In physics, a system is a name given to a collection of objects that we want to observe and analyze. If we are to describe the motion of an object by conserving energy, then the system must include the object of interest and all the other objects with which it interacts. When defining a system, we draw a line around the things that matter to us, leaving out the things that don’t. The type of things that we do not include within the system are generally called the system surroundings, ignoring that some parts of the surroundings will inevitably influence the system and therefore make our calculations less accurate. However, it is not unworthy to do this. In fact, understanding the effects you need to describe is just as important as knowing which effects you can ignore.

The law of conservation of energy declares that the total amount of energy in an isolated physical system remains unchanged over time. However, that energy can be transformed into another form of energy. In analytical mechanics, it can be shown that the principle of conservation of energy is a consequence of the fact that the dynamics of the evolution of systems is governed by the same characteristics at each instant of time. This leads to the temporal translation being a symmetry that leaves the evolution equations of the system invariant, which is why Noether’s theorem leads to the existence of a conserved quantity, energy.

**Conservation of Energy in Classical Mechanics**

**Newtonian Mechanics:**In Newtonian mechanics, a special instance of the law of conservation of energy is formulated: the conservation of mechanical energy, which is as follows: The total mechanical energy of a closed system of bodies, between which only conservative forces act, remains constant. To put it simply, the absence of dissipative forces (for example, frictional forces), mechanical energy does not arise out of nowhere and cannot disappear anywhere. For instance, in a nuclear power plant, the mechanical energy that drives the turbines does not come from nothing; it is generated from the thermal energy contained in the water vapor. Previously, this same energy was the internal energy of the atoms, specifically, nuclear energy.

**Thermodynamics:**In thermodynamics, historically, the conservation law is formulated as the first principle of thermodynamics: “The change in the internal energy of a thermodynamic system during its transition from one state to another is equal to the sum of the work of the external forces on the system and the amount of heat transferred to the system, and it does not depend on the way in which this transition takes place ”.Alternatively: “The amount of heat received by the system is used to change its internal energy and perform work against external forces.” The law of conservation of energy, in particular, states that there are no perpetual motion machines of the first type; that is, such processes are impossible. The only result of this would be the production of work without any change in other bodies.

**Hydrodynamics:**In the hydrodynamics of an ideal fluid, the law of conservation of energy is traditionally formulated in Bernoulli’s equation:

- points 1 and 2 lie on a streamline,
- the fluid has constant density,
- the flow is steady, and
- there is no friction.

Although these restrictions may sound severe, the Bernoulli equation is very useful, in part because it is simple to use but also because it can give great insight into the balance between pressure, velocity and elevation.

**Electrodynamics:**In electrodynamics, the law of conservation of energy is historically formulated in the form of the Poynting theorem (sometimes also called the Umov – Poynting theorem), which relates the density of electromagnetic energy flux to the density of electromagnetic energy and the density of Joule’s losses.

In verbal form, the theorem can be formulated as follows: A change in the electromagnetic energy enclosed in a specific volume during a certain interval of time is equal to the flux of electromagnetic energy through the surface that limits a given volume and the amount of thermal energy released in a given volume taken with the opposite sign.

**Energy Conservation in Everyday Life**

The operation of a light bulb is another possible example, a certain amount of electrical energy is received by the light bulb when the switch is operated and transforms it into light energy and thermal energy as the light bulb heats up. The total amount of electrical, thermal, and light energy is the same, but it has been transformed from electrical into light and thermal.