Understanding Hamiltons Principle & the Variational Formulation of Mechanics

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In summary, the conversation is discussing the introduction of the variational formulation of mechanics in a book. The speaker is having trouble understanding a specific part that involves a differential quantity, δJ, and the requirement for the Hamiltonian integral to be zero. They question the necessity of introducing this new notation, instead of using partial derivatives. They are seeking a more detailed explanation of the reasoning behind it.
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I have some trouble understanding how my book introduces the variational formulation of mechanics. On the attached file I have stipulated with red a part, which I do not understand at all.
What is the idea behind introducing this differential quantity:
δJ = dJ/dα * dα and writing the requirement for the hamiltonian integral to be zero in the way described by the last equation. Surely all that is required is that the euler lagrange equation 2.11 is satisfied? For me it is just weird to start talking about these things. Can someone explain the idea behind it in some detail?
 

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They just introduce new notation. So that instead of writing the partial derivatives, one could just write those variations.
 

FAQ: Understanding Hamiltons Principle & the Variational Formulation of Mechanics

What is Hamilton's Principle?

Hamilton's Principle is a fundamental principle in classical mechanics that states that the motion of a system can be described by minimizing the action, which is the integral of the Lagrangian over time.

What is the variational formulation of mechanics?

The variational formulation of mechanics is a mathematical framework that uses the calculus of variations to derive the equations of motion for a system. It is based on the idea that the motion of a system can be described by minimizing the action, and it is closely related to Hamilton's Principle.

How is Hamilton's Principle related to Newton's laws of motion?

Hamilton's Principle is an alternative formulation of classical mechanics, and it is equivalent to Newton's laws of motion. This means that any system that can be described by Newton's laws can also be described by Hamilton's Principle.

What is the role of the Lagrangian in Hamilton's Principle?

The Lagrangian is a mathematical function that describes the energy of a system in terms of its generalized coordinates and velocities. In Hamilton's Principle, the Lagrangian is integrated over time to calculate the action, which is then minimized to determine the equations of motion for the system.

How is Hamilton's Principle used in practical applications?

Hamilton's Principle is used in many practical applications, such as in engineering and physics, to derive equations of motion for systems that can be described by Newton's laws. It is also used in variational methods to solve optimization problems and in the study of dynamical systems.

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