[Paper Review] Lagrangians and Hamiltonians for High School Students

John W. Norbury|ArXiv.org|Apr 14, 2000

Experimental and Theoretical Physics Studies2 references3 citations

TL;DR

This paper introduces Lagrangian and Hamiltonian mechanics to advanced high school students using intuitive explanations and basic calculus, avoiding the need for calculus of variations. It demonstrates that the Euler-Lagrange and Hamilton's equations reproduce standard Newtonian equations of motion, offering a deeper, energy-based understanding of classical dynamics without advanced mathematics.

ABSTRACT

A discussion of Lagrangian and Hamiltonian dynamics is presented at a level which should be suitable for advanced high school students. This is intended for those who wish to explore a version of mechanics beyond the usual Newtonian treatment in high schools, but yet who do not have advanced mathematical skills.

Motivation & Objective

• To provide advanced high school students with accessible entry points into Lagrangian and Hamiltonian dynamics beyond standard Newtonian mechanics.
• To present these formulations without requiring calculus of variations, making them approachable for students with introductory calculus knowledge.
• To demonstrate that the Euler-Lagrange and Hamilton's equations yield the same equations of motion as Newton’s laws, reinforcing conceptual understanding.
• To inspire intellectual curiosity by showing how energy-based formulations unify and generalize classical mechanics.
• To serve as a self-study resource or classroom supplement for motivated high school students interested in theoretical physics.

Proposed method

• Defines the Lagrangian as $ L(x,\dot{x}) = \frac{1}{2}m\dot{x}^2 - U(x) $, combining kinetic and potential energy.
• Introduces partial derivatives informally to compute $ \frac{\partial L}{\partial x} $ and $ \frac{\partial L}{\partial \dot{x}} $, linking the latter to momentum $ p = m\dot{x} $.
• Derives the Euler-Lagrange equation $ \frac{\partial L}{\partial x} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) $ as the fundamental equation replacing $ F = ma $.
• Defines the Hamiltonian as $ H(p,x) = p\dot{x} - L(x,\dot{x}) $, showing it equals total energy $ T + U $.
• Derives Hamilton’s equations: $ \frac{\partial H}{\partial p} = \dot{x} $ and $ \frac{\partial H}{\partial x} = -\dot{p} $, which replace Newton’s second law.
• Applies both formulations to the harmonic oscillator, verifying consistency with Newtonian results through symbolic derivation.

Experimental results

Research questions

• RQ1 Can Lagrangian and Hamiltonian formulations of classical mechanics be taught to advanced high school students without requiring calculus of variations?
• RQ2 How do the Euler-Lagrange and Hamilton's equations reproduce the same equations of motion as Newton’s second law in simple systems?
• RQ3 What is the conceptual advantage of formulating mechanics in terms of energy rather than forces for students with basic calculus knowledge?
• RQ4 How can partial derivatives be introduced intuitively to high school students to enable understanding of Lagrangian mechanics?
• RQ5 To what extent can the harmonic oscillator serve as a pedagogical model to demonstrate equivalence between Newtonian, Lagrangian, and Hamiltonian approaches?

Key findings

• The Euler-Lagrange equation $ \frac{\partial L}{\partial x} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) $ correctly reproduces Newton’s equation $ -\frac{dU}{dx} = m\ddot{x} $ for the harmonic oscillator.
• For the harmonic oscillator, the Lagrangian $ L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 $ leads to the correct equation of motion $ -kx = m\ddot{x} $ via the Euler-Lagrange equation.
• The Hamiltonian for the harmonic oscillator is $ H = \frac{p^2}{2m} + \frac{1}{2}kx^2 $, derived by expressing $ \dot{x} $ in terms of $ p $, confirming it as total energy.
• Hamilton’s equations $ \dot{x} = \frac{p}{m} $ and $ \dot{p} = -kx $ reproduce the equation of motion $ m\ddot{x} = -kx $ when differentiated.
• The three-dimensional generalization of the Lagrangian and Hamiltonian formulations is shown to yield the component-wise Newtonian equations $ F_i = m\ddot{x}_i $ for $ i = x,y,z $.
• The paper confirms that energy-based formulations (Lagrangian and Hamiltonian) are equivalent to Newtonian mechanics in standard cases, offering a deeper conceptual framework without advanced mathematics.