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zheng89120
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Why is it that only Canonical transformations preserve the Hamilton's equations? Or what makes non-canonical transformations not preserve the Hamilton's equations?
A canonical transformation is a mathematical transformation that preserves the canonical form of Hamilton's equations of motion. It is used to transform one set of coordinates and momenta into a new set of variables that describe the same physical system.
Canonical transformations are important because they allow us to simplify the equations of motion for a physical system. By transforming to new coordinates and momenta, we can often find simpler and more elegant expressions for the dynamics of the system.
A coordinate transformation only changes the way we describe a system mathematically, while a canonical transformation also preserves the physical dynamics of the system. This means that the equations of motion will remain the same, even after the transformation.
No, not every transformation is a canonical transformation. For a transformation to be canonical, it must satisfy certain conditions, such as preserving the Poisson brackets between the coordinates and momenta of the system.
Canonical transformations are closely related to symmetries in physics. In fact, every symmetry in a physical system corresponds to a canonical transformation that leaves the equations of motion unchanged. This is known as Noether's theorem.