A spring between objects of mass 1 kg and 3 kg on a frictionless floor is compressed and released with both objects initially at rest.
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自然長に戻ったとき、3 kgの速さは1 kgの何倍か？
右を固定すると、ばねが自然長に戻ったときの左の速さは、固定しない場合の何倍か？
The equations explain the phenomenon
1Consider the case where L does not depend on position∂L∂x=0⇒p=∂L∂x=constant……
2Write the Lagrangian L for the 2 objectsL=½ m₁ x₁²+½ m₂ x₂²− U(x₂ − x₁)……
3The 2 objects are displaced by the same distance a in the same directionx₁→x₁+a，x₂→x₂+a⇒∂L∂x₁+∂L∂x₂=0……
4The Euler–Lagrange equations for the 2 objects are addedd(p₁ + p₂)dt=0……
5The result applies to 2 objects initially at restm₁ v₁+m₂ v₂=0⇒|v₂||v₁|=m₁m₂……
6A candidate conserved quantity H is defined and its time dependence is examineddHdt=px+p x− (∂L∂x) x− (∂L∂x) x− ∂L∂t……
7The canceling terms are eliminated∂L∂t=0⇒H=∑ᵢ pᵢ xᵢ − L=constant……
8Conservation of momentum and energy determines the 2 speeds½ k d²=½ m₁ v₁²+½ m₂ v₂²……
9The object on the right is fixed to the floorWith the right object fixed:p₁+p₂ is not constant，Conservation of energy gives½ m₁ v₁²=½ k d²……
Further topics: rotational symmetry, external work, phase, and friction
10L is invariant under rotation∂L∂θ=0⇒m r² θ=constant……
11The spring constant varies with timek(t)=k₀(1 + ε cos 2ω₀t)⇒dHdt=½ (dkdt) x²，Generally nonzero……
12The symmetries correspond to conserved quantitiesContinuous symmetry→Corresponding conserved quantity……
13The wave's phase is shiftedFor the Schrödinger field:Phase symmetry of the action→Conservation of total probability……
14The case with friction is revisitedMechanical Energy→Internal energy of the floor and objects……
What this chapter has shown
Changing only the time while holding position and velocity fixed leaves L unchanged, so dH/dt=0. For this apparatus, H=T+U, so the total energy is conserved. In the unfixed case, translating the entire apparatus also leaves L unchanged, so the total momentum of the 2 objects is conserved as well.
1/3Speed ratio |v₂|/|v₁|
Fixing the object on the right prevents conservation of the total momentum of the 2 objects. For the same spring compression, the object on the left moves about 1.15 times as fast.
In everyday lifeWhen 2 people push off each other on ice, the lighter person moves away faster.
Conservation laws are not isolated rules to memorize. They arise from 3 symmetries: the laws of physics are the same at every position, in every direction, and at every time. These symmetries lead to conservation of momentum, angular momentum, and energy, respectively. If the object on the right is fixed to the floor, the system consisting of the 2 objects and the spring experiences an external force from the support, so its total momentum is no longer conserved. Energy is still conserved, however, because the support does no work and L has no explicit time dependence. The conservation laws encountered in high school are all manifestations of this 1 underlying idea.