Why do things always want to reduce their energy?

Question

Every object wants to reduce its potential energy, but why is that so? Does it have an explanation, or is it just a law we accept?

Answer 1

Objects have no "wants". The empirical fact that objects are often found in a state in which their potential energy has a local minimum is a consequence of the second law of thermodynamics.

Objects (or system of objects) can have potential energy and kinetic energy. Because of friction, damping, air drag etc. (all consequences of the second law of thermodynamics), the kinetic energy of an object tends to dissipate into the surrounding environment, leaving it with only potential energy. Once it is left with only potential energy then it must be at a local minimum or local maximum of potential energy. But a state in which it is at a local maximum of potential energy is unstable. Therefore objects tend (given enough time) to be found in states that are a local minimum of potential energy.

The qualification "given enough time" is important. Dissipation of kinetic energy may take a very long time indeed - the solar system has been in a state that is a long way from minimum potential energy for billions of years.

Answer 2

Objects do not want. Objects lose energy as they roll up a hill (potential), so they don't all gain energy all of the time. They change velocity when a force acts on it. At the top of a hill, the component of gravitational force parallel to the ground is zero, but along the slopes, the force is downhill, therefore the object gains speed as it moves down the hill. When it gets to the bottom, the downward force stops and it has the maximum kinetic energy that it can gain from the hill. Therefore, we say that the top of the hill has the greatest potential for gaining kinetic energy and the bottom of the hill has the lowest.

Systems may lose energy by dissipation which results in heat being released or emitting light. If it has no way to gain that energy back, it will have less energy and will be less able to move up a potential. Due to frictional forces, an object can not make it all of the way up the next hill, if the hill is the same height as or more than the one it started on. So, objects tend to do more falling than rising, hence objects lose energy more than they gain it, a process called erosion.

Answer 3

Why do things always want to reduce their energy?

Things do not always want to do this.

For example, in classical mechanics, the total energy $H=T+U$ of a system is often constant. I.e., the total energy is never reduced and never increased, it is always the same.

This follows directly from the equations of motion, the definition of the potential energy, and the homogeneous quadradic nature of the kinetic energy. (This latter point can be used to show that the conserved Hamiltonian energy $H=\frac{\partial L}{\partial \dot q}\dot q - L$ is equal to the total energy $T+U$.)

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In classical thermodynamics, a thermodynamics potential called the "entropy" and denoted by $S(U,V,N)$ is usually introduced along with an entropy maximization principle. The entropy maximization principle can be Legendre transformed into a type of energy minimization principle. For example, the Helmholtz Free Energy $F = U - TS$ obeys a minimization principle.

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With some caveats out of way, why then, does it seem like things "want" to reduce their energy?

In classical mechanics, this is because real systems are not isolated and their energy can dissipate to the surrounding environment. Of course, the total energy of the entire system is still constant, but we often neglect the changes in the surrounding environment and thus it appears that energy is being lost (since if energy is lost from the small system of interest and the change in the large system is ignored, then it appears that energy is lost overall, but it is not, we are just not keeping track of it.)

So, for example, when there is a friction force acting on a classical body, the system is non-conservative, therefore the Lagrangian equations of motion are: $$ \frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i} = Q^{fric}_i\;, $$ where $Q_i^{fric}$ is the non-conservative friction force, and any conservative forces have been accounted for by a potential $U$ where $L=T-U$. Thus, even if there is no explicit time dependence, we still have: $$ \frac{dH}{dt} = \sum_i Q^{fric}_i\dot q_i\;, $$ so the energy can change.

For example, a simple harmonic oscillator, subject to a friction-like force $-\mu \dot q$ (an actual friction force would not look exactly like this, but this illustrates the point) would obey the equation of motion: $$ m\ddot q + k q = -\mu \dot q\;, $$ which is solved, for example, by: $$ q(t) = A e^{\alpha t}\sin(\beta t)\;, $$ where $$ \alpha = \frac{-\mu}{2m}\;. $$

So, if $\mu > 0$, as we expect it to be, the overall energy dissipates. (Of course, if $\mu=0$ there is no dissipation and the system just oscillates back and forth forever, the total energy being conserved.)

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Wrapping up, in classical mechanics, we can have dissipative systems, but the model is rather phenomenological. For example, we have to assert that friction dissipates energy rather and adding energy.

In classical thermodynamics, we can have dissipative systems, but we have to assert the existence of thermodynamics potentials that are functions of state and that obey phenomenological minimization or maximization principles.

In classical statistical mechanics, we have theorems like the H-theorem that are used to justify the thermodynamic potential behavior (e.g., Free Energy minimization).