The training of large language models (LLMs) faces significant computational cost challenges, limiting their scalability toward artificial general intelligence (AGI) and broader adoption. With model sizes doubling approximately every 3.4 months and training costs escalating from 64 million USD for GPT-4 in 2020 to 191 million USD for Gemini Ultra in 2023, the economic burden has become unsustainable. While techniques such as quantization offer incremental improvements, they fail to address the fundamental computational bottleneck. In this work, we introduce DS-LLM, a novel framework that leverages dynamical system (DS)-based machines, which exploit Natural Annealing to rapidly converge to minimal energy states, yielding substantial efficiency gains. Unlike traditional methods, DS-LLM maps LLM components to optimization problems solvable via Hamiltonian configurations and utilizes continuous electric current flow in DS-machines for hardware-native gradient descent during training. We mathematically demonstrate the equivalence between conventional LLMs and DS-LLMs and present a method for transforming a trained LLM into a DS-LLM. Experimental evaluations across multiple model sizes demonstrate orders-of-magnitude improvements in speed and energy efficiency for both training and inference while maintaining consistent accuracy. Additionally, we provide an in-depth analysis of the challenges and potential solutions associated with this emerging computing paradigm, aiming to lay a solid foundation for future research.