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Qmechanic ♦
Qmechanic ♦
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  1. user318039'suser318039's answer is exactly right: An action principle only has to reproduce the given equations of motion (EOMs) as its Euler-Lagrange (EL) equations.

  2. The Lagrangian does not have to have dimension of energy. For instance, if one scales a Lagrangian with a dimensionfull (non-zero) constant, the EOMs will stay the same. See also e.g. this & this Phys.SE posts.

  3. By the way: OP's paraxial wave equation (1) has the same form as the TDSE, cf. e.g. this Phys.SE post.

  1. user318039's answer is exactly right: An action principle only has to reproduce the given equations of motion (EOMs) as its Euler-Lagrange (EL) equations.

  2. The Lagrangian does not have to have dimension of energy. For instance, if one scales a Lagrangian with a dimensionfull (non-zero) constant, the EOMs will stay the same. See also e.g. this & this Phys.SE posts.

  3. By the way: OP's paraxial wave equation (1) has the same form as the TDSE, cf. e.g. this Phys.SE post.

  1. user318039's answer is exactly right: An action principle only has to reproduce the given equations of motion (EOMs) as its Euler-Lagrange (EL) equations.

  2. The Lagrangian does not have to have dimension of energy. For instance, if one scales a Lagrangian with a dimensionfull (non-zero) constant, the EOMs will stay the same. See also e.g. this & this Phys.SE posts.

  3. By the way: OP's paraxial wave equation (1) has the same form as the TDSE, cf. e.g. this Phys.SE post.

Source Link
Qmechanic ♦
Qmechanic ♦
  • 213k
  • 48
  • 590
  • 2.3k

  1. user318039's answer is exactly right: An action principle only has to reproduce the given equations of motion (EOMs) as its Euler-Lagrange (EL) equations.

  2. The Lagrangian does not have to have dimension of energy. For instance, if one scales a Lagrangian with a dimensionfull (non-zero) constant, the EOMs will stay the same. See also e.g. this & this Phys.SE posts.

  3. By the way: OP's paraxial wave equation (1) has the same form as the TDSE, cf. e.g. this Phys.SE post.