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Read Lagrange's Mécanique analytique (English translation: Analytical Mechanics). The book is split up into two parts: statics and dynamics. The first chapter, "The Various Principles of Statics," is a beautiful historical overview. Lagrange works out many problems; for example, he has a chapter entitled "The Solution of Various Problems of Statics." But, ...

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I have finished reading a great book called "The Theoretical Minimum" by Leonard Susskind (a famous string theorist) and George Hrabovsky. It's about classical mechanics but mainly talks about both the Lagrangian formulation and the Hamiltonian formulation of classical mechanics. It is great for beginners in physics or just about anyone. It also reviews the ...

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In the context of a canonical transformation (CT) $$z^I~=~(q^i,p_i)~\longrightarrow ~(Q^j,P_j)~=~Z^J~=~f^J(z,t),$$ the matrix $$\textbf{M}^J{}_I~:=~\frac{\partial Z^J}{\partial z^I}$$ is the Jacobian matrix of the CT. Here the indices $$i,j~\in~\{1,\ldots, n\} \quad\text{and}\quad I,J~\in~\{1,\ldots, 2n\}.$$ If the CCR reads $$... 0 This is really straight forward, once you get used to the notation. (Don't you hate it when people say that?)$$[\pi (\vec{x},t), (-\nabla^{2} +m^{2})\phi (\vec{y},t)] , Here you need to remember that $\nabla^2$ acts on the $\phi(\vec{y},t)$ only, so $\pi$ can pass right through this wave operator. Now when you evaluate the commutator you'll end up with ...

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Page 60-64 Goldstein, Poole and Safko (3rd Edition) goes into a really nice derivation and description of the Energy Function. In the footnotes it states that this is equivalent to the Hamiltonian (it is just not in the correct generalized coordinates for the Hamiltonian). If this function is derived from scleronomous (equations of constraints are time ...

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FOR THE GENERATING FUNCTION F1(q,Q) dF1/dq = p AND p = Q/q THEN F1 = Qlnq + K1(Q)........(1) also dF1/dQ =-P ................(2) and P=ln(Q/q)-(1/2)ln(Q).............(a) substituting equation (a) in (2) and integrating F1 =-(1/2)(QlnQ-Q) + Qlnq + K2(q)..........................(3) comparing (1) and (3) we get F1 = -(1/2)(QlnQ-Q) + Qlnq with ...

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No, the principle of least action started well before quantum mechanics. It is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics. this new formulation of ...

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I think the Hamiltonian is not necessarily the energy for the following reason: you can demonstrate that the Lagrangian may be deduced from the D'alembert principle which is linked to the concept of force, etc. but it may be also deduced from the Hamilton's principle which is a pure mathematical concept applied to physics (a certain quantity has to be an ...

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