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经典力学在摇摆舞中应用

17/11/2014

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经典力学在摇摆舞中应用 by Darren


摘要:要想深入了解经典力学对于摇摆舞的作用,并且利用它来学好摇摆舞,必须先来了解一下基础知识“经典力学”。

经典力学(Classical mechanics)或称牛顿力学,是力学的一个分支。是以牛顿运动定律(Newton's laws of motion)为基础,在宏观世界和低速状态下,研究物体运动的基要学术。在物理学里,经典力学是最早被接受为力学的一个基本纲领。经典力学又分为静力学(描述静止物体)、运动学(描述物体运动)、和动力学(描述物体受力作用下的运动)。在十六世纪,伽利略·伽利莱(Galileo Galilei)就已采用科学实验和数学分析的方法研究力学。他为后来的科学家提供了许多豁然开朗的启示。艾萨克·牛顿(Isaac Newton)则是最早使用数学语言描述力学定律的科学家。以下分别阐述牛顿的三大运动定律。

1.第一定律(First Law):当物体不受外力作用,或所受合力为零时,原先静止的物体将保持静止状态,原先运动着的物体则沿着直线作等速度运动——简言之,即在物体的净外力为零时,物体将保持原先的运动状态。因此,该定律又被称为“惯性定律”。

2.第二定律(Second Law):物体的加速度,与所受的净外力成正比,与物体的质量成反比。加速度的方向与净外力的方向相同。由于后文要讨论的经典力学在摇摆舞中的实际应用基本都在定性的范围内讨论,所以仅将第二定律抽象成初等数学表达式:

a=F/m

其中:a表示物体的加速度;

F表示物体受到的净外力;

m表示物体的质量。

3.第三定律(Third Law):当两物体相互作用时,彼此互以力作用于对方,两者大小相等,方向相反,但作用在不同的物作用体上。该定律又称为“作用力与反作用力定律”。

由上述三大定律可以看出,第一定律侧重于静力学,第二定律侧重于运动学和动力学,而第三定律则侧重于静力学和动力学。 了解了牛顿三大运动定律的内容后,我们可以以八拍的Lindy hop中最为基础也是最为标志性的动作——Swingout为例,来看一下,在这个我们已经习以为常的摇摆舞动作中,伟大的运动定律是如何一拍一拍地发挥作用的。

前提:舞蹈涉及了包括势能、动能和生物能等多种能量的转换,摇摆舞又是一种以男生(Lead)为主导的舞蹈,故本文主要从男生的角度阐述如何施加正确的力来引导女生(Follow),从而忽略生物能等其他能量的转化。又假定舞者是正常健康的自然人,可以正确控制自身的身体平衡,故不考虑外力所产生的力矩对于身体平衡的影响,可忽略身高,而将人体看作一个质点。

第0拍:确切地说应该是上个八拍的第8拍之后,而新一轮八拍还未开始的那一瞬间。这是个力从不平衡到平衡的阶段。在完成上一个八拍的动作之后,男女双方需要在此时迅速建立平衡,从而为接下来的动作做好准备。
图片
此时男女双方的大致位置如图1所示,红色的力为女生的受力,蓝色的为男生的。由于手是男女双方唯一的接触点(男生的左手与女生的右手),所以不难看出:这个点就是力的作用点。男生作用于女生手上的力为F1,根据第三定律,此时女生的手上也有个力F2作用于男生手上,大小与F1相等,方向与F1相反。由此时并未发生水平位移,所以可知,男女双方的脚底与地面的静摩擦力f1与f2分别于F1和F2组成了两对大小相等[1],方向相反的平衡力。另外,舞者自身受地球引力影响而产生并且作用在其重心上的的重力G1和G2与地面给予其的支撑力N1和N2有各为两对作用力与反作用力。由于此时的男女双方都处于一个相对静止的稳定状态,所以我们以女生为质点,由第一定律可以得出下列关系式:


F1=f1
G11=N1


第1拍:这是一个力逐渐发生变化的过程。力的变化也就意味着原先平衡状态将会被改变。此时,男生施加在女生手上的力F1会逐渐增大,而随着F1的增大,静摩擦力f1也会相应增大,直至f1与滑动摩擦力fs1相等[2]的那一瞬间,女生就会向前运动。

第2拍:如果你明白了牛顿第一定律,那这将是最轻松的一拍。男生的左手不需要对女生施加任何外力,只是用来控制女生行走的方向,要做的只是等女生顺着自己肩膀的自然反应做出转向的动作。这一拍中,男女双方没有任何力的相互作用,是一个平衡的过程

第3~4拍:男生此时要伸右手放在女生的后背以阻碍其沿着原运动方向前进。男女双方再次产生力的相互作用。
图片
如图2所示,此时力的作用的点在女生的后背上,同时又再次达到一个类似的匀速圆周运动的平衡状态。
图片
如俯视图图3所示,此时男女两人的重力G依然与地面的支撑力N相互抵消;同样,男生的手在女生后背产生的摩擦力Ft1与地面对于产生的切向摩擦力ft1相互抵消;但由于离心作用的存在,男生施加在女生后背上的力将大于地面向心方向上的摩擦力fC1,即FC1>fC1。而它们的差值就充当了向心力,即FC1-fC1=m1aC1,其中aC1为此刻女生的向心加速度。根据向心加速度的表达式a=vt2/R,其中vt为圆周运动的切向速度,R为圆周运动的运动半径,我们不难推导出下列关系式:

vt12=R(FC1-fC1)/m1



其中R和m1为定值;在运动过程切向的滑动摩擦力ft1也是定值。所以不难看出男生施加在女生后背上的压力FC1的大小决定了女生运动速度vt1的快慢。换句话说,男生用的力越大,女生转得就越快;如果男生用相同的力,那么质量较小的女生转得会比质量较大的女生快。也有不少男生会在第3、4拍的时候把右手放更远,以增大R来换取女生更大的运动速度,而且这样所需要用到的力就会比较小,这是个很讨巧省力的方法。

第5~6拍:随着女生绕着男生做圆周运动到了α点,男生在此时放手,撤掉了施加在女生后背上的作用力,根据第一定律,在净外力为零情况,女生将沿着原先的运动方向继续前进,即图3中vt1所指的切向方向。就男生这个施力物体来说,这也是相对比较轻松的两拍,但要做好难度较大,关键在于放手的时机的把握。不难看出,放手时机的不同很大地决定了女生后面的运动走向,真可谓“失之毫厘,差之千里”啊。

第7~8拍:在中规中矩的Swingout中,男生在这两拍Triple Step的动作看似不多,但力的作用却发生了极为复杂的变化。

图片
此刻,男女双方的距离越来越远,图4中女生受到的滑动摩擦力f1方向与其运动方向相反,与受到男生施加在其手上的力F1方向相同。也就是说女生在水平方向上受到的净外力F=F1+fS1,根据第二定律,也就产生了一个与女生运动方向相反的加速度a1,从对运动的作用效果来看,它实为一个减速度,这个减速度使v1越来越小,直至为零。此时,女生会停在某一点上,并开始产生向力F1方向运动的趋势,滑动摩擦力消失,取而代之的是与F1方向相反的静摩擦力——如图1所示。于是,又回到了之前第0拍的那一瞬间。

至此,我们完成了一个完整的Lindyhop Swingout。

小结:通过完整地分析在Swingout过程中各个时段女生的受力状况,我们可以得出以下2点:

1.给女生:除了克服摩擦力和控制自身平衡的力以外,不要做出任何施力动作。换句话说,就是除了走路和保持平衡外,不需要任何力气。一个很主动去施加力的女生会给舞伴造成很大的困扰。

2.给男生:不要额外地施加力——尤其在第2、5、6拍的时候。额外的力就是意味着平衡被打破,打破平衡就意味着缺少美感,就需要额外的动作再去建立平衡。即使是其他的令人眩目的花哨的非常规动作中,平衡也是时刻存在的,或者说,那些动作本身就是在建立平衡。多余的力同时也会对你的舞伴造成误导,做出你原本并不希望她做的动作。由牛顿第二定律我们可以看出,学会根据不同女生的m去调整自己所需要施加的力才是上策。而且,省去了那些多余施加的力,你会发现跳舞比以往轻松不少,也许跳一支舞只需要消耗原来一半的体能。在一个美女如云,男生资源又稀缺的舞池里,这意味着什么,不言而喻吧。




注释:

[1] 当水平放置于地面的物体产生运动趋势但未发生位移时,其与地面之间就产生了静摩擦力,大小与施加在物体上的水平净外力相等,方向与物体的运动趋势相反。

[2] 理论上研究时通常直接认为当静摩擦力与滑动摩擦力相等时,物体就会发生位移,即f1MAX=fS1。但实际中,静摩擦力必须大于滑动摩擦力,物体才会发生位移,即f1MAX>fS1。

[3] 此时滑动摩擦力依然存在,但是由于人体的生物能产生的动力与之抵消,因此我们能将此刻看作是平衡状态。



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The Proactive Follower

15/11/2014

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The Proactive Follower

By Bobby White

As a teacher, I quickly realized the advice “just follow” is almost never helpful. As far as teaching advice goes, it’s vague and doesn’t give the follower an actionto do, which is exactly what any student needs in order to get better.

But more importantly, it’s wrong.

Great followers never “just follow.” They are constantly being proactive in many different ways in order to make the dance successful and contribute their voice to what is being created.

Let’s talk with a few of the world’s greatest followers to find out how they do that.

Great followers are…

Proactive in their RHYTHM

Followers should always strive to have good rhythm and be proactive about keeping that good rhythm. Followers should also be proactive about keeping that solid rhythm and pulse in non-closed positions, during turns, traveling — well, all the time. (Sometimes you’ll see followers who grow timid in their rhythm the more disconnected they are from their leaders or if they are in the middle of turns.)

“Follows tend to wait for the leader to ‘set the beat’ and they follow his rhythm,” says Laura Keat. “I think that follows need to be more responsible for demonstrating the rhythm and ‘dancing’ in every dance. Therefore if the leader makes a mistake or is off, the follow can still dance rhythmically. We, follows, can still follow the leader’s timing and shapes, but the beat/rhythm/flow of the music is carried in each of our minds and is each of our responsibility to represent through dance.”

“The dancers could just as well be new instruments, as long as the whole is harmonious,” says Annie Trudeau.

“And that’s the key for the follower… finding that harmonious rhythm in movement and in movement relative to your partner. Because rhythms in dancing happen in many dimensions, not only up and down or with our feet.”

Laura Glaess stresses how she thinks about being proactive not only in terms of knowing where the rhythm is but also in terms of where the overall phrasing of the music is. “Depending on the leader and how in sync I am with him, we might be doing crazy rhythmic variations that require me to sacrifice my bounce ["pulse", steady on-every-beat body rhythm]. However, I have the understanding of where the phrase or chorus is, and the understanding that when that occurs, a change in this rhythm will occur, like a return to something basic.”

Proactive in their POSTURE

Great followers are proactive in using their posture, whether it be specifically to match and connect with their leader and/or aesthetic reasons. (This might seem obvious, but a surprising amount of followers don’t control their posture.)

Laura Glaess thinks of it as shape in general. “Basically it includes my arms and my feet. My body. I feel like I know how I want my body to be aligned and the things that it does comfortably.”

But great followers are also proactive about not allowing a leader, or themselves, to change their posture so much that it negatively changes their dancing (like allowing a leader to get them off balance) or affects their safety (like throwing themselves into dips.)

“At all times,” says Annie Trudeau, “posture should be chosen for good balance, good dynamics, good swing dance aesthetics and intentions, and good connection to ourselves to allow the two bodies to move as a whole in harmony and synergy.”

Proactive in their FLOW

When I was a dancer who was good enough to know what good flow was but not good enough to do it, I had the opportunity to dance with some original dancers, like SoCal Lindy Hopper/Bal-Swing dancerAnne Mills. I was struck by how well the dance flowed, even though I clearly wasn’t contributing much myself to it. I then experienced the same thing the first times I danced with Sylvia Sykes.

In hindsight, this was because Anne (and Sylvia) never sacrificed her flow, regardless of what I did with mine. In fact, it seemed like it was almost part of a bigger personal dancing philosophy of hers — she wouldn’t allow me to make her not look elegant. Yes, if I asked her to move faster or slower, she would — but those speed changes would transition incredibly smoothly.

I have since realized this is a also trait of many modern great followers.

“Both partners are responsible for flow,” says Sylvia Sykes, “but often the follow needs to be a bit more on top of keeping it going. No matter how angular or start/stop the lead is, the follow can/should attempt to keep the flow and round the edges without back leading.”

Nick Williams is known to many as an incredible swing dancing leader. What people also don’t realize is how good he is as a follower. (Though it can be strange moving around someone with the body density of a refrigerator.) When I went looking for a great male follower to interview, his professional expertise and the experiences I’ve had leading him made him a clear choice.

“Being a leader primarily, I know what I like in my followers, which is what I try to create when I’m in the follower role,” he said. “Because of this, flow and rhythm become my most important tools…It’s always surprising to me how much is not actually communicated to the follower, even though it’s the leaders intention, so most of my focus is trying to make the dance work, paying attention to both the big picture as well as the subtleties.”

Proactive in their LEARNING

Many teachers reinforce the “just follow” mentality in their students. It’s easy for followers to not receive a lot of direction in classes (often a teacher’s fault, in my opinion.) But great followers did not just wait around to be taught variations.

As we have mentioned, great following is not “just following.” But if it isn’t that, what is it? It is interpreting and reacting to signals and forces, often with the goal to do so in creative ways. It is the feelings and forces a follower gets in the body, the image and actions of the leader seen with the eyes, and the music heard with the ears, that a follower interprets. It is knowing how to act with their body that they react. And it is in playing with how they do so that they creatively interpret.

This, quite simply, takes a lot of practice time and hard work.

“The first year or two I was dancing, the good leads in my city had an obvious preference of dancing with the best follows; therefore it was hard for me to dance with them since there were so many extra follows who were really good,” said Laura Keat.

“I began creating exercises that I could practice at home by myself or with one other follow, since there were so many of us, to get the muscle memory of ‘creativity within connection and following.’ There was no way I was going to get in enough practice time to catch up by dancing with actual leads since I wasn’t their preferred follow. So I found ways to clock my own hours and catch up with the follows that those guys preferred. No one knew that I was spending 2-3 hours a day by myself at home practicing. They all just thought I was a ‘natural’ since I was as enjoyable to dance with as some of those preferred follows within a year or two.”

Proactive in their CHOICES

This is one Laura Glaess personally added to the list, which definitely belongs on it.

“If my lead doesn’t clearly communicate what to do, or if I don’t clearly get the message,” she said, “I still have to choose something. Even after I discover that my choice might be wrong, to a degree, I need to stick with it and work within it.”

“In my invitational J&J with Dax at ILHC [2013], we did some kind of break away, and I don’t think he had a clear idea of where to go [from] there. I didn’t offer any solutions or make any choices. That was a huge learning moment. If I had made a clear choice, he would have had something to work with and we would have continued the dance from there. In later dances, if I make a choice that doesn’t work with the leader’s choice, and least there’s something talk about.”

On the subject of making choices in moments of doubt, Kate Hedin added that followers have several paths and don’t necessarily have to choose an additive one. “I can choose to fill that space by asserting the presence of the dancer I want to portray — and sometimes that means more silence and less noise — but it’s still an active choice. I choose to only put into the dance those things that I can stand by, only those qualities I am proud of.”


Proactive in their PRESENCE

No one has ever described incredible swing dance followers as “meek” and “timid.”

Even though they are followers, they are dancers first. They step, travel through space, and make every extra movement with commitment.

To quote the great Lindy Hop and Balboa follower Marie Nahnfeldt Mattsson, “I love this step. I lovethis step. I love this step…”

Sylvia Sykes points out there are many ways to do this. “You don’t have to be loud or flashy…a calm, confident, consistent presence can be very effective as well.”

Kate Hedin had a very personal experience learning what it meant to have presence.

“I’ve always had clear opinions about the dance, about the music, about the rhythm, as well as ideas [about] how to use the mechanics of the dance to synthesize and execute them,” she said. “However, I was never really good at getting those ideas across visually — as least not in the beginning. So, while I was being the dancer I wanted to be internally for myself, I was not sharing or exuding that dancer outwardly.”

After working on it for several years, she became one of the most powerful voices in Bal-Swing.

“It’s kind of like making your words match your actions. Some people talk a big talk but they don’t have the actions to back that. And others may have good ideas/substance, but don’t know how to explain themselves or never get heard. I want those two parts of myself to be consistent and self-reinforcing.”


Proactive in having a WELL-BALANCED CONVERSATION


Imagine a conversation where one person talks and the other person spends the whole time nodding their head (which, technically, makes it more of a lecture at that point). Or, imagine a conversation where one person tries to say their part, but the other person keeps butting in and interrupting.

Some followers concentrate so much on following that they are the dancing equivalent of people who just nod their heads in a conversation. Other followers are so excited to express themselves that they are the person who doesn’t seem to pay attention to what’s going on in the conversation and keep interrupting or going off on tangents. (And, of course, there are just as many leaders who do the same.)

Great followers are proactive about keeping the dance conversation lively and as well-balanced as possible. They both listen and talk, and they try to keep the conversation lively.

“I like to think that the music has it all, and the leader builds the foundations, the dirt roads,” says Annie Trudeau. “The follower uses this base to create with the leader on top. So together, they will add the bricks, the wood and doors on the foundations, the stonework, and the flowers on the side of the dirt road.”

(Annie also has a pleasing painting analogy — the leader brings the canvas, and they both paint the picture.)

Proactive in their DANCING

Occasionally I see followers who, if they accidentally lost connection with their leader or were led in something unexpected or strange, would kind of start to sputter and stop dancing as if they had run out of gas.

I’d recommend that followers, however, try to be proactive about their dancing — if they lose connection, they become solo dancers until they meet back up again; if a strange move happens, they are still dancing even though they are surprised or don’t know what they are supposed to do. They don’t have to stop looking for connection and dance as if they’re alone — they can calm their dancing (such as their pulse) in order to concentrate on getting back in sync with their partner — but they still want todance.

Sylvia Sykes sums it up quite well: “Along with keeping rhythm, flow, and general technique, one should dance; not just execute figures ‘correctly’…but move with joy.”


Proactive in INVENTION

While social dancing, or practicing, or in general creating their voice, great followers are proactive in the process of inventing content.

“Inventing in the dance is so hard if you have any insecurities or are afraid of disrupting your lead,” Laura Keat said, when asked about advice in follower invention. “I found that I had to focus on sharing a conversation with my lead instead of ‘practicing inventing’ to make it more pleasant for both of us. I focused on making the dance enjoyable by watching my leader’s reactions for positive responses instead of focusing on avoiding mistakes or being discouraged by perceived negative responses.”

Kate Hedin added this for followers to think about as they begin inventing: “It’s fantastic that followers are creating new content,” she said. “However, sometimes those new ideas don’t look like they came from the same dancer. And, more importantly, sometimes those new ideas are breaking fundamental mechanics. As a proactive follower, I need to understand that creating something new doesn’t mean ignoring what’s there.”

When working on choreographies or moves, they might look for where they might take moves themselves, or they simply might keep their creative minds open to possibilities. (This is something my partner Kate Heidn really brought home. When we work on choreographies, she gets a large amount of say in what moves we do, as it’s something she doesn’t get to decide in social dancing.)


Final Thoughts

When looking over this list of skills a “proactive” follower has, Sylvia Sykes mentioned that this was a list just as much for leaders as well.

These statements get us to a very important point. When one thinks of these skills, one sees, overall, the description not necessarily of a great follower, but of a great dancer, period. Following just happens to be the medium these dancers use to be incredible.


ABOUT THE ARTICLE

I want to drop special thanks first to those incredible dancers who allowed me to interview them.

Second, to my teaching and dancing partner, Kate, who is an amazing example of a follower who can both follow in its most literal dancing sense and yet do so incredibly proactively. (I also thought of her immediately when I wrote the sentence about some of the old-timers having the philosophy that they will never allow anyone to not make them look elegant.)

When I asked her what being a proactive follower meant to her, this is what she said:

To be a proactive follower means:

1. to be equally invested in the success of the dance, and to be an active contributor to that goal, not just be passively carried through the dance.
2. to have an idea of the dancer you want to be, and to actively uphold that idea, through content and character.
3. to assert your identity as a dancer.
4. all of this, without negating the primary role of a follower.

I personally think there is a lot of inspiration in that description.

Also, I’ve learned a great deal through conversations with many different people (many of them allowed me to interview them for the article) but wanted to give a shout-out specifically to someone who isn’t quoted in the article, David Rehm. I’ve learned quite a lot from conversations with him, especially from the language he uses on the subject, and in observing his classes.



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