1 / 65

Relativistic Classical Mechanics

Relativistic Classical Mechanics. Albert Abraham Michelson (1852 – 1931). Edward Williams Morley (1838 – 1923). James Clerk Maxwell (1831-1879). XIX century crisis in physics: some facts Maxwell : equations of electromagnetism are not invariant under Galilean transformations

Télécharger la présentation

Relativistic Classical Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Relativistic Classical Mechanics

  2. Albert Abraham Michelson (1852 – 1931) Edward Williams Morley (1838 – 1923) James Clerk Maxwell (1831-1879) XIX century crisis in physics: some facts • Maxwell: equations of electromagnetism are not invariant under Galilean transformations • Michelson and Morley: the speed of light is the same in all inertial systems

  3. 7.1 Postulates of the special theory • 1) The laws of physics are the same to all inertial observers • 2) The speed of light is the same to all inertial observers • Formulation of physics that explicitly incorporates these two postulates is called covariant • The space and time comprise a single entity: spacetime • A point in spacetime is called event • Metric of spacetime is non-Euclidean

  4. 7.5 Tensors • Tensor of rank n is a collection of elements grouped through a set of n indices • Scalar is a tensor of rank 0 • Vector is a tensor of rank 1 • Matrix is a tensor of rank 2 • Etc. • Tensor product of two tensors of ranks m and n is a tensor of rank (m + n) • Sum over a coincidental index in a tensor product of two tensors of ranks m and n is a tensor of rank (m + + n – 2)

  5. 7.5 Tensors • Tensor product of two vectors is a matrix • Sum over a coincidental index in a tensor product of two tensors of ranks 1 and 1 (two vectors) is a tensor of rank 1 + 1 – 2 = 0 (scalar): scalar product of two vectors • Sum over a coincidental index in a tensor product of two tensors of ranks 2 and 1 (a matrix and a vector) is a tensor of rank 2 + 1 – 2 = 1 (vector) • Sum over a coincidental index in a tensor product of two tensors of ranks 2 and 2 (two matrices) is a tensor of rank 2 + 2 – 2 = 2 (matrix)

  6. 7.4 7.5 Metrics, covariant and contravariant vectors • Vectors, which describe physical quantities, are called contravariantvectors and are marked with superscripts instead of a subscripts • For a given space of dimension N, we introduce a concept of a metric – N x N matrix uniquely defining the symmetry of the space (marked with subscripts) • Sum over a coincidental index in a product of a metric and a contravariant vecor is a covariantvector or a 1-form (marked with subscripts) • Magnitude: square root of the scalar product of a contravariant vector and its covariant counterpart

  7. 7.4 7.5 3D Euclidian Cartesian coordinates • Contravariant infinitesimal coordinate vector: • Metric • Covariant infinitesimal coordinate vector: • Magnitude:

  8. 7.4 7.5 3D Euclidian spherical coordinates • Contravariant infinitesimal coordinate vector: • Metric • Covariant infinitesimal coordinate vector: • Magnitude:

  9. 7.4 7.5 David Hilbert (1862 – 1943) Hilbert space of quantum-mechanical wavefunctions • Contravariant vector (ket): • Covariant vector (bra): • Magnitude: • Metric:

  10. 7.4 7.5 4D spacetime • Contravariant infinitesimal coordinate 4-vector: • Metric • Covariant infinitesimal coordinate vector:

  11. 7.4 7.5 4D spacetime • Magnitude: • This magnitude is called differential interval • Interval (magnitude of a 4-vector connecting two events in spacetime): • Interval should be the same in all inertial reference frames • The simplest set of transformations that preserve the invariance of the interval relative to a transition from one inertial reference frame to another: Lorentz transformations

  12. 7.2 Hendrik Antoon Lorentz (1853 – 1928) Lorentz transformations • We consider two inertial reference frames S and S’; relative velocity as measured in S is v : • Then Lorentz transformations are: • Lorentz transformations can be written in a matrix form

  13. 7.2 Lorentz transformations

  14. 7.2 Lorentz transformations • If the reference frame S‘ moves parallel to the x axis of the reference frame S: • If two events happen at the same location in S: • Time dilation

  15. 7.2 Lorentz transformations • If the reference frame S‘ moves parallel to the x axis of the reference frame S: • If two events happen at the same time in S: • Length contraction

  16. 7.3 Velocity addition • If the reference frame S‘ moves parallel to the x axis of the reference frame S: • If the reference frame S‘‘ moves parallel to the x axis of the reference frame S‘:

  17. 7.3 Velocity addition • The Lorentz transformation from the reference frame S to the reference frame S‘‘: • On the other hand:

  18. 7.4 Four-velocity • Proper time is time measured in the system where the clock is at rest • For an object moving relative to a laboratory system, we define a contravariant vector of four-velocity:

  19. 7.4 Four-velocity • Magnitude of four-velocity

  20. 7.1 t t’ Hermann Minkowski (1864 - 1909) x’ x Minkowski spacetime • Lorentz transformations for parallel axes: • How do x’ and t’ axes look in the x and t axes? • t’ axis: • x’ axis:

  21. 7.1 t x Minkowski spacetime • When • How do x’ and t’ axes look in the x and t axes? • t’ axis: • x’ axis:

  22. 7.1 t t’ x’ x Minkowski spacetime • Let us synchronize the clocks of the S and S’ frames at the origin • Let us consider an event • In the S frame, the event is to the right of the origin • In the S‘ frame, the event is to the left of the origin

  23. 7.1 t t’ x’ x Minkowski spacetime • Let us synchronize the clocks of the S and S’ frames at the origin • Let us consider an event • In the S frame, the event is after the synchronization • In the S‘ frame, the event is before the synchronization

  24. 7.1 Minkowski spacetime

  25. 7.4 Four-momentum • For an object moving relative to a laboratory system, we define a contravariant vector of four-momentum: • Magnitude of four-momentum

  26. 7.4 Four-momentum • Rest-mass: mass measured in the system where the object is at rest • For a moving object: • The equation has units of energy squared • If the object is at rest

  27. 7.4 Four-momentum

  28. 7.4 Four-momentum • Rest-mass energy: energy of a free object at rest – an essentially relativistic result • For slow objects: • For free relativistic objects, we introduce therefore the kinetic energy as

  29. 7.9 Non-covariant Lagrangian formulation of relativistic mechanics • As a starting point, we will try to find a non-covariant Lagrangian formulation (the time variable is still separate) • The equations of motion should look like

  30. 7.9 Non-covariant Lagrangian formulation of relativistic mechanics • For an electromagnetic potential, the Lagrangian is similar • The equations of motion should look like • Recall our derivations in “Lagrangian Formalism”:

  31. 7.9 Non-covariant Lagrangian formulation of relativistic mechanics • Example: 1D relativistic motion in a linear potential • The equations of motion: • Acceleration is hyperbolic, not parabolic

  32. Useful results

  33. 7.9 8.4 Non-covariant Hamiltonian formulation of relativistic mechanics • We start with a non-covariant Lagrangian: • Applying a standard procedure • Hamiltonian equals the total energy of the object

  34. 7.9 8.4 Non-covariant Hamiltonian formulation of relativistic mechanics • We have to express the Hamiltonian as a function of momenta and coordinates:

  35. More on symmetries • Full time derivative of a Lagrangian: • Form the Euler-Lagrange equations: • If

  36. 7.9 8.4 Non-covariant Hamiltonian formulation of relativistic mechanics • Example: 1D relativistic harmonic oscillator • The Lagrangian is not an explicit function of time • The quadrature involves elliptic integrals

  37. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan A • So far, our canonical formulations were not Lorentz-invariant – all the relationships were derived in a specific inertial reference frame • We have to incorporate the time variable as one of the coordinates of the spacetime • We need to introduce an invariant parameter, describing the progress of the system in configuration space: • Then

  38. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan A • Equations of motion • We need to find Lagrangians producing equations of motion for the observable behavior • First approach: use previously found Lagrangians and replace time and velocities according to the rule:

  39. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan A • Then • So, we can assume that • Attention: regardless of the functional dependence, the new Lagrangian is a homogeneous function of the generalized velocities in the first degree:

  40. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan A • From Euler’s theorem on homogeneous functions it follows that • Let us consider the following sum

  41. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan A • If three out of four equations of motion are satisfied, the fourth one is satisfied automatically

  42. 7.10 Example: a free particle • We start with a non-covariant Lagrangian

  43. 7.10 Example: a free particle • Equations of motion

  44. 7.10 Example: a free particle • Equations of motion of a free relativistic particle

  45. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan B • Instead of an arbitrary invariant parameter, we can use proper time • However • Thus, components of the four-velocity are not independent: they belong to three-dimensional manifold (hypersphere) in a 4D space • Therefore, such Lagrangian formulation has an inherent constraint • We will impose this constraint only after obtaining the equations of motion

  46. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan B • In this case, the equations of motion will look like • But now the Lagrangian does not have to be a homogeneous function to the first degree • Thus, we obtain freedom of choosing Lagrangians from a much broader class of functions that produce Lorentz-invariant equations of motion • E.g., for a free particle we could choose

  47. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan B • If the particle is not free, then interaction terms have to be added to the Lagrangian – these terms must generate Lorentz-invariant equations of motion • In general, these additional terms will represent interaction of a particle with some external field • The specific form of the interaction will depend on the covariant formulation of the field theory • Such program has been carried out for the following fields: electromagnetic, strong/weak nuclear, and a weak gravitational

  48. 7.10 Covariant Lagrangian formulation of relativistic mechanics: plan B • Example: 1D relativistic motion in a linear potential • In a specific inertial frame, the non-covariant Lagrangian was earlier shown to be • The covariant form of this problem is • In a specific inertial frame, the interaction vector will be reduced to

  49. 7.10 7.6 Example: relativistic particle in an electromagnetic field • For an electromagnetic field, the covariant Lagrangian has the following form: • The corresponding equations of motion:

  50. 7.10 7.6 Example: relativistic particle in an electromagnetic field • Maxwell's equations follow from this covariant formulation (check with your E&M class)

More Related