The form of the Lagrangian also makes the relativistic action functional proportional to the proper time of the path in spacetime.
In covariant form, the Lagrangian is taken to be:[1][2] where σ is an affine parameter which parametrizes the spacetime curve.
If a system is described by a Lagrangian L, the Euler–Lagrange equations retain their form in special relativity, provided the Lagrangian generates equations of motion consistent with special relativity.
It is instructive to look at the total relativistic energy of a free test particle.
Expanding in a power series, the first term is the particle's rest energy, plus its non-relativistic kinetic energy, followed by higher order relativistic corrections; where c is the speed of light in vacuum.
The differentials in t and τ are related by the Lorentz factor γ,[nb 1] where · is the dot product.
Following the non-relativistic approach, we expect the derivative of this seemingly correct Lagrangian with respect to the velocity to be the relativistic momentum, which it is not.
The definition of a generalized momentum can be retained, and the advantageous connection between cyclic coordinates and conserved quantities will continue to apply.
Integrating py and pz obtains similarly where Y and Z are arbitrary functions of their indicated variables.
Since the functions X, Y, Z are arbitrary, without loss of generality we can conclude the common solution to these integrals, a possible Lagrangian that will correctly generate all the components of relativistic momentum, is where X = Y = Z = 0.
Alternatively, since we wish to build a Lagrangian out of relativistically invariant quantities, take the action as proportional to the integral of the Lorentz invariant line element in spacetime, the length of the particle's world line between proper times τ1 and τ2,[nb 1] where ε is a constant to be found, and after converting the proper time of the particle to the coordinate time as measured in the lab frame, the integrand is the Lagrangian by definition.
Either way, the position vector r is absent from the Lagrangian and therefore cyclic, so the Euler–Lagrange equations are consistent with the constancy of relativistic momentum, which must be the case for a free particle.
Also, expanding the relativistic free particle Lagrangian in a power series to first order in (v/c)2, in the non-relativistic limit when v is small, the higher order terms not shown are negligible, and the Lagrangian is the non-relativistic kinetic energy as it should be.
For the case of an interacting particle subject to a potential V, which may be non-conservative, it is possible for a number of interesting cases to simply subtract this potential from the free particle Lagrangian, and the Euler–Lagrange equations lead to the relativistic version of Newton's second law.
Although this has been shown by taking Cartesian coordinates, it follows due to invariance of Euler Lagrange equations, that it is also satisfied in any arbitrary co-ordinate system as it physically corresponds to action minimization being independent of the co-ordinate system used to describe it.
For example, it is also true that if the Lagrangian is explicitly independent of time and the potential V(r) independent of velocities, then the total relativistic energy is conserved, although the identification is less obvious since the first term is the relativistic energy of the particle which includes the rest mass of the particle, not merely the relativistic kinetic energy.
The extension to N particles is straightforward, the relativistic Lagrangian is just a sum of the "free particle" terms, minus the potential energy of their interaction; where all the positions and velocities are measured in the same lab frame, including the time.
In particular, if this other observer moves with constant relative velocity then Lorentz transformations must be used.
A seemingly different but completely equivalent form of the Lagrangian for a free massive particle, which will readily extend to general relativity as shown below, can be obtained by inserting[nb 1] into the Lorentz invariant action so that where ε = −m0c2 is retained for simplicity.
Although the line element and action are Lorentz invariant, the Lagrangian is not, because it has explicit dependence on the lab coordinate time.
Still, the equations of motion follow from Hamilton's principle Since the action is proportional to the length of the particle's worldline (in other words its trajectory in spacetime), this route illustrates that finding the stationary action is asking to find the trajectory of shortest or largest length in spacetime.
[3] For a particle, either massless or massive, the Lorentz invariant action is (abusing notation)[4] where lower and upper indices are used according to covariance and contravariance of vectors, σ is an affine parameter, and uμ = dxμ/dσ is the four-velocity of the particle.
This results in the following equation of motion: Which, given initial conditions of results in the position of the particle as a function of time being From Euler-Lagrange equation we have Integrating with respect to time: Where
: Which simplifies to This is expected solution to the equation of motion to the Newtonian particle subject to a constant force:
In special relativity, the Lagrangian of a massive charged test particle in an electromagnetic field modifies to[9][10] The Lagrangian equations in r lead to the Lorentz force law, in terms of the relativistic momentum In the language of four-vectors and tensor index notation, the Lagrangian takes the form where uμ = dxμ/dτ is the four-velocity of the test particle, and Aμ the electromagnetic four-potential.
The Euler–Lagrange equations are (notice the total derivative with respect to proper time instead of coordinate time) obtains Under the total derivative with respect to proper time, the first term is the relativistic momentum, the second term is then rearranging, and using the definition of the antisymmetric electromagnetic tensor, gives the covariant form of the Lorentz force law in the more familiar form, The Lagrangian is that of a single particle plus an interaction term LI Varying this with respect to the position of the particle xα as a function of time t gives This gives the equation of motion where is the non-gravitational force on the particle.
Rearranging gets the force equation where Γ is the Christoffel symbol, which describes the gravitational field.
For a charged particle in an electromagnetic field, the Lagrangian is given by If the four spacetime coordinates xμ are given in arbitrary units (i.e. unitless), then gμν is the rank 2 symmetric metric tensor, which is also the gravitational potential.
Instead, the energy-momentum relation appears as the equation of motion for the auxiliary field
After the equation of motion has been derived, one must gauge fix the auxiliary field