# Dynamic System Simulations and Derivations-12/12
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.

> [Cloud Computing's powerful role in making complex physics and engineering simulations accessible and interactive via the web](https://viadean.notion.site/Dynamic-System-Simulations-and-Derivations-21f1ae7b9a3280a7a7d7dd08e434ec85?source=copy_link). Instead of needing specialized software or high-performance local machines, cloud resources enable the dynamic visualization and analysis of intricate systems—from multi-component mechanical oscillators to the nuanced behaviors of fluids and vibrating structures described by partial differential equations. This democratizes scientific exploration and enhances understanding through immediate, visual feedback.

### :clapper: Animated result and Interactive web
#### a dynamic visual representation of the undamped double oscillator

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#### Displacement Plot of Undamped Double Oscillator
<iframe width="332" height="249" src="https://www.youtube.com/embed/dFB80dOy4Og" title="Displacement Plot of Undamped Double Oscillator" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

#### Vibrating String with Obstacle

<iframe width="332" height="249" src="https://www.youtube.com/embed/KJtPGa0swG4" title="Vibrating String with Obstacle" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

#### Continuity Equation Demonstration

<iframe width="332" height="249" src="https://www.youtube.com/embed/dtaKgi2Uo60" title="Continuity Equation Demonstration" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

#### Traveling Wave Solutions

<iframe width="332" height="249" src="https://www.youtube.com/embed/yEGkiYBbmVM" title="Traveling Wave Solutions" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

#### Vibrating Rod Simulation

<iframe width="332" height="249" src="https://www.youtube.com/embed/uIpHmY_ZUY0" title="Vibrating Rod Simulation" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

## Latest update

<iframe src="https://viadean.notion.site/ebd/21f1ae7b9a3280a7a7d7dd08e434ec85" width="100%" height="600" frameborder="0" allowfullscreen />