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Let $$L~=~-mc^2\sqrt{1- \frac{|\textbf{v}|^2}{c^2} },$$ where $\textbf{v}$ is the usual velocity of the particle in a fixed inertial frame. Then, this is the Lagrangian for a relativistic free particle. Now what does it mean by "the conserved quantity for a Lorentz boost"? Does it mean that the particle is boosted by some fixed velocity and there comes out a quantity that is preserved? I cannot get the exact meaning of the phrase. Could anyone please explain to me?

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This is easiest to see from the Hamiltonian formulation, cf. this Phys.SE post. Below we will also give the non-relativistic expressions for comparison, because it is interesting and somewhat subtle, cf. this Phys.SE post, and because OP asked the analogous non-relativistic question earlier.

I) The Hamiltonian is the kinetic energy, i.e the energy minus the rest energy$^1$ $$ H~=~p^0c-mc^2~=~\sqrt{{\bf p}^2c^2+m^2c^4}-mc^2 \quad\longrightarrow\quad \frac{{\bf p}^2}{2m}\quad\text{for}\quad c\to \infty. \tag{1} $$

The 3 boost generators $B^i$ are part of the 6 Lorentz generators $$B^i~=~\frac{J^{0i}}{c} ~=~tp^i-x^i\frac{p^0}{c} \quad\longrightarrow\quad tp^i-mx^i\quad\text{for}\quad c\to \infty,\qquad i~\in~\{1,2,3\}.\tag{2}$$ The pertinent infinitesimal quasi-symmetry transformations are generated by the boosts $$\delta x^i ~=~\{ x^i , {\bf B}\cdot \delta {\bf v} \} ~=~t~\delta v^i - \frac{p^i}{p^0c}{\bf x}\cdot \delta {\bf v} \quad\longrightarrow\quad t~\delta v^i\quad\text{for}\quad c\to \infty, \tag{3}$$ $$ \delta p^i ~=~\{ x^i , {\bf B}\cdot \delta {\bf v} \} ~=~\frac{p^0}{c}\delta v^i \quad\longrightarrow\quad 0\quad\text{for}\quad c\to \infty,\tag{4}$$ $$ \delta t~=~0.\tag{5}$$ The Hamiltonian Lagrangian $$ L_H ~=~{\bf p}\cdot \dot{\bf x} - H \quad\longrightarrow\quad {\bf p}\cdot \dot{\bf x} - \frac{{\bf p}^2}{2m} \quad\text{for}\quad c\to \infty \tag{6}$$ has a quasi-symmetry $$\delta L_H~=~\frac{d}{dt}\left( \frac{m^2c}{p^0} {\bf x}\cdot \delta {\bf v} \right)\quad\longrightarrow\quad \frac{d}{dt}\left( m {\bf x}\cdot \delta {\bf v} \right) \quad\text{for}\quad c\to \infty. \tag{7}$$ One may check that the corresponding Noether charges are precisely the boost generators (2).

II) The corresponding Lagrangian formulation$^1$ $$L~=~mc^2\left(1-\sqrt{1-\frac{\dot{\bf x}^2}{c^2}}\right)\quad\longrightarrow\quad \frac{1}{2}m\dot{\bf x}^2\quad\text{for}\quad c\to \infty,\tag{8}$$ has infinitesimal boost quasi-symmetry $$\delta x^i ~=~t~\delta v^i - \frac{\dot{x}^i}{c^2}{\bf x}\cdot \delta {\bf v} \quad\longrightarrow\quad t~\delta v^i\quad\text{for}\quad c\to \infty, \tag{9}$$ $$ \delta t~=~0,\tag{10}$$ and conserved boost charges $$B^i~=~m\frac{t\dot{x}^i-x^i}{\sqrt{1-\frac{\dot{\bf x}^2}{c^2}}} \quad\longrightarrow\quad m(t\dot{x}^i-x^i)\quad\text{for}\quad c\to \infty.\tag{11}$$ This is left as an exercise to the reader. One way is to integrate out the 3-momentum ${\bf p}$ from the Hamiltonian formulation in section I. See also this related Phys.SE posts and links therein.

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$^1$ We have removed the rest energy, which is a constant, i.e. a total time derivative, in order to be able to go to the non-relativistic limit.

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