Can I prove conservation of mechanical energy?

In summary, the conservation of mechanical energy for objects can be proven using the laws of kinematics and the definition of potential energy/kinetic energy in the case of Newtonian gravity. This can be shown through lengthy calculations and the use of the time-derivative of the total energy. However, this proof is problematic in General Relativity due to the lack of a universal time. The potential energy is defined in a way that ensures conservation of energy in the system, and potential energy is not explicitly needed in the Newtonian formulation of mechanics.
  • #1
Bipolarity
776
2
I am trying to consolidate my understanding of kinematics and mechanics.

If there is no such thing as friction, and the only force in the universe is gravity, then can we prove using the laws of kinematics and the definition of potential energy/kinetic energy that the mechanical energy of every object is conserved?

BiP
 
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  • #2
With Newtonian gravity, it is possible. You can use that every force on an object has a corresponding force on another object, and after some lengthy calculation (at least I think it was lengthy), you can show that the time-derivative of the total energy is 0, which means conservation of energy. This is not restricted to gravity, you can include the electromagnetic interaction as well.
With General Relativity, it is problematic - as you have no universal time which is the same everywhere, what is meant with "the energy at a specific time"?
 
  • #3
I see. For the sake of curiosity, I am interested in seeing the mathematical proof in the case of Newtonian gravity, obeying the laws of Newtonian mechanics. Do you know where this proof can be found?

BiP
 
  • #4
Perhaps starting with a simple long (several meters) pendulum and a stop watch can show that T (kinetic energy)+ V (potential energy) is a constant to a small fraction of a percent.
[added] You also have to monitor the amplitude of the pendulum as a function of time.
 
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  • #5
I saw one on a blackboard and I think that some introductory physics books should have that.

Anyway, I can derive it here, too:
With Newtonian gravity, each pair of objects i,j has a potential [itex]V_{ij}=m_i m_j \frac{G}{|r_{ij}|}[/itex] where rij is the distance between the objects.

The total energy of the system is then given by
[tex]E=\frac{1}{2} \sum_k m_k\, v_k^2 + \frac{1}{2} \sum_{i,j,i \ne j} V_{ij}[/tex]
where the second 1/2 is required as the sum counts each pair twice. The time-derivative is then given by
[tex]\frac{dE}{dt}=\sum_i m_i\, v_i \cdot a_i + \frac{1}{2} \sum_{i,j,i \ne j} m_i\, m_j\, G \frac{d}{dt}\frac{1}{|r_{ij}|}[/tex]
The dot indicates a scalar product. The chain rules now gives
[tex]\frac{d}{dt}\frac{1}{|r_{ij}|}
= \frac{1}{|r_{ij}|^2} \frac{d}{dt} |r_{ij}|
= \frac{1}{|r_{ij}|^3} (x_i-x_j) \cdot (v_i-v_j)[/tex]

The force on an object is now given by the gravitational forces of all other objects, which is the negative gradient of the potential.
[tex]m_i a_i = - \sum_{j \ne i} m_i\, m_j\, G\; \nabla \left(\frac{1}{|r_{ij}|}\right) = - \sum_{j \ne i} m_i\, m_j\, G \frac{x_i-x_j}{|r_{ij}|^3}[/tex]

If you insert both expressions in the expression for dE/dt, you will see that both parts just cancel and you get 0.

This is not restricted to potentials which are proportional to 1/r. In the general case of V(|r|), you get [tex]\frac{d}{dt}V(r)=\left(\frac{d}{d|r|}V(|r|)\right) \frac{1}{|r|} (x_i-x_j)\cdot(v_i-v_j)[/tex] and [tex]m_i a_i = - \sum_{j \ne i} m_i\, m_j\, G \left(\frac{d}{d|r|}V(|r|)\right) \frac{x_i-x_j}{|r_|}[/tex]
 
  • #6
Truly amazing! To think that what you just wrote forms the basis of all of classical mechanics!

BiP
 
  • #7
It should be noted that in fact potential energy is in a sense defined so that overall energy of the system is conserved. On the level of Newtonian formulation of mechanics, there is no explicit need for concept of potential energy. The proof given above is precisely speaking proving only the thing that we have chosen right form for the potential energy so that it indeed does reproduce the Newton law of the gravitation.
 

FAQ: Can I prove conservation of mechanical energy?

1. What is conservation of mechanical energy?

Conservation of mechanical energy is a fundamental principle in physics that states that the total amount of mechanical energy in a closed system remains constant over time. This means that the sum of kinetic energy (energy of motion) and potential energy (energy of position) remains the same, even as the forms of energy may change.

2. How is conservation of mechanical energy proven?

Conservation of mechanical energy can be proven through experiments and mathematical calculations. In an isolated system, energy can only be transferred between different forms, but the total amount of energy remains constant. This can be demonstrated through experiments such as a pendulum or a ball rolling down a hill, where the total energy remains the same at all points throughout the motion.

3. What are some real-life examples of conservation of mechanical energy?

There are many real-life examples of conservation of mechanical energy, such as a swinging pendulum, a rollercoaster ride, or a bouncing ball. In all of these cases, the total amount of mechanical energy remains the same, even as the forms of energy change. Another example is a falling object, where potential energy is converted into kinetic energy as it falls, but the total energy remains constant.

4. Are there any exceptions to conservation of mechanical energy?

Conservation of mechanical energy is a fundamental law of physics and is considered to hold true in all cases. However, in certain situations, external forces such as friction or air resistance may cause some energy to be lost from the system. In these cases, the total mechanical energy may not remain constant, but the principle still holds true.

5. Why is conservation of mechanical energy important?

Conservation of mechanical energy is important because it is a fundamental law of nature that governs the behavior of energy in the physical world. It allows us to predict and understand the motion and behavior of objects in various situations. It also has practical applications, such as in designing efficient machines and understanding energy conservation in everyday activities.

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