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To demonstrate whether a dynamical system can be Hamiltonian, you check a small set of strict mathematical conditions. Below is a clear, technical checklist, from fastest rejection to constructive proof. Start from the system form  \[ \dot{x} = f(x), \qquad x \in \mathbb{R}^{2n}. \] The system is Hamiltonian if and only if it can be written as \[ \dot{x} = J \nabla H(x), \] where the canonical symplectic matrix is  \[ \dot{x} = J \nabla H(x), \] where \[ J = \begin{pmatrix} 0 & I_n \\ - I_n & 0 \end{pmatrix} \] is the canonical symplectic matrix, and \( H(x) \) is a scalar Hamiltonian function.  Necessary condition: zero divergence    Hamiltonian flows preserve phase--space volume (Liouville's theorem): \[ \nabla \cdot f(x) = 0. \] If \( \nabla \cdot f(x) \neq 0 \), the system is not Hamiltonian. If \( \nabla \cdot f(x) = 0 \), Hamiltonian structure is possible. Example: damped pendulum \[ \dot q = \frac{p}{m\ell^2}, \qquad \dot...

  Hi everyone 👋 I’ll be honest upfront: I’m not a huge fan of Microsoft Word — I usually work in LaTeX , where formatting and citations behave exactly the way I want without any hacks 😄 That said, we all occasionally find ourselves back in Word , whether for collaborations, journal templates, or institutional requirements. And when that happens, a few quirky tricks can go a long way. If you’re using Zotero with Microsoft Word and want to improve the visual style of your document, this small trick can be surprisingly useful. In many scientific papers and technical documents, in-text citations appear in blue , similar to hyperlinks. Unfortunately, Zotero does not provide a built-in option to automatically change citation colors in Word. The good news? You can achieve this easily using a simple Word VBA macro  no need to abandon Zotero or manually format citations one by one. i originally found this in zotero forms this >  Post  < Below is a simple,...

Introduction Modern robotics demands actuators that aren’t just strong, but also smart — capable of precise control, smooth motion, and safe human interaction. Traditional servos are great for small robots, but they can be stiff, noisy, and limited in range or torque. This project demonstrates a custom robot actuator built around a 5010 360 KV brushless DC (BLDC) motor , controlled by an ESP32 running the SimpleFOC library. The goal is to create a compliant joint — one that can be moved by hand, but automatically returns to its home position with adaptive stiffness. 🧠 What Makes This Actuator Special Unlike a typical servo, this actuator behaves intelligently : You can turn it by hand — it feels soft and back-drivable. When you release it, the motor returns to its initial position smoothly. If you twist it harder (e.g., due to gear reduction), it becomes stiffer , resisting displacement more strongly. It’s powered by 12 V and controlled by a simple ESP32 boar...

Voltage Regulator Current Boost Circuits Voltage regulators, such as the LM317, 78xx series, or other linear regulators , are commonly used for providing a stable output voltage. However, these regulators have a maximum current limit , often in the range of 1A to 1.5A for typical ICs. To increase the current handling capability, external pass transistors are added in a current boost configuration. How Current Boost Circuits Work The circuits shown in the image use NPN power transistors ( 2N6133, 2N6124 ) to handle additional current while the regulator maintains voltage regulation. First Circuit (Single NPN Pass Transistor) The voltage regulator drives the base of Q1 (2N6133) . The transistor acts as a current amplifier . R1 limits the base current to Q1, ensuring proper operation. The regulator supplies only a small portion of the total current , while Q1 handles most of the load current . Second Circuit (Darlington Configuration with Current Limit) Uses an additional PN...

  Remove the sstate cache directory # Delete sstate cache direcotry $ rm -rf sstate-cache/ Delete tmp directory # Delete tmp direcotry $ rm -rf tmp/  Bitbake cleanall command  #for complete clean the recipe  $ bitbake -fc cleanall <recipe_name> Bitbake clean command  # clean the output of the recipe $ bitbake -c clean <recipe_name>

In order to find which device tree your machine is using, you can extract such information using the -e flag on bitbake and then grep: bitbake -e virtual/kernel | grep "^KERNEL_DEVICETREE="

  Sometimes, while working on the terminal, especially during long-running processes, it’s useful to have an audible notification when the task is finished. In this guide, we’ll show you how to make your terminal play a sound after any command completes, ensuring you're alerted without constantly checking the terminal. Why Do This? This trick can save time and improve workflow, especially when: You’re running lengthy build processes or installations. You’re waiting for large file transfers. You're programming embedded systems and need confirmation when flashing is done. Let’s dive into how to do this on a Linux system. Step-by-Step Guide to Playing a Sound After a Command 1. Using paplay or aplay to Play Sounds On most Linux distributions, you can use paplay or aplay to play sound files directly from the terminal. Here's how you can append a sound notification after a command. Basic Command Structure : PC:~$ your_command && paplay /path/to/soundfile.oga  Or, if...