Classical Mechanics from Energy Conservation or: Why not Momentum?

This paper argues that Newtonian mechanics can be derived directly from energy conservation without defining work, demonstrating that both special relativity and quantum theory are intrinsically Hamiltonian and suggesting that the historical progression of mechanical formulations should be reversed to prioritize the Hamiltonian approach over Newtonian and Lagrangian methods.

The Gist

Imagine you are trying to teach someone how to drive a car.

The Traditional Way (The "Newton" Method):
Most physics textbooks start by telling you about the engine, the steering wheel, and the concept of "force." They say, "If you push the gas, the car moves. If you push the brake, it stops." They treat "force" as the main character of the story. It's like trying to understand a movie by starting with the special effects crew and the camera angles, rather than the plot. The author of this paper, Christian Baumgarten, argues that this is backwards, confusing, and actually a bit of a "nightmare" for logical thinkers.

The Paper's Proposal (The "Energy" Method):
Baumgarten suggests we flip the script. Instead of starting with "Force," we should start with Energy.

Think of Energy like money in a bank account.

• Position (Potential Energy): This is money sitting in a savings account (like a ball held high up on a shelf).
• Motion (Kinetic Energy): This is money you are spending to buy speed (like the ball falling and gaining speed).

The most important rule of the universe, according to this paper, is the Law of Conservation of Energy. It's like a strict bank rule: You can't create money out of thin air, and you can't destroy it. You can only move it from your savings account (position) to your spending account (motion), or vice versa. The total amount in the bank always stays the same.

The Big Twist: Why "Momentum" Matters More Than "Speed"

Here is where the paper gets really interesting. It argues that to understand the universe correctly (especially when things move very fast, near the speed of light), we need to change how we count our "money."

1. The Old Way (Speed):
In everyday life, we think of motion based on speed (how fast you are going). If you double your speed, you double your "motion money." This works fine for driving a car or throwing a ball. It leads to Newton's famous $F=ma$ (Force = Mass $\times$ Acceleration).

2. The New Way (Momentum):
The paper says that "speed" is actually a bit of a trick. The real, fundamental quantity is Momentum.

• Think of Momentum as the "heaviness" of your motion.
• In the everyday world, Momentum and Speed are best friends; they move together perfectly.
• But in the "Relativistic" world (where things move near the speed of light), they start to drift apart. As you get faster, your "Momentum" gets heavier and harder to push, even if your speed doesn't increase much.

The Analogy:
Imagine you are running on a treadmill.

• Newton's View: You just run faster, and you get tired faster. Simple.
• The Paper's View: As you run faster, the treadmill starts to get heavier and heavier, like it's made of lead. You can keep pushing (adding momentum), but the treadmill (your speed) refuses to go much faster.

The paper argues that if you start with Energy and Momentum (instead of Speed), you can naturally derive the rules of Einstein's Special Relativity without needing complicated "thought experiments" about light beams or time travel. It's like finding the master key that opens both the door to everyday physics and the door to the universe's fastest speeds.

Why the Order of Teaching is Wrong

The author is frustrated with how physics is taught in schools.

• Current Order: Newton $\rightarrow$ Lagrange $\rightarrow$ Hamilton.
  • Analogy: This is like learning to bake a cake by first studying the history of the oven, then the chemistry of flour, and finally realizing, "Oh, I need to mix the ingredients." It's illogical.
• Proposed Order: Energy Conservation $\rightarrow$ Hamilton (Momentum) $\rightarrow$ Lagrange $\rightarrow$ Newton.
  • Analogy: This is like starting with the recipe (the total ingredients/energy), understanding how they interact (momentum), and then realizing that for simple cakes, you can just use a shortcut (Newton's laws).

The "Least Action" Mystery

There is a famous principle in physics called the "Principle of Least Action." It sounds very deep and philosophical, like nature is a lazy genius that always takes the easiest path.

• The Paper's Take: The author thinks this principle is overhyped. It's not the cause of how things move; it's just a result.
• The Metaphor: Imagine you are walking through a field. You naturally take the path of least resistance. The "Principle of Least Action" is like a philosopher saying, "The universe is smart and chose the best path!"
• Baumgarten says: "No, the universe isn't making a conscious choice. It's just following the simple rule of Energy Conservation. The 'lazy path' is just what happens when you do the math correctly starting with Energy."

The Bottom Line

This paper is a plea to stop teaching physics like a history lesson and start teaching it like a logical puzzle.

1. Start with the Bank Rule: Energy is conserved. It just changes forms.
2. Use the Right Currency: Use Momentum, not just Speed, to describe motion.
3. Let the Rest Fall into Place: If you do this, Newton's laws appear naturally for slow things, and Einstein's laws appear naturally for fast things.

The author believes that by doing this, we stop confusing students with abstract concepts like "Force" and "Absolute Space" and instead give them a clear, logical picture of how the universe actually works: It's all about the flow of energy.

Technical Summary

1. Problem Statement

The paper addresses two primary issues in the pedagogy and logical structure of classical mechanics:

1. Pedagogical Misordering: The standard historical progression of teaching mechanics (Newtonian $\to$ Lagrangian $\to$ Hamiltonian) is argued to be illogical and misleading. It treats specific cases (Newtonian) as foundational, whereas the more general Hamiltonian formalism is presented as a derivative "formulation."
2. The Definition of Energy and Momentum: Recent attempts to derive mechanics solely from energy conservation by treating energy as a function of position and velocity ($E(x, v)$) fail to produce relativistic mechanics. The author argues that energy must fundamentally be treated as a function of position and momentum ($E(x, p)$) to accommodate both classical and relativistic regimes without ad-hoc adjustments.
3. The Role of Work: The paper challenges the notion that the concept of "work" should be banned from physics, arguing instead that it provides a clear, non-circular starting point for defining energy and deriving Newton's laws.

2. Methodology

The author employs a deductive approach, starting from the Principle of Energy Conservation and the concept of Work, rather than Newton's Laws or the Principle of Least Action (PLA). The methodology proceeds in three stages:

• Stage A: The Velocity-Based Derivation (Newtonian Limit):

  • Assumes energy $E$ is the sum of positional work $V(x)$ and motional work $T(v)$.
  • Applies the conservation condition $\frac{dE}{dt} = 0$.
  • Uses the chain rule to separate variables, assuming acceleration is independent of velocity.
  • Integrates to recover $T = \frac{1}{2}mv^2$ and $F=ma$.
  • Result: This successfully recovers Newtonian mechanics but fails for relativistic speeds because it presumes a specific functional form for kinetic energy dependent only on velocity.

• Stage B: The Momentum-Based Derivation (General Case):

  • Assumes energy $E$ is a function of position $x$ and a generalized "quantity of motion" $p$ (momentum), where $p$ is a monotonic function of velocity $v$.
  • Applies $\frac{dE}{dt} = \frac{\partial E}{\partial p}\dot{p} + \frac{\partial E}{\partial x}\dot{x} = 0$.
  • Derives Hamilton's first equation ($v = \frac{dT}{dp}$) and the force law ($\dot{p} = -\frac{\partial V}{\partial x}$) without assuming a specific form for $T(p)$.
  • This approach leaves the relationship between $p$ and $v$ open, allowing for different physical theories.

• Stage C: Deriving Relativistic Mechanics:

  • Introduces the concept of Inertia ($N = p/v$) and postulates that inertia increases with kinetic energy ($dN = \chi dT$).
  • Combines this with the Hamiltonian definition of velocity ($v = dT/dp$) to derive the relativistic momentum-velocity relation and the relativistic kinetic energy equation.
  • Demonstrates that the transition from Newtonian to Einsteinian mechanics requires only a change in the inertia assumption, not a change in the fundamental conservation principle.

• Stage D: Reversing the Lagrangian/Hamiltonian Order:

  • Uses the Legendre transform to derive the Lagrangian ($L = vp - H$) from the Hamiltonian, rather than the reverse.
  • Argues that the Principle of Least Action (PLA) is an output of the theory (a consequence of energy conservation and the Hamiltonian structure) rather than an input axiom.

3. Key Contributions

• Derivation of Newton's Second Law from Energy: The paper demonstrates that $F=ma$ and the form of kinetic energy ($T=\frac{1}{2}mv^2$) can be derived directly from the conservation of energy and the definition of work, without assuming Newton's laws a priori.
• Resolution of the Relativistic Paradox: It clarifies why treating energy as $E(x, v)$ leads to incorrect relativistic equations. The error lies in assuming acceleration is independent of velocity. By shifting the independent variable to momentum ($p$), the formalism naturally accommodates the velocity-dependent inertia of special relativity.
• Hamiltonian Primacy: The paper argues that Special Relativity and Quantum Mechanics are intrinsically Hamiltonian theories. The description of dynamics using coordinates and momenta is more fundamental than using coordinates and velocities.
• Pedagogical Reversal: It proposes a new teaching order:
  1. Energy Conservation & Work (Observation).
  2. Hamiltonian Mechanics (General Dynamics).
  3. Derivation of Newtonian Mechanics (Low-velocity limit).
  4. Derivation of Relativistic Mechanics (High-energy limit).
  5. Derivation of Lagrangian Mechanics (via Legendre transform).
• Critique of the Principle of Least Action (PLA): The author contends that the PLA is often taught as a deep, teleological principle (nature "chooses" the optimal path). The paper argues this is a philosophical relic; in practice, Lagrangians are postulated because they yield correct equations, not the other way around. Energy conservation is presented as the more robust, "real-world" foundation.

4. Results

• Mathematical Derivation:
  • From $E = T(p) + V(x)$ and $\dot{p} = -\partial V/\partial x$, the author derives the general equation of motion.
  • By assuming $dN = \chi dT$ (inertia proportional to energy), the derivation yields:
    $$p = \frac{mv}{\sqrt{1-v^2/c^2}}$$
    $$T(p) = \sqrt{m^2c^4 + p^2c^2} - mc^2$$
  • This derivation achieves the core equations of Special Relativity without invoking the postulate of the constancy of the speed of light or Lorentz transformations between inertial frames as a starting point.
• Logical Consistency: The paper shows that the "paradox" between the velocity-based and momentum-based derivations is resolved by recognizing that in the relativistic regime, acceleration is a function of both position and velocity, whereas in the Newtonian limit, it is effectively independent of velocity.
• Lagrangian Recovery: The relativistic Lagrangian for a free particle is successfully recovered as $L = -mc^2\sqrt{1-v^2/c^2}$ via the Legendre transform from the derived Hamiltonian, proving that the Lagrangian is a secondary construct.

5. Significance

• Theoretical Unification: The paper provides a unified framework where classical and relativistic mechanics are seen as limits of a single energy-conservation principle, distinguished only by the functional relationship between momentum and velocity (inertia).
• Educational Reform: It offers a compelling argument for restructuring physics curricula. By starting with energy conservation (a concept observable in simple systems like pendulums) and moving to the general Hamiltonian formalism, students avoid the logical pitfalls of "force-free" bodies in absolute space and the teleological confusion of the Principle of Least Action.
• Conceptual Clarity: It redefines "inertia" not as a constant mass, but as the ratio of momentum to velocity ($N=p/v$), which varies with energy. This provides a clearer physical intuition for relativistic effects (e.g., why particles cannot be accelerated to $c$ despite infinite momentum increase).
• Philosophical Shift: It challenges the "mythical" status of variational principles, suggesting that energy conservation is the true "universal currency" of physical processes, while variational principles are merely mathematical tools derived from it.

In conclusion, Baumgarten argues that the Hamiltonian formalism, rooted in energy conservation and the momentum variable, is the most natural and general foundation for mechanics, rendering the traditional Newtonian-first approach logically inferior and pedagogically confusing.