Question

Calculation of end lagrangian of relativistic particle

Answer 1

If a system is described by a Lagrangian L, the Euler–Lagrange equations

{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}={\frac {\partial L}{\partial \mathbf {r} }}} \frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{r}}} = \frac{\partial L}{\partial \mathbf{r}}

retain their form in special relativity, provided the Lagrangian generates equations of motion consistent with special relativity. Here r = (x, y, z) is the position vector of the particle as measured in some lab frame where Cartesian coordinates are used for simplicity, and

{\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}={\frac {d\mathbf {r} }{dt}}=\left({\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)} \mathbf{v} = \dot{\mathbf{r}} = \frac{d\mathbf{r}}{dt} = \left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)

is the coordinate velocity, the derivative of position r with respect to coordinate time t. (Throughout this article, overdots are with respect to coordinate time, not proper time). It is possible to transform the position coordinates to generalized coordinates exactly as in non-relativistic mechanics, r = r(q, t). Taking the total differential of r obtains the transformation of velocity v to the generalized coordinates, generalized velocities, and coordinate time

{\displaystyle \mathbf {v} =\sum _{j=1}^{n}{\frac {\partial \mathbf {r} }{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} }{\partial t}}\,,\quad {\dot {q}}_{j}={\frac {dq_{j}}{dt}}} \mathbf{v} = \sum_{j=1}^n \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_j +\frac{\partial \mathbf{r}}{\partial t} \,, \quad \dot{q}_j = \frac{dq_j}{dt}

remains the same. However, the energy of a moving particle is different to non-relativistic mechanics. It is instructive to look at the total relativistic energy of a free test particle. An observer in the lab frame defines events by coordinates r and coordinate time t, and measures the particle to have coordinate velocity v = dr/dt. By contrast, an observer moving with the particle will record a different time, this is the proper time, τ. Expanding in a power series, the first

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