Is minimizing the action same as minimizing the energy?

Wrong. Consider a nonstationary external force (potentials). The energy is not conserved at all, but the equations of motion with the initial conditions describe the system behaviour.

No it is not the same. First of all, it is not the same to "minimize the energy" and "conserve the energy" as you seem to equate. In fact they are contradictory statements in some sense, since if the total energy is conserved its value is constant and can't be neither minimized nor increased without externel intervention. "Conservation of energy" is the first law of thermodynamics and the "principle of minimum energy" is a way of thinking about the second law of thermodynamics (which is a completely different concept from the first).

So if they are non-compatible as I said previously why they both are considered true in thermodynamics? Because in one case we are considering a closed system and in the other we are not. Let me explain; suppose you have a single spring floating in space. If the spring is standing still is trivial to say that energy is conserved and if the spring is oscillating (expanding and contracting periodically) you have an exchange between kinetic and elastic potential energy: when the spring is highly compressed or highly extended it has a lot of potential energy acumulated and little kinetic energy (since its speed is diminishing before changeing direction), and when the string passes its point of relaxation is moving faster (because inertia moves it towards the compressed or extend state), thus it has high kinetic energy even if there are no elastic forces trying to compress or expand it at that specific instant (low or zero potential energy). But this porcess doen't change the total energy (kinetic + potential) of the system. Here we can't show the principle of least energy, just that energy is conserved, because the system is contained and doesn't exchange energy with the environment.

But now immagine a gas of hundreths of springs floating in space and bouncing between each other. Sometimes a spring that was oscillating with low amplitude (with little total energy) would collide with another spring and receive some impulse that will increase its amplitude. Obviously because of conservation of energy the other spring will lose some of the energy on this transfer. Each collision allows for an event where the energy is absorbed or transfered to another spring. IF you intially have one spring with a lot of energy and the rest not-oscillating, it is reasonable to think that because of the collisions the energy of that one spring will be "diluted" and shared with the others little by little. It would be very improbably that the entirety of the energy would end up transfered to another spring, in general it will be distributed among all the springs (this is an increase in entropy which is just a consequence of statistics). In the end you will have that, on average, the enery of the initial spring will be equally shared among all the springs. If you consider the entire set of springs your physical system, it is clear that the energy would be conserved (there's no exchange between the spring gas and the rest of space), but if you consider only one spring as your physical system (which is an open system able to exchange energy with the sorrounding springs), then you must assume energy is not conserved in this particular place. The interesting thing is, that because the energy gets shared between the springs you end up seeing the energy of the initial spring getting lower and lower until it reaches a minumum. That's both principles more or less simply explained.

The principle of least action is also a totally different concept (to adress your question). The principle of least action differs both from the conservation of energy and the least energy principle. Conservation of energy can be deduced from the principle of least action under simple conditions but htey are not the same. So, what the action is, in the case of a single spring? I would say (in simple words) that the action of the spring is the total sum of the energy exchanges between kinetic and potential between two instants. It is not the energy but a measure of the exchange between different types of energy. It turns out that in nature, while energy is conserved in a single oscillating spring, the way the kinetic energy gets converted into potential (and viceversa) throughout the event is always the way that makes the minimal overall exchanges between the two. So as you can see, those are completely different concepts. Not only the least energy is different from the least action, but the concept of energy is totally different form the concept of action.

The title of your submission does not match the content of your submission. I'll come back to that, first let me answer the content.

Also, first a general observation: in any logical system there is great freedom to interchange axiom and theorem, without any change of the content. Generally, derivations can be walked in both directions, what you really are doing is demonstrating consistency.

Action

Now to the concept of Action.
The principle of least action and the work-energy theory theorem are one and the same thing. They have a different appearance of course but they are the same thing. One way to see that they have to be the same thing: both take kinetic energy and potential energy as inputs, and when applied to a problem they both produce the same equation of motion as when the newtonian formulation is applied. Same input, same output; there is no room for the two to be anything but one and the same thing.

(For a demonstration that the two are one and the same thing see the answer I gave to a question that is summerized as: Is the action purely a mathematical tool, or does it also have a physical interpretation? )

The work-energy theorem gives that the amount of change of potential energy will always be the same as the amount of change of kinetic energy. Therefore the sum of potential energy and kinetic energy will be constant.

This gives a conservation law that is valid with the restriction that you have to avoid conversion to any type of energy other than potential energy and kinetic energy.

In the case of 'minimizing the action': the point in variation space where the action is stationary is the point in variation space where trajectory continuously satisfies the work-energy theorem.

So you can see various interconnections, and it seems to me that the work-energy theorem is at the hub of it.

Generally, if you can express some law, but you find you have to specify some restriction (valid only if ... ), then it is unlikely to be a fundamental statement. Conversely, if the law can be expressed without any need to specify restriction, it is likely to be fundamental.

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About the title of your submission: in the content of your submission you state the relevant question: paraphrased by me: "If I specify the constraint that the sum of potential energy and kinetic energy must remain constant, is that mathematically equivalent to 'minimizing the action'?"

But the title of your submission doesn't reflect that intention. On stackechange changing the title is available. When you click the 'edit' button then in the edit window you can also edit the title of the question.