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Let $Q$ denote the set of all possible configurations of the system (the configuration manifold). Consider a point $q_0\in Q$. For the sake of conceptual clarity, and to make contact with physics notation, let's work in some local coordinate patch around $q_0$. Suppose that $q_0$ represents the position of the system under consideration at time $t_0$. ...

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I) OP essentially asked (v1): If two Lagrangian densities ${\cal L}$ and $\tilde{\cal L}$ have the same eqs. of motions, must they necessarily differ by a total divergence? Answer: No, one e.g. can always multiply a Lagrangian density ${\cal L}$ with a constant factor $\tilde{\cal L}=\lambda {\cal L}$ different from one $\lambda\neq 1$ without altering ...

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The principle of Least (Stationary) Action (aka Hamilton's Principle) is derived from Newton's axioms plus D'Alembert's principle of virtual displacements. Because D'Alembert's principle allows to account for the (reactions of the) bonds between the components of a system in a transparent way, the Lagrangian and Hamiltonian formulations are possible. ...

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Let's assume a typical fermionic mass-term (interacting leptons and quarks are spin 1/2-particles): $$\tag 1 \bar{\Psi}\Psi = \bar{\Psi}\left(\frac{1 + \gamma_{5}}{2} + \frac{1 - \gamma_{5}}{2}\right)\Psi = \left| \bar{\Psi}\left( 1 \pm \gamma_{5} \right) = \left( (1 \mp \gamma_{5})\Psi\right)^{\dagger}\gamma_{0} \right| =$$ $$=\bar{\Psi}_{L}\Psi_{R} + ... 2 Classical lagrangians of fermions are always constructed out of Grassmann numbers. No exception. In both of OP's cases, the mass term is nonvanishing: In the first case the mass term is proportional to \psi_1^*\psi_2^*-\psi_2^*\psi_1^* = 2\psi_1^*\psi_2^* \neq 0. In the second case, I write in the chiral basis:$$\gamma^0=\begin{pmatrix}& \sigma^0 ...

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I'll assume you want to know the equilibrium points. The Lagrangian tells you everything you need to know about the system. Because variation of generalized momentum is: $$\frac{dp_k}{dt} = Q_k + \frac{\partial T}{\partial q_k} = -\frac{\partial V}{\partial q_k}+ \frac{\partial T}{\partial q_k} = \frac{\partial L}{\partial q_k}$$ Then: $$... 2 It is interesting to look at a linearized version of gravity, with g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} If you choose the Lorentz gauge :$$\partial^\mu \bar h_{\mu\nu}=0 \quad\quad \bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} h^i_i \,\eta_{\mu\nu} \tag{0}$$the equations of movement in the vaccuum are simply :$$\square \bar h_{\mu\nu}=0 \tag{1}$$... 2 The key point in all of this is that general relativity is a gauge theory, and, as the saying goes, "the gauge always hits twice" (apparently attributed to Claudio Teitelboim). What this means is that (1) you have an arbitrary freedom in defining your evolution, corresponding to the ability to make gauge transformations, and (2) some of the evolution ... 2 The "associated scalar equation" is just the formula for the time evolution of the scalar magnitude of the displacement, r, rather than all its vector components. It really only makes sense to write such an equation if the right-hand side can be expressed in terms of r only, and not \mathbf{r}. Then you can use it to analyze the evolution of r in ... 1 In civil engineering they use it for structures, and strength of materials in the elastic realm. It goes by the name of the enegy method. Google books might give an indication. Some authors are Beer and the mechanical engineer Stephen Timoshenko. This is for some what "static" indeterminant structures. So, there is no time element. But, I am sure it ... 1 I'm a electrical engineer, and have never used either one in over 30 years of designing circuits. I vaguely remember that we went over them briefly in school, but since I haven't used them (knowingly) since, I can't tell you what the physical meaning of either is, which of course perpetuates the fact that I'm not going to use them. 1 First expand the product in \mathcal{L}_0:$$\mathcal{L}_0=-\frac{1}{2}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu}\right)-\frac{1}{2}m^2A_{\mu}A^\mu. Now in the term $\partial_\mu A_\nu\partial^\mu A^\nu$, the vector fields $A$ have the same index, hence the metric tensor in the first term of $D_{\rho ... 1 Not the exact title, but stripping some non-essential bits of the search to crash course lagrangian dynamics pdf, I found a 12-page document titled Crash Course in Discrete Lagrangian Dynamics (found here). This is likely not what you are looking for, but it could be useful nonetheless. 1 To see that this is the simplest possible non-relativistic quantum field theory for fermions, it's useful to derive the dynamics. The canonical momentum for$\psi(x,y,z)$is the Lagrangian's derivative with respect to$\partial_\tau \psi(x,y,z)$– and it is$\psi(x,y,z)^\dagger$(up to signs and$\pm i$which depend on conventions). At any rate, the ... 1 In short: virtual displacement is "pretend you are moving, but don't really move". In other words - you move by such a small amount that you don't change the state of the system - but it gives you insight (through work done etc) in what would happen if you did move. In other words - if the system is really moving, you can look at an interval$dt$to see how ... 1 Tong is alluding to the standard trick in the derivation of Noether's theorem by promoting the (infinitesimal)$x$-independent parameter$\epsilon$to become$x\$-dependent, see e.g. this Phys.SE post.

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