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Leaving differential equations in the same form

    To preserve differential equations in the same form regardless of the coordinate system we choose is the reason why we develop the theory of tensors. This is the most fruitful area in which physicists and mathematicians work together to achieve their common goals. The advantages of tensors in physics and intrinsic differential geometry are tremendous and continue to increase without cease. For given differential equations, we must answer two questions: First, how do we express a physical or geometrical quantity in the tensor form? Second, What is the allowable transformation group [Kre, p.101, l.16]?
  1. A methodology for leaving Lagrange's equations in the same form.

    2. Lagrange's equations have the same form regardless of

    1. coordinate system (Cartesian, Cylindrical or spherical) [Sym, p.366, l.20-l.22; Coh, p.1485, l.-15]
    2. the type of system (a particle or a rigid body [Coh, pp.1482-1485] and
    3. the type of external field (a particle in a central potential [Coh, p.1488, §4.a] or a charged particle placed in an electromagnetic field [Coh, p.191, §4.b]).

    Hamilton's equations are more advantageous than Lagrange's equations because

    1. Hamilton's equations are simpler and more symmetric [Lan1, p.132, l.6-l.10]
    2. Hamilton's equations retain the same form under a wider range of transformations [Lan1, p.143, l.20-p.144, l.11 & p.144, l.-4-l.-1]. Therefore, we adopt Hamilton's equations as the equations of motion in quantum mechanics.

    Remark.

The transformation L→ L' leaves

  • Benefits of leaving Maxwell's equations in the same form.

    • By searching for a transformation that leaves the form of Maxwell’s equations unaltered [Rob, p.2, l.13-l.28], we establish the relationship between time and space [Lan2, p.10, (4.3)].

    Remark.

    Choices of vector and scalar potential:

    The transformation (A,f )→ (A',f ') leaves E and H invariant.
  • Gauge Invariance.
        Suppose a system includes the electromagnetic field. Active or passive Galilean transformations leave the time-dependent, nonrelativistic Schrödinger equation in its original form [Mer2, (4.101), (4.113), (4.118)].
    Remark. Keys to consistency:
    1. Modify the phase of the wave function [Mer2, p.76, l.15].
    2. Refine potentials [Mer2, p.76, (4.112)].

  • Quantization and second quantization leave the equation of motion in the same form [Schi, p.351, l.6].
    Remark. Key to consistency: [Mer2, p.350, (46.6)].

  • Canonical transformations are used to solve mechanical problems via the following two methods:
    1. Search for a set of cyclic coordinates for which the Hamiltonian is a constant motion [Go2, p.379, l.18].
    2. Seek a canonical transformation from (the coordinates and momenta, (q, p), at time t) to a new set of constant quantity [Go2, p.438, l.7]. For example, if we let the transformed Hamiltonian be 0 [Go2, p.438, l.-4], then we obtain Hamilton-Jacobi equation [Go2, p.439, (10-3)].

  • A transformation is canonical if and only if [Go2, p.393, l.-1] its Jacobian matrix satisfies the symplectic condition [Go2, p.393, (9-55)].

  • Infinitesimal canonical transformations [Go2, p.394, (9-61c)].
    1. In terms of a generator: [Go2, p.395, (9-63c).
    2. In terms of the Poisson bracket: [Go2, p.407, (9-100)].
    3. Agreement with the traditional sense in the case of time evolution: [Go2, p.407, (9-101)].

    4. Infinitesimal rotation operators [Coh, p.697, (48)] vs. finite rotation operators [Coh, p.698, (55)].
      Remark 1. 1-(i/ħ)daLz = exp(-idaLz/ħ).
      Remark 2. The exponential function is a group homomorphism from the Lie algebra (th real line, +) onto the Lie group of the unit circle. Similarly, the exponential function can be considered a group homomorphism from the Lie algebra generated by angular momenta onto the Lie group of rotations. In this mathematical sense, we have two equivalent views to describe rotations in mechanics. From the viewpoint of the Lie algebra, we use angular momenta. From the viewpoint of the Lie group, we use rotations. If one fails to establish a relationship between the Lie algebra and mechanics [Che, chap. IV, §II & §III], one certainly cannot see the big picture of theory of Lie groups. In my opinion, none of the examples of Lie algebras in [Che, chap. IV, §III] are physically significant.

  • Tensors are used as a device which from a theory's beginning keeps track of checking each logical step to ensure that all physical quantities and operations in a formula remain covariant (or contravariant) with coordinate transformations. As a consequence, in general relativity, if we have a formula in tensor form in the free fall frame, then the same formula will automatically be valid in any other accelerated frame. In differential geometry, see [Lau, p.21, l.18-l.21].
    Example 1. [Ken, p.189, (B.3)-(B-5); p.190, (B-6); P.191, l.1, l.8, and l.-10].
    Example 2. Arc lengths [Lau, p.26, l.-10-l.-9], angles, and areas [Lau, p.27, Exercise 3.4.4] are invariant under parameter transformations.

  • A canonical transformation is defined as the transformation that preserves the symmetrical form of Hamilton's equations [Ches, p.198, l.15-l.18].
    1. Necessary and sufficient conditions for [Ches, p.198, (8-118)] to be canonical
          [Ches, p.198, (8-118)] is canonical
      Û[Ches, p.199, (8-123), (8-124) & (8-125)]
      Û[Ches, p.200, (8-132), (8-133) & (8-134)] (in terms of Lagrange brackets)
      Û[Ches, p.201, (8-137), (8-138) & (8-139)] (in terms of Poisson brackets).
    2. Using generating functions to effectively construct canonical transformations [Ches, §8-16].