Lagrange-D’Alembert Principle and random ODE

In summary, The conversation discusses the Lagrange-D'Alembert Principle and its generalization to ODEs in Banach spaces. An application is also mentioned, specifically the discussion of geodesics in infinite dimensional manifolds and a random ODE with nonholonomic constraint. The conversation also touches on the possibility of a Lagrange-D'Alembert principle for Quantum Mechanics, which is further reviewed in a separate paper.
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wrobel
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Here is my paper. A criticism and other comments are welcome.

Abstract: The Lagrange-D'Alembert Principle is one of the fundamental tools of classical mechanics. We generalize this principle to mechanics-like ODE in Banach spaces.
As an application we discuss geodesics in infinite dimensional manifolds and a random ODE with nonholonomic constraint.

https://arxiv.org/abs/2112.05276v3
 
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Interesting.

In Quantum mechanics, the time evolution given by the Schrodinger equation can be restated as a infinite dimensional (classical) Hamiltonian system (at least for the discrete spectrum case). So I wonder if you could state a Lagrange-D'Alembert principle for QM

I give a review of the above statement in https://arxiv.org/abs/2107.07050
 
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1. What is the Lagrange-D’Alembert Principle?

The Lagrange-D’Alembert Principle is a fundamental principle in classical mechanics that states that the motion of a system can be described by the minimization of a quantity called the action, which is the integral of the Lagrangian over time. This principle is used to derive the equations of motion for a system and is based on the principle of least action.

2. How is the Lagrange-D’Alembert Principle applied to random ODEs?

The Lagrange-D’Alembert Principle can be applied to random ODEs by considering the random forces as constraints on the system. This means that the action is modified to include terms related to the random forces, and the equations of motion are derived by minimizing this modified action.

3. What is the significance of the Lagrange-D’Alembert Principle in random ODEs?

The Lagrange-D’Alembert Principle is significant in random ODEs because it provides a systematic way to incorporate random forces into the equations of motion. This allows for the study of systems with stochastic behavior, which is important in many fields such as physics, engineering, and biology.

4. Can the Lagrange-D’Alembert Principle be applied to all types of random ODEs?

Yes, the Lagrange-D’Alembert Principle can be applied to all types of random ODEs as long as the random forces can be treated as constraints on the system. This includes both linear and nonlinear random ODEs.

5. Are there any limitations to the use of the Lagrange-D’Alembert Principle in random ODEs?

One limitation of the Lagrange-D’Alembert Principle in random ODEs is that it assumes that the random forces are independent of the system's state variables. This may not always be the case in real-world systems, and in such cases, alternative methods may need to be used.

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