And that energy requires time and space.

“…It takes energy to overcome resistance or inertia in nature or to make physical changes to a physical system or to convert mass to pure energy or to convert pure energy to mass, and so on... And that energy requires time and space.”

“to overcome resistance or inertia in nature” is the same as “ to make physical changes to a physical system” . However in Matter cannot be some conversion of “mass to pure energy” or some conversion of “pure energy to mass” . Since in Matter there is no some “pure energy”, i.e. which is outside material objects. Or, more correctly, possible Matter’s Creator possibly some “outside energy” has, but in what form, i.e. in what changing non-material informational pattern this energy is storaged, humans have no data about now.

Though, of course, any energy “is pure” always, since Energy is absolutely fundamental Quantity, which is necessary to change any information, includimg, of course, to change informational patterns “material objects/systems”; and which exists, in this sense, as something “external” to any information.

“…And that energy requires time and space.”

The absolutely fundamental Quantity “Energy” doesn’t require time and space, the absolutely fundamental Rules/Possibilities “Time” and “Space” are another members, as the set’s member Energy is also, of the set of absolutely fundamental Rules, Possibilities, Quantities, Actions, etc. “Logos”, and all of Logos set’s members are necessary for any information could exist and change.

However they are necessary “in parallel” .I.e. that is any information, which requires "independently, " time, space, and energy . Including the informational system “Matter” exists and constantly changes in the absolute [5]4D Euclidian spacetime [concrete realization/action of the rules/possibilities “space” and “time”], and at Beginnong some initial portion of energy created in the spacetime some fixed informational pattern and further an additional portion forced this pattern to become be constantly changing Matter.

As well as we don’t know how energy acts in the fundamentally non-material informational system “human’s consciousness”, which is dynamical system also, and, for example, thinking requires some energy. All what is known is that the consciusness exists and changes basing on fundamentally different from Matter’s system of laws and links and in some essentially other then Matter’s one spacetime.

More see the SS post above and papers that are linked in the posts; besides relating to the “Space” and “Time” it could be useful to read SS posts in https://www.researchgate.net/post/Is_time_an_illusion?view=5ae9beda10569fc4a335e8f5

Cheers

Dear Sergey Shevchenko

To my opinion You are one of the few who do understand the physics of nature - congratulation.

Kurt Wraae

Thanks,

Cheers