Determining the hamiltonian

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  • #1
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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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  • #2
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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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