IB Mathematics Internal Assessment

From Pendulums to Potential Fields: A Comparative Analysis of Critical Points in Physical Systems Across Dimensions

Word Count: 3927

Table of Contents

1. Introduction

2. Differentiation of One-Variable Functions

•  2.1. Examples of 1D Curves and 2D Surfaces in Physics

•  2.2. Defining Critical Points in 1D Curves

– 2.2.1. Maxima and Minima in Physical Systems

– 2.2.2. Examples of Critical Points in Mechanical Energy

– 2.2.3. Inflection Points and Transition States

– 2.2.4. Examples of Concavity Change in Motion

2.3. Calculation and Classification of Critical Points

• 2.3.1. Calculating Maxima and Minima in Potential Energy

• 2.3.2. First Derivative Test for Physical Equilibrium

•  2.3.3. Example: Simple Pendulum Analysis

•  2.3.4. Calculating Inflection Points in Kinematics

• 2.3.5. Finding All Critical Points in Oscillatory Systems

•  2.3.6. The Second Derivative Test for Stability

3. Transitioning to Analyzing Critical Points in 2D Surfaces

•  3.1. Visualization of 2D Physical Surfaces

• 3.2. Defining Critical Points in Potential Energy Landscapes

• 3.3. Finding Critical Points in 2D Physical Systems

— 3.3.1. Example: Particle in Magnetic Field

3.4. Classifying the Nature of Critical Points

• 3.4.1. The Hessian Matrix in Physical Context

• 3.4.2. The Determinant and Stability Criterion

•  3.4.3. Example: Saddle Points in Electrostatic Potentials

• 3.4.4. Experimenting with Different Physical Potentials

• 3.4.5. Example: Gravitational Potential Analysis

4. Conclusion

• 4.1. Comparing the Dimensions in Physical Contexts

• 4.2. Generalization to Higher-Dimensional Physical Systems

Bibliography

1 Introduction

The study of critical points in calculus finds profound applications in understanding physical phenomena, from the simple oscillation of a pendulum to the complex energy landscapes of molecular systems. My fascination with this topic began during physics laboratory sessions, where I observed how mathematical concepts directly translate to physical behavior.

This investigation explores how critical point analysis extends from one-dimensional me- chanical systems to two-dimensional potential fields, addressing the research question: How does the mathematical framework for critical point analysis evolve when tran- sitioning from one-dimensional to two-dimensional physical systems, and what new physical insights emerge from this dimensional expansion?

Through concrete physical examples and mathematical rigor, this paper demonstrates the powerful connection between abstract calculus and tangible physical reality.

2    Differentiation of One-Variable Functions

2.1 Examples of 1D Curves and 2D Surfaces in Physics

Figure 1: Potential energy curve of a simple pendulum V (θ) = mgl(1 - cosθ)

2.2    Defining Critical Points in 1D Curves

2.2.1    Maxima and Minima in Physical Systems

In physical contexts, critical points represent equilibrium positions:

• Minima: Stable equilibrium (system returns after small displacement)

• Maxima: Unstable equilibrium (system moves away after small displacement)

2.2.2    Examples of Critical Points in Mechanical Energy

For a mass-spring system with potential energy V (x) = 1/2kx2 :

V ′ (x) = kx = 0 ⇒ x = 0 (1)

V ′′(x) = k > 0 ⇒ stable equilibrium (2)

2.2.3    Inflection Points and Transition States

In  physical  systems,  inflection  points  often  represent  transition  states  between  different regimes of behavior.

2.3    Calculation and Classification of Critical Points

2.3.1    Calculating Maxima and Minima in Potential Energy

Consider a particle in an anharmonic oscillator with potential:

(3)

2.3.2    First Derivative Test for Physical Equilibrium

The first derivative test identifies where net force vanishes:

(4)

2.3.3 Example: Simple Pendulum Analysis

As detailed in the theoretical framework, the simple pendulum demonstrates clear physical interpretation of mathematical critical points.

2.3.4    Calculating Inflection Points in Kinematics

In motion analysis, inflection points in displacement-time graphs indicate acceleration changes.

3 Transitioning to Analyzing Critical Points in 2D Sur-faces

3.1 Visualization of 2D Physical Surfaces

Figure 2: Two-dimensional potential energy surface showing multiple critical points

3.2    Defining Critical Points in Potential Energy Landscapes

In two dimensions, critical points satisfy:

(5)

3.3 Finding Critical Points in 2D Physical Systems

3.3.1    Example: Particle in Magnetic Field

Consider a charged particle in a magnetic field with potential:

(6)

Finding critical points:

(7)

(8)

4    Classifying the Nature of Critical Points

4.1    The Hessian Matrix in Physical Context

The Hessian matrix encodes curvature information crucial for stability analysis:

(9)

4.2 The Determinant and Stability Criterion

The Hessian determinant classifies critical points:

• D > 0, Vxx  > 0:  Stable equilibrium (minimum)

• D > 0, Vxx  < 0: Unstable equilibrium (maximum)

• D < 0: Saddle point (mixed stability)

4.3 Example: Saddle Points in Electrostatic Potentials

Consider the electrostatic potential:

(10)

Critical point at (0, 0) with Hessian:

This saddle point represents an unstable equilibrium where the potential decreases in y-direction but increases in x-direction.

4.4    Experimenting with Different Physical Potentials

4.4.1    Gravitational Potential Analysis

For a mass in a gravitational field with additional quadrupole moment:

(12)

This complex potential demonstrates multiple critical points with different stability char- acteristics.

5 Conclusion

5.1    Comparing the Dimensions in Physical Contexts

The transition from one-dimensional to two-dimensional analysis reveals fundamental in- sights:

Table 1: Comparison of Critical Point Analysis in Physical Systems

5.2    Generalization to Higher-Dimensional Physical Systems

The principles established extend naturally to higher dimensions:

• Three-dimensional potential fields in electromagnetism

•  Multi-dimensional configuration spaces in statistical mechanics

•  High-dimensional energy landscapes in machine learning

The emergence of saddle points in two dimensions represents a crucial conceptual ad- vancement, enabling understanding of transition states and anisotropic stability that are ubiquitous in real physical systems.

Bibliography

1.  Goldstein, H., Poole, C., & Safko, J. (2002).  Classical Mechanics  (3rd ed.).  Addison- Wesley.

2.  Marion, J. B., & Thornton, S. T. (2004).  Classical Dynamics of Particles and Systems (5th ed.). Brooks/Cole.

3.  Stewart, J. (2015).  Calculus:  Early  Transcendentals  (8th ed.).  Cengage Learning.

4.  Feynman,  R.  P.,  Leighton,  R.  B.,  &  Sands,  M.  (2005).    The  Feynman  Lectures  on Physics. Addison-Wesley.

5.  Kibble, T. W. B., & Berkshire, F. H. (2004).  Classical Mechanics  (5th ed.).  Imperial College Press.