Hamiltonian For A Relativistic Particle

Abstract The intend of this paper is to formulate a Hamiltonian for the most simple and general mechanical system in Space-time—a single particle. The method used is the simple application of the Euler-Lagrange equation to the Lagrangian, and Hamilton's equations to the Hamiltonian, for an arbitrary metric tensor, . The Legendre transformation is used on the Lagrangian for a single particle to get the desired Hamiltonian. To assure that the expression for the Hamiltonian is valid, Hamilton's equations are applied to find the equation of motion and show that they are the same as the equation of motion that is produced by the Lagrangian. The relevance of this application is fundamental for the understanding of the behavior of a single particle that behaves macroscopically (i.e. non-quantum-mechanically). Accepting that General Relativity is the model of Spacetime that is as close to the truth as currently possible, then knowing how a single particle behaves mechanically under its boundaries is very important. It is very reassuring that the applications of the Hamiltonian and Lagrangian framework are relevant and valid even for General Relativity.

Hamiltonian For A Relativistic Particle The Lagrangian for a relativistic particle with arbitrary metric tensor is: (1) where

is the mass of the particle,

is the metric tensor,

the particle, also called 'proper-time' , which can take values of

;

is the time measured in the rest frame of

is the proper time derivative of the contravariant coordinate ;

;

.

Applying the Euler-Lagrange equation to this Lagrangian yields the following equation of motion: (2) where

and

. This single equation contains the four equations of motion in the time

and space components (i.e.

,1,2,3). For details about the derivation, see Appendix A.

The following analysis will show that the Hamiltonian formalism leads to the same equations of motion as in eq. (2). The Lagrangian is dependent on the particle's contravariant position in Space-time and its contravariant 4-velocity. The Hamiltonian is dependent on the particle's contravariant position in Spacetime and covariant 4-momenta. The difference between a covariant and contravariant coordinates system is in the consistency of way they behaves after a transformation. A Legendre transformation from momentum

is equal to

to . Solving for

will be performed. The canonical conjugate and plugging into the Hamiltonian-Lagrangian

relation. We obtain: (3)

For details about the derivation, see Appendix B. This Hamiltonian is analogous to

for classical

mechanics. However, instead of summing over three coordinates (i.e. x,y,z ), time must be summed over as well, (t,x,y,z). The next step is to apply Hamilton's equations to this Hamiltonian:

. The set of

equations that come out these equalities are:

For details about derivation, see appendix C. We can solve for the canonical conjugate momentum in (4b), take one proper-time derivative to the result and plug it into (4a), expressing everything as a function of 's and its proper-time derivatives. The resulting equation becomes: (5) This equation does not look like the equations of motion that come out of the Lagrangian, eq. (2). This is because equations (2) involve only derivatives of covariant expressions of the metric tensor. In the right hand side of equations (5), there is a derivative of a contravariant expression. The following identity is required in order to express equations (5) as derivatives of covariant expressions. Applying this to eq. (5). We get:

which is exactly the same as eq. (2), bravo! For details regarding the derivative of a contravariant identity and derivation of the equations of motion, see appendix C. This concludes the project. Demonstrating that the Lagrangian and Hamiltonian framework are completely consistent with each other. The idea for this project came naturally. In working with the Hamiltonian, it seemed logical to compare it to working with the Lagrangian. The derivations flowed smoothly except for understanding whether the canonical conjugate momentum needed to be covariant or contravariant. However, following the logic of the indices is tricky but helps maintain the validity of the analysis. The other major difficulty was demonstrating that equation (5) and equation (2) are equal. The key to the solution is the identity for the derivative of the inverse of a matrix.

Appendix A Finding the equation of motion from the Lagrangian for a relativistic particle (Geodesic Equation)

Appendix B Deriving the Hamiltonian from the Lagrangian.

Appendix C Finding the equation of motion from the Hamiltonian for a relativistic particle.

Appendix A Lagrangian for relativistic particle

Eq. (1)

Euler-Lagrange Equation

(re-labeling indices and using the fact that the metric is symmetric,

Chain rule on

)

has been used.

(Multiplying by Using the fact that

)

. Eq.(2)

Appendix B Legendre Transformation

(Adding and subtracting

)

Finding Hamiltonian We need to find

and express

as a function of

(Plugging this in B.1)

Eq. (3)

Appendix C Hamilton's Equations Eq. (4a)

Eq. (4b) Proof for

Finding Equations of motion

Taking proper-time derivative (C.1) Now we are done with

. We go to

and express it as a function of

's

(C.2) Writing C.1 equal to C.2 yields Eq. (5) Applying the identity derived for

to C.2 yields (C.3)

Now making C.1 equal to C.3 (re-labeling indices and moving everything to one side) (Multiplying by (Equations of motion)

) Eq.(2)