Let V (r1→, ..., rM→) be the potential energy of a system of M massive particles which has the scaling property: V (α r1→, ..., α rM→) = αk V (r1→, ..., rM→) (k is usually an integer, α an arbitrary constant.) Prove that, if the Lagrangian is to remain invariant (except for multiplication by a constant), and all distances are scaled by a factor α, then the time must be scaled by a factor: β = α1−k/2
Let V (r1→, ..., rM→) be the potential energy of a system of M massive particles which has the scaling property: V (α r1→, ..., α rM→) = αk V (r1→, ..., rM→) (k is usually an integer, α an arbitrary constant.) Prove that, if the Lagrangian is to remain invariant (except for multiplication by a constant), and all distances are scaled by a factor α, then the time must be scaled by a factor: β = α1−k/2
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Let V (r1→, ..., rM→) be the potential energy of a system of M massive particles which has the scaling property:
V (α r1→, ..., α rM→) = αk V (r1→, ..., rM→)
(k is usually an integer, α an arbitrary constant.) Prove that, if the Lagrangian
is to remain invariant (except for multiplication by a constant), and all distances are scaled by a factor α, then the time must be scaled by a factor:
β = α1−k/2
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