Energy formulation of classical mechanics?

In summary, when comparing the different formulations of classical mechanics, the commonly mentioned ones are Newtonian, Lagrangian, and Hamiltonian mechanics. While energy conservation is often used in Newtonian mechanics to simplify working with conservative forces, it is not considered a separate formulation as it does not provide enough information to fully describe mechanics. Instead, additional rules must be added to reach the level of Lagrangian or Hamiltonian mechanics.
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Energy "formulation" of classical mechanics?

When discussing the different formulations of classical mechanics, those that are always mentioned are Newtonian, Lagrangian and Hamiltonian mechanics. However, it is my understanding that when dealing with conservative forces in Newtonian mechanics, we choose to work with energy instead of forces because it makes things easier. Isn't this energy approach a separate formulation of classical mechanics just as much as Lagrangian and Hamiltonian mechanics?
 
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  • #2
Not really (although it depends on what exactly you count as a formulation). Energy conservation can solve some problems, but it not enough information to fully describe mechanics.

Or, the answer could be yes, but you must add rules until you have Lagrangian or Hamiltonian mechanics.
 

1. What is the energy formulation of classical mechanics?

The energy formulation of classical mechanics, also known as Hamiltonian mechanics, is a mathematical formulation of classical mechanics that describes the motion of a system in terms of its total energy. It is based on the principle of conservation of energy, where the total energy of a system remains constant over time.

2. How is the energy formulation different from other formulations of classical mechanics?

The energy formulation differs from other formulations, such as the Lagrangian formulation, in that it uses the concept of energy to describe the motion of a system, rather than the concept of forces. It also uses a different set of equations, known as Hamilton's equations, to solve for the motion of a system.

3. What is the role of the Hamiltonian in the energy formulation of classical mechanics?

The Hamiltonian, denoted by H, is a mathematical function that represents the total energy of a system in the energy formulation of classical mechanics. It is defined as the sum of the kinetic and potential energies of the system, and plays a central role in determining the equations of motion for the system.

4. How is the energy formulation used in practical applications?

The energy formulation of classical mechanics is used in a wide range of practical applications, such as in engineering, physics, and astronomy. It is used to model the behavior of complex systems, such as celestial bodies, and to predict their future motion. It is also used in the design of mechanical systems, such as machines and vehicles.

5. Are there any limitations to the energy formulation of classical mechanics?

While the energy formulation of classical mechanics is a powerful tool for describing the motion of many systems, it does have some limitations. It cannot accurately describe the motion of systems that involve quantum effects, such as atoms and subatomic particles. Additionally, it does not take into account relativistic effects at high speeds, and must be modified to accurately describe these systems.

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