16 questions linked to/from What canonical momenta are the "right" ones?
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### Hamiltonian definitions in the presence of boundary term [duplicate]

Consider a Lagrangian of the form $$L(q,\dot{q})=L_1(q,\dot{q})+\frac{d L_2(q,\dot{q})}{dt}$$ I understand that $\dot{L_2}$ does not modify the equations of motion, ...
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### Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me ...
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### Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{...
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### Are Maxwell's equations “physical”?

The canonical Maxwell's equations are derivable from the Lagrangian $${\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ by solving the Euler-Lagrange equations. However: The Lagrangian above is ...
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### Why is dependence on derivatives not a problem in the definition of canonical energy-momentum tensor?

Let $\mathcal L(\phi,\partial\phi)$ be a Lagrangian for a field $\phi$. It is known that the Lagrangian $\mathcal L$ and the Lagrangian $\mathcal L+\partial_\mu K^\mu$ produces the same physics, ...
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### How does the Hamiltonian change if $L\to L + \frac{dF}{dt}$? [duplicate]

The Hamiltonian is defined as the Legendre transform of the Lagrangian $$H = p\dot{q} -L .$$ In the Lagrangian formalism we are free to add the total derivative of an arbitrary function $F=F(q,t)$ to ...
I want to show that the gauge transformation $$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$ corresponds to a canonical transformation of the Hamiltonian \$H(p, q, ...