Is Symmetry Required for Determining the Hamiltonian?

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The discussion centers on the necessity of symmetry in determining the Hamiltonian, particularly in the context of matrix M. It is established that for the Hamiltonian to be correctly derived, matrix M must be symmetric, meaning M equals its transpose (M^T = M). This conclusion is supported by the properties of the Lagrangian, which remains invariant under transposition (L^T = L). Therefore, the symmetry of matrix M is essential for accurate Hamiltonian formulation.

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My book writes a 5-step recipe for detemining the hamiltonian, which I have attached. However I see a problem with arriving at the last result. Doesn't it only follow if the matrix M is a symmetric matrix - i.e. the transpose of it is equal to itself.
 

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I am assuming the tilde above an object implies taking a transpose. If that is the case, then M is indeed a symmetric matrix. One can see this by looking at the Lagrangian. Since the Lagrangian is a number, we can take a transpose of L and we'll get back the same number, i.e. [itex]L^T=L[/itex], this will rewrite the Lagrangian in terms of [itex]M^T[/itex]. But since these are equal, we must have [itex]M^T=M[/itex]
 

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