To derive equations of motion with PDEs when the system is continuous, and its state variables change not only over time but also across spatial dimensions. PDEs provide the mathematical framework to describe how these spatially distributed properties interact and evolve.

The choice between using an Ordinary Differential Equation (ODE) or a Partial Differential Equation (PDE) to describe the "equation of motion" fundamentally depends on the nature of the physical system you are trying to model.

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Here's why we sometimes need to derive the equation of motion using a Partial Differential Equation (PDE):

Describing Continuous Systems (Fields)

• ODEs for Discrete Systems: When we talk about the motion of a single particle or a rigid body, its position (and possibly orientation) can be described by a finite number of coordinates, and these coordinates change only with respect to time. For example, the position of a pendulum bob can be described by a single angle $\theta(t)$, which is a function of time only. The equation of motion for such systems typically results in an Ordinary Differential Equation (ODE), like $F=m a$ or Newton's second law for rotation.
• PDEs for Continuous Systems: Many physical systems are not made of discrete particles but are continuous and distributed over space. In these systems, a physical property (like displacement, temperature, density, pressure, or electric field) can vary from point to point within the system and also change over time.
• For such systems, the "state" is a function not just of time ( $t$ ) but also of one or more spatial variables (e.g., $x, y, z$ ).
• For example, the displacement of a vibrating string $u(t, x)$ depends on both time $t$ and its position $x$ along the string. The temperature in a heat-conducting body $T(t, x, y, z)$ depends on time and three spatial coordinates.
• To describe how these spatially varying properties evolve over time, we need equations that involve partial derivatives with respect to both time and space. These are Partial Differential Equations.

Capturing Spatial Interactions and Propagation

PDEs are necessary because they can model:

• Propagation: How disturbances (like waves, heat, or fluid flow) travel through space.
• Diffusion: How properties (like heat, concentration) spread out over time.
• Dispersion: How different frequencies of waves travel at different speeds.
• Interactions between adjacent points: The behavior at one point in a continuous medium is influenced by its neighboring points. PDEs inherently capture these local interactions through spatial derivatives.

Examples of Equations of Motion Derived as PDEs:

Here are some common physical phenomena whose equations of motion are PDEs:

1. Wave Equation:
   • System: Vibrating string, sound waves in air, electromagnetic waves.
   • PDE: $\frac{\partial^2 u}{\partial t^2}=c^2 \nabla^2 u$ (where $\nabla^2$ is the Laplacian, covering $u_{x x}, u_{y y}$, etc.).
   • Why PDE? The displacement $u$ varies along the string's length ( $x$ ) and over time ( $t$ ).
2. Heat Equation (Diffusion Equation):
   • System: Temperature distribution in a solid, diffusion of a chemical.
   • PDE: $\frac{\partial T}{\partial t}=k \nabla^2 T$.
   • Why PDE? The temperature $T$ varies spatially $(x, y, z)$ and over time $(t)$.